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\begin{document}

\maketitle
\date{}

\begin{abstract}
We address the problem of estimating and comparing measures of concordance that arise as Pearson's linear correlation coefficient between two random variables transformed so that they follow the so-called concordance-inducing distribution.
The class of such transformed rank correlations includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient as special cases.

To answer which transformed rank correlation is best to use, we propose to compare them in terms of their best and worst asymptotic variances on a given set of copulas.
A criterion derived from this approach is that concordance-inducing distributions with smaller variances of squared random variables are more preferable.
In particular, we show that Blomqvist's beta is the optimal transformed rank correlation, and Spearman's rho outperforms van der Waerden's coefficient.
Moreover, we find that Kendall's tau also attains the optimal asymptotic variances that Blomqvist's beta does, although Kendall's tau is not a transformed rank correlation.
\end{abstract}

\hspace{0mm}\\
\emph{MSC classification:} 62H20, 62G20, 62H10, 62H12\\
\noindent \emph{Keywords:} Measures of concordance, Spearman's rho, Blomqvist's beta, Kendall's tau, asymptotic variance.

\section{Introduction}\label{sec:introduction}

A random vector $(X,Y)$ is said to be more \emph{concordant (discordant)} if large values of one random variable are more likely to correspond to large (small) values of the other random variable.
When $X$ and $Y$ are continuous, concordance and discordance are the properties of the $\emph{copula}$ $C$ of $(X,Y)$, which is a bivariate distribution function with standard uniform univariate marginal distributions.
A measure of concordance quantifies concordance or discordance of $C$ by a single number $\kappa$ in $[-1,1]$; see Definition~\ref{def:axioms:MOC} below.

In this paper we consider $G$-\emph{transformed rank correlations}, a subclass of measures of concordance which can be represented as Pearson's linear correlation coefficient $\rho$ between two random variables transformed so that they follow the so-called \emph{concordance-inducing distribution} $G$; namely,
\begin{align*}
\kappa_G(U,V)=\rho(G^\i(U),G^\i(V))\quad \text{ where } \quad (U,V)\sim C,
\end{align*}
and $G^\i$ is the generalized inverse of the distribution function $G$.
This class contains popular measures of concordance, such as Spearman's rho $\rho_{\text{S}}$, Blomqvist's beta $\beta$ and van der Waerden's coefficient $\zeta$; see Example~\ref{ex:transformed:rank:correlations} below.
Such a correlation representation of $\kappa$ is of great benefit to intuitively understand and explain the construction and ideas behind $\kappa$, construct its estimators and investigate their asymptotic properties, and analyze robustness \citep{raymaekers2019fast} and matrix compatibility of $\kappa$ \citep{hofert2019compatibility}.
Moreover, \cite{hofert2019compatibility} showed that the class of transformed rank correlations exhausts all measures of concordance of the form $\kappa(U,V)=\rho(g_1(U),g_2(V))$ for two left-continuous functions $g_1,\ g_2: (0,1)\rightarrow \IR$.

\begin{comment}
Since the work of \cite{embrechtsmcneilstraumann2002}, copulas have been widely
adopted in insurance and risk management to model dependence between random variables.
To quantify dependence in terms of a single number, measures of concordance, such as Spearman's rho and Kendall's tau, have been widely used as alternatives to Pearson's linear correlation coefficient since measures of concordance do not depend on the marginal distributions of the underlying random variables whereas Pearson's correlation does not possess this desirable property of a measure of dependence.
Interestingly, popular measures of concordance such as Spearman's rho and Blomqvist's beta can be represented as Pearson's linear correlation $\rho$ between random variables transformed by some real-valued functions $g_1,\ g_2$.
\citet{hofert2019compatibility} showed that, for such a transformed correlation coefficient to be a measure of concordance, the two functions $g_1$ and $g_2$ necessarily have to be generalized inverses of nondegenerate radially symmetric (cumulative) distribution functions $G_1,\ G_2:\IR \rightarrow [0,1]$ of the same type with finite second moments, and that one can take $G_1=G_2=:G$ without loss of generality.
For such a distribution function $G$ with $G^\i$ being the generalized inverse of $G$, the resulting measure
\begin{align*}
\kappa_{G}(C)=\kappa_{G}(U,V)=\rho(G^\i(U),G^\i(V))
\end{align*}
for a copula $C$ and $(U,V)\sim C$, is called the $G$-\emph{transformed rank correlation coefficient} with \emph{concordance-inducing distribution} $G$.
Thanks to this representation via Pearson's correlation, this class of measures of concordance has various appealing properties, such as interpretability and ease of studying compatibility and attainability problems; see \citet{hofert2019compatibility}.
Another advantage of this class is its ease of estimation since one can estimate $\kappa_G$ by the sample correlation of pseudo-observations from $C$ transformed by $G^\i$ for a given $G$.
\end{comment}

For the class of transformed rank correlations, natural questions are how to compare them and which concordance-inducing distribution is best to use.
\citet{de2016comparing} compared Pearson's linear correlation coefficient and Spearman's rho by numerical experiments in terms of bias, variance and robustness to outliers.
Various measures of concordance were compared in terms of their power in tests of independence; see, for example, \citet{bhuchongkul1964class}, \citet{behnen1971asymptotic}, \citet{behnen1972characterization}, \citet{luigi1999asymptotic}, \citet{rodel2004linear} and \citet{genest2005locally}.

The present paper tackles the problem of comparing $G$-transformed rank correlations from theoretical point of view concerning statistical estimation of $\kappa_G$.
In our proposed approach, a concordance-inducing distribution $G$ is more preferable than another one $G'$ if the worst and best asymptotic variances $\sigma_G^2(C)$ of a canonical estimator $\hat \kappa_G$ of $\kappa_G$ on a given set of copulas $\mathcal D$ are smaller than those of $G'$.
In other words, $G$ is more preferable than $G'$ if $\hat \kappa_G$ tends to estimate $\kappa_G$ more accurately than $\hat \kappa_{G'}$ estimates $\kappa_{G'}$ if $\mathcal D$ is the set of underlying copulas which one wants to quantify and compare their dependence.
Estimation of $\kappa$ is often inevitable since an explicit form of $\kappa(C)$ is not always available.
Therefore, this preference order of $\kappa_G$ fits the purpose of using $\kappa$ for measuring and comparing concordance among models of interest.

From this approach for comparing $\kappa_G$, we derive a criterion that a concordance-inducing distribution $G$ with smaller variance $\Var_G(X^2)$, where $X\sim G$, is more preferable.
Therefore, heavy-tailed concordance-inducing distributions, such as Student $t$ distributions, are not recommended as choice of $G$ in comparison to normal ones, which lead to van der Waerden's coefficient $\zeta$.
We also find that $\rho_{\text{S}}$ outperforms $\zeta$, and Beta distribution-based transformed rank correlations are more preferable than $\rho_{\text{S}}$.
In particular, we prove that Blomqvist's beta satisfies $\Var_G(X^2)=0$, and thus is the unique optimal transformed rank correlation attaining the optimal best and worst asymptotic variances under certain conditions on $\mathcal D$.
As stated by \citet{schmid2007nonparametric}, one of the advantages of $\beta$ over other measures of concordance is that it admits an explicit form if the copula can be written explicitly (this advantage, in fact, can be passed on to a wider class of discrete concordance-inducing distributions; see Appendix~\ref{app:sec:class:discrete:concordance:inducing:functions}).
The optimality of $\beta$ provides its additional advantage in terms of accuracy of its estimation.

Despite the optimality of $\beta$, its major drawback is that it depends only on the local value $C(1/2,1/2)$ of the underlying copula $C$.

Considering this drawback, we also compare $\kappa_G$ with Kendall's tau $\tau$.
Based on the representation of $\tau$ in terms of Pearson's linear correlation coefficient, we find that $\tau$ also attains the optimal best and worst asymptotic variances that $\beta$ does.

Therefore, $\tau$ can be a good alternative to $\beta$ in terms of the accuracy of its estimation although the optimality of $\tau$ is violated if the sample size required to construct its estimator is taken into account (see Remark~\ref{remark:comparability:kappa:g:tau} below).
Finally, in a simulation study, we find that the choice of concordance-inducing distribution $G$ and the strength of dependence of the underlying copula $C$ affect the asymptotic variance of $\hat \kappa_G$ more than the model of dependence.

This paper is organized as follows.
In Section~\ref{sec:correlation:based:mocs}, we review measures of concordance, the class of $G$-transformed rank correlations $\kappa_G$ and their basic properties.
In Section~\ref{sec:estimation:kappa:g:comparison} we introduce the approach for comparing $\kappa_G$ in terms of its asymptotic variance $\sigma_G^2(C)$.
A canonical estimator of $\kappa_G$ is presented in Section~\ref{subsec:canonical:estimator:kappa:G}, and the properties of $\sigma_G^2(C)$ are studied in Section~\ref{subsec:properties:asymptotic:variance}.
Section~\ref{subsec:optimal:location:shift} addresses effects of location-scale transforms of $G$ on $\sigma_G^2(C)$.
In Section~\ref{sec:optimal:concordance:inducing:distributions}, we study optimal best and worst asymptotic variances and their attaining concordance-inducing distributions on $\mathcal D$.
The case when $\mathcal D$ is a set of fundamental or Fr\'echet copulas is analyzed in Section~\ref{subsec:asymptotic:variance:fundamental:copulas}.
The optimality of Blomqvist's beta is proved in Section~\ref{subsec:optimality:Blomqvist}, and its uniqueness is discussed in Section~\ref{subsec:uniqueness:optimality:beta}.

Kendall's tau and $\kappa_G$ are compared in Section~\ref{sec:comaprison:kendall:tau}.
In Section~\ref{sec:simulation:study}, a simulation study is conducted to compare $\sigma_G^2(C)$ for various parametric copulas and concordance-inducing distributions.
Section~\ref{sec:concluding:remark} concludes this work with discussions about directions for future research.

\section{Correlation-based measures of concordance}\label{sec:correlation:based:mocs}

Let $\mathcal C_2$ be the set of all bivariate copulas, that is, all bivariate distribution functions with standard uniform marginal distributions.

We call $C' \in \mathcal C_2$ \emph{more concordant} than $C \in \mathcal C_2$, denoted by $C\preceq C'$, if $C(u,v)\leq C'(u,v)$ for all $(u,v) \in [0,1]^2$.
The survival function of $C$ is denoted by $\bar C(u,v)=\Prob(U>u,V>v)$, $(u,v)\in[0,1]$ where $(U,V)\sim C$.
The \emph{comonotonicity}, \emph{counter-monotonicity} and \emph{independence} copulas are denoted by $M(u,v)=\min(u,v)$, $W(u,v)=\max(u+v-1,0)$ and $\Pi(u,v)=uv$, $(u,v)\in [0,1]^2$, respectively.
By the \emph{Fr\'echet--Hoeffding inequalities}, it holds that $W\preceq C \preceq M$ for all $C \in \mathcal C_2$.
For any map $\kappa:\mathcal C_2\rightarrow \IR$, we identify $\kappa(C)$ with $\kappa(U,V)$ for a random vector $(U,V)\sim C$ defined on a fixed atomless probability space $(\Omega,\mathcal A,\Prob)$.
A map $\kappa$ on $\mathcal C_2$ is called a measure of concordance if it satisfies the following seven axioms proposed in \citet{scarsini1984}.

\begin{definition}[Axioms for measures of concordance]\label{def:axioms:MOC}
A map $\kappa:\mathcal C_2 \rightarrow \IR$ is called a \emph{measure of concordance} if it satisfies the following seven axioms.
  \begin{enumerate}
  \item \emph{Domain:} $\kappa(C)$ is defined for any $C \in \mathcal C_2$.
  \item \emph{Symmetry:} $\kappa(V,U)=\kappa(U,V)$ for any $(U,V)\sim C \in \mathcal C_2$.
  \item \emph{Monotonicity:} If $C\preceq C'$ for $C,C' \in \mathcal C_2$, then $\kappa(C)\le\kappa(C')$.
  \item \emph{Range:} $-1 \leq \kappa(C)\leq 1$ for any $C \in \mathcal C_2$, $\kappa(M)=1$ and $\kappa(W)=-1$.
  \item \emph{Independence:} $\kappa(\Pi)=0$.
  \item \emph{Change of sign:} $\kappa(U,1-V) =-\kappa(U,V)$ for any $(U,V)\sim C \in \mathcal C_2$.
  \item \emph{Continuity:} Let $C_n \in \mathcal C_2$, $n\in\IN$, and
    $C  \in \mathcal C_2$ with $C_n$ converging
    pointwise to $C$ as $n\rightarrow \infty$. Then $\lim_{n\rightarrow \infty}\kappa(C_n)=\kappa(C).$
 \end{enumerate}
  \end{definition}

Consider a class of maps on $\mathcal C_2$ written as $\kappa_{g_1,g_2}(U,V) = \rho(g_1(U),g_2(V))$ for two left-continuous functions $g_1,\ g_2:(0,1)\rightarrow \IR$.
\citet{hofert2019compatibility} showed that for $\kappa_{g_1,g_2}$ to be a measure of concordance, it must be the so-called $G$-transformed rank correlation defined as follows.
For a distribution function $G: \IR \rightarrow [0,1]$, the \emph{generalized inverse} of $G$ is defined by
\begin{align*}
 G^\i(p)=\inf\{x\in\IR:G(x)\ge p\},\quad p\in(0,1).
\end{align*}
A multivariate distribution $H$ on $\IR^d$ is called \emph{radially symmetric} if $\bX - \bmu \deq \bmu-\bX$ for $\bX \sim H$ and some $\bmu \in \IR^d$.

\begin{definition}[$G$-transformed rank correlation]
For a nondegenerate radially symmetric distribution function $G:\IR\rightarrow [0,1]$ with finite second moment, the $G$-\emph{transformed rank correlation} of $C \in \mathcal C_2$ is defined by
\begin{align*}
\kappa_{G^\i,G^\i}(C)=\rho(G^\i(U),G^\i(V))\quad\text{where}\quad (U,V)\sim C.
\end{align*}
We call $G$ the \emph{concordance-inducing distribution}, and the set of all concordance-inducing distributions is denoted by $\mathcal G$.
  \end{definition}

The class of $G$-transformed rank correlations includes popular measures of concordance as special cases.

\begin{example}[Examples of $G$-transformed rank correlations]
\label{ex:transformed:rank:correlations}
\hspace{2mm}
\begin{enumerate}
\item[1)]\label{example:spearman:rho} \emph{Spearman's rho}:
$\kappa_G$ reduces to \emph{Spearman's rho} $\rho_{\text{S}}(U,V)=12\E[UV]-3$ \citep{spearman1904general} if $G$ is the standard uniform distribution $\U(0,1)$.
\item[2)]\label{example:blomqvist:beta} \emph{Blomqvist's beta}:
$\beta(C)=4C(1/2,1/2)-1$ is called Blomqvist's beta (\citet{blomqvist1950measure}, also known as \emph{median correlation}), which is a $G$-transformed rank correlation with $G$ being a symmetric Bernoulli distribution $\Bern(1/2)$ on $\{0,1\}$.
\item[3)]\label{example:van:der:Waerden}
\emph{van der Waerden's coefficient}:
When $G$ is the standard normal distribution $\N(0,1)$, then $\kappa_G$ is known as \emph{van der Waerden's coefficient} (\citet{sidak1999theory}, also known as \emph{normal score correlation} and \emph{Gaussian rank correlation}) $\zeta(U,V)=\rho(\Phi^\i(U),\Phi^\i(V))$  where $\Phi$ is the distribution function of $\N(0,1)$.
\end{enumerate}
\end{example}

The following proposition summarizes basic properties of $\kappa_G$; see \citet{hofert2019compatibility}.

  \begin{proposition}[Basic properties of $\kappa_G$]\label{prop:properties:g:transformed:correlations}
  For any $G\in \mathcal G$, the $G$-transformed rank correlation $\kappa_G$ satisfies the following properties.
  \begin{enumerate}
\item\label{item:kappa:g:is:moc} $\kappa_G$ is a measure of concordance.
\item\label{item:kappa:g:invariance:location:scale:transform} $\kappa_G$ is invariant under location-scale transforms of $G$, that is, $\kappa_{G_{\mu,\sigma}}(C)=\kappa_G(C)$ for all $C\in \mathcal C_2$, where $\mu\in \IR$, $\sigma>0$ and $G_{\mu,\sigma}(x)=G\left(\frac{x-\mu}{\sigma}\right)$.
\item\label{item:linearity:kappa:g} For $n \in \IN$, let $C_1,\dots,C_n \in \mathcal C_2$ and
  $\alpha_1,\dots,\alpha_n$ be non-negative numbers such that
  $\alpha_1+\cdots+\alpha_n = 1$. Then
  \begin{align*}
    \kappa_G\biggl(\,\sum_{i=1}^{n}\alpha_i C_i \biggr)=\sum_{i=1}^{n}\alpha_i \kappa_G(C_i).
  \end{align*}
  \end{enumerate}
  \end{proposition}

\section{Estimation of $\kappa_G$ and their comparison}\label{sec:estimation:kappa:g:comparison}

In this section, we propose a novel approach for comparing $G$-transformed rank correlations to answer the question which concordance-inducing distribution is best to be used.
In the proposed approach, transformed rank correlations are compared in terms of the asymptotic variances of their canonical estimators, and one concordance-inducing distribution $G\in \mathcal G$ is considered better than another $G'\in \mathcal G$ if the best and worst asymptotic variances of an estimator $\hat \kappa_{G}$ of $\kappa_{G}$ among a set of copulas $\mathcal D \subseteq \mathcal C_2$ are smaller than those of $\kappa_{G'}$.

\subsection{Canonical estimator of $\kappa_G$}\label{subsec:canonical:estimator:kappa:G}

Based on Proposition~\ref{prop:properties:g:transformed:correlations} Part~\ref{item:kappa:g:invariance:location:scale:transform}, we first consider standardized concordance-inducing distributions $G$ such that
$\E_G[X]=0$ and $\Var_G(X)=1$ where $X\sim G$.

\begin{definition}[Canonical estimator of $\kappa_G$]\label{def:canonical:estimator:kappa:G}
Let
\begin{align*}
\mathcal G_4=\{G \in \mathcal G: \E_G[X]=0,\ \Var_G(X)=1\ \text{and}\ \E_G[X^4]<\infty\}.
\end{align*}
For $G \in \mathcal G_4$, the \emph{canonical estimator} of $\kappa_G$ is given by
\begin{align*}
\hat \kappa_G=\hat \kappa_G(C;n)=\frac{1}{n}\sum_{i=1}^n G^\i(U_i)G^\i(V_i)\quad\text{where} \quad (U_1,V_1),\dots,(U_n,V_n)\iidsim C, \quad  n \in \IN.
\end{align*}
For $C \in \mathcal C_2$, denote by $\sigma_G^2(C)=\Var(G^\i(U)G^\i(V))$ the \emph{asymptotic variance} of $\hat \kappa_G$ such that
\begin{align*}
\sqrt{n}\left\{\hat \kappa_G - \kappa_G(C)\right\} \darrow \N(0,\sigma_G^2(C)),\quad \text{as}\ n \rightarrow \infty.
\end{align*}
\end{definition}

We compare different concordance-inducing distributions in terms of the optimal asymptotic variances defined as follows.
We define arginf and argsup to be the empty set if infima and suprema are not attainable.

\begin{definition}[Optimal best and worst asymptotic variances for $\kappa_G$]
\label{Def:optimal:best:worst:asymptotic:variances}
For $G \in \mathcal G_4$ and $\mathcal D\subseteq \mathcal C_2$, the \emph{best and worst asymptotic variances} and their \emph{attaining copulas} on $\mathcal D$ are given by
\begin{align*}
\underline \sigma_G^2(\mathcal D)=\inf_{C \in \mathcal D} \sigma_G^2(C),\quad
\overline \sigma_G^2(\mathcal D)=\sup_{C \in \mathcal D} \sigma_G^2(C),
\quad
\underline C_G(\mathcal D)=\arginf_{C \in \mathcal D} \sigma_G^2(C),\quad
\overline C_G(\mathcal D)=\argsup_{C \in \mathcal D} \sigma_G^2(C),
\end{align*}
respectively.
For $\mathcal H \subseteq \mathcal G_4$ and $\mathcal D\subseteq \mathcal C_2$, the \emph{optimal best and worst asymptotic variances} and the sets of \emph{optimal concordance-inducing distributions} on $(\mathcal H,\mathcal D)$ are defined, respectively, by
\begin{align*}
\underline\sigma_{\ast}^2(\mathcal H,\mathcal D)=\inf_{G \in \mathcal H}\underline \sigma_G^2(\mathcal D),
\quad
\overline\sigma_{\ast}^2(\mathcal H,\mathcal D)=\inf_{G \in \mathcal H}\overline \sigma_G^2(\mathcal D),
\quad
\underline G_{\ast}(\mathcal H,\mathcal D)=\arginf_{G \in \mathcal H}\underline \sigma_G^2(\mathcal D),
\quad
\overline G_{\ast}(\mathcal H,\mathcal D)=\arginf_{G \in \mathcal H}\overline \sigma_G^2(\mathcal D).
\end{align*}

\end{definition}

\begin{definition}[Preference and optimal concordance-inducing distribution]
\label{def:preference:optimal:concordance:inducing:distribution}
We say that $G \in \mathcal G_4$ (or $\kappa_G$) is more \emph{preferable} to $G' \in \mathcal G_4$ (or $\kappa_{G'}$) on $\mathcal D \subseteq \mathcal C_2$, denoted by $G' \leq_{\mathcal D}G$ (or $\kappa_{G'}\leq_{\mathcal D} \kappa_G$), if
\begin{align*}
\underline \sigma_G^2(\mathcal D)\leq \underline \sigma_{G'}^2(\mathcal D)
\quad\text{and}
\quad
\overline \sigma_G^2(\mathcal D)\leq \overline \sigma_{G'}^2(\mathcal D).
\end{align*}
For $\mathcal H \subseteq \mathcal G_4$ and $\mathcal D\subseteq \mathcal C_2$, the set of \emph{optimal concordance-inducing distributions} is given by
\begin{align*}
G_{\ast}(\mathcal H,\mathcal D)=\underline G_{\ast}(\mathcal H,\mathcal D)\cap \overline G_{\ast}(\mathcal H,\mathcal D).
\end{align*}
\end{definition}

By definition, the preference order $G' \leq_{\mathcal D}G$ is a partial order.
Moreover, it is a total order for certain choices of $\mathcal D$; see Corollary~\ref{cor:optimal:concordance:inducing:functions:fundamental:frechet:copulas} below.
If $G \in G_{\ast}(\mathcal H,\mathcal D)$, then $G' \leq_{\mathcal D} G$ for all $G' \in \mathcal H$ and thus $G$ is considered as the best choice among $\mathcal H\subseteq \mathcal G_4$ to accurately estimate $\kappa_G$ if one believes that $\mathcal D$ is the set of underlying copulas for which one wants to quantify and compare dependence.

Although an ideal choice of $\mathcal H$ is $\mathcal H = \mathcal G_4$, other choices can be of interest; for example, one can consider $\mathcal H=\mathcal G_4^{\text{c}}$ where $\mathcal G_4^{\text{c}}$ is the set of continuous concordance-inducing distributions in $\mathcal G_4$, and $\mathcal H=\mathcal G_4^{\text{b}}$ where $\mathcal G_4^{\text{b}}$ is the set of concordance-inducing distributions in $\mathcal G_4$ with bounded supports.
Note that one-sided distributions $X \sim G$ such that $\esssup(X)=\infty$ and $\essinf(X)<\infty$, or $\esssup(X)<\infty$ and $\essinf(X)=-\infty$, cannot be concordance-inducing since they cannot be radially symmetric.
Therefore, $\mathcal G_4\backslash \mathcal G_4^{\text{b}}$ is a set of concordance-inducing distributions with $\IR$ as support.

\subsection{Properties of the asymptotic variance}
\label{subsec:properties:asymptotic:variance}

We study properties of the asymptotic variance and its optimal best and worst counterparts.
Let $\nu_1,\ \nu_2 : \mathcal C_2 \rightarrow \mathcal C_2$ be partial reflections of copulas defined, respectively, by
\begin{align*}
\nu_1(C)(u,v)  =v-C(1-u,v)\quad\text{and}\quad
\nu_2(C)(u,v)  =u-C(u,1-v),\quad C \in \mathcal C_2.
\end{align*}
Their composition is given by $\nu_1\circ \nu_2(C)(u,v)=u+v-1+C(1-u,1-v)$.
Denote by $\iota:\mathcal C_2 \rightarrow \mathcal C_2$ the identity $\iota(C)=C$.
For $\varphi  \in \{\iota, \nu_1,\nu_2, \nu_1\circ \nu_2\}$, let $C_{\varphi} = \varphi(C)$.

Denote by
\begin{align*}
(U_{\nu_1},V_{\nu_1})\aseq (1-U,V)\sim C_{\nu_1},\quad
(U_{\nu_2},V_{\nu_2})\aseq (U,1-V)\sim C_{\nu_2},\quad
(U_{\nu_1\circ \nu_2},V_{\nu_1\circ \nu_2})\aseq (1-U,1-V)\sim C_{\nu_1\circ \nu_2},

\end{align*}
for $(U,V)\sim C$, where ``a.s.'' stands for \emph{almost surely}.

\begin{proposition}[Basic properties of $\sigma_G^2(C)$]
\label{prop:basic:properties:asymptotic:variance}
Let $G \in \mathcal G_4$.
Then the map $C \mapsto \sigma_G(C)$ satisfies the following properties.
\begin{enumerate}
\item\label{item:covariance:formula:asymptotic:variance}
\emph{Covariance formula}:
For $(U,V)\sim C \in \mathcal C_2$ and $(X,Y)=(G^\i(U),G^\i(V))$, we have that
\begin{align}\label{eq:covariance:formula:asymptotic:variance}
\sigma_G^2(C)=\Cov(X^2,Y^2)+1 - \Cov(X,Y)^2.
\end{align}
\item\label{item:fundamental:copulas:asymptotic:variance}
\emph{Values at fundamental copulas}:
$\sigma_G^2(\Pi)=1$ and $\sigma_G^2(M)=\sigma_G^2(W)=\Var_G(X^2)$.
\item\label{item:bounds:asymptotic:variance}
\emph{Bounds}: $0\leq \sigma_G^2(C)\leq 1+\Var_G(X^2)$ for $X \sim G$.
In particular, $\sigma_G^2\left(\frac{M+W}{2}\right)= 1+\Var_G(X^2)$.
\item\label{item:boundedness:asymptotic:variance}
\emph{Boundedness}:
$\sigma_G^2(C)<\infty$ for all $C \in \mathcal C_2$.
\item\label{item:reflection:invariance:asymptotic:variance}
\emph{Reflection invariance}:
$\sigma_G^2(C)=\sigma_G^2(C_{\nu_1})=\sigma_G^2(C_{\nu_2})=\sigma_G^2(C_{\nu_1\circ \nu_2})$.
\end{enumerate}
\end{proposition}

\begin{proof}
\noindent \ref{item:covariance:formula:asymptotic:variance})
For $(U,V)\sim C$, write $X=G^\i(U)$ and $Y=G^\i(V)$.
Since $G \in \mathcal G_4$, we have that $\Cov(X^2,Y^2)=\E[(XY)^2]-\E[X^2]\E[Y^2]=\E[(XY)^2]-1$ and $\Cov(X,Y)=\E[XY]-\E[X]\E[Y]=\E[XY]$.
Therefore,
\begin{align*}
\sigma_G^2(C)=\Var(XY)=\E[(XY)^2]-\E[XY]^2=\Cov(X^2,Y^2)+1 - \Cov(X,Y)^2.
\end{align*}

\noindent \ref{item:fundamental:copulas:asymptotic:variance})
If $(U,V)\sim \Pi$, then $(X^2,Y^2)$ and $(X,Y)$ are both independent random vectors, and thus $\Cov(X^2,Y^2)=\Cov(X,Y)=0$.
Therefore $\sigma_G^2(\Pi)=0 + 1- 1^2 = 1$ by~\eqref{eq:covariance:formula:asymptotic:variance}.
If $C=M$, then we have that $\Cov(X,Y)=\kappa_G(M)=1$ and that $(X^2,Y^2)\deq(G^\i(U)^2,G^\i(U)^2)$ for $U\sim \U(0,1)$.
When $C=W$, we have that $\Cov(X,Y)=\kappa_G(W)=-1$ and that
\begin{align*}
(X^2,Y^2)\deq(G^\i(U)^2,G^\i(1-U)^2)=(G^\i(U)^2,(-G^\i(U))^2)=(G^\i(U)^2,G^\i(U)^2),\quad U\sim \U(0,1),
\end{align*}
by radial symmetry of $G \in \mathcal G_4$.
Therefore, in either case, one has $\rho(X^2,Y^2)=1$ and thus
\begin{align*}
\Cov(X^2,Y^2)= \rho(X^2,Y^2)\operatorname{SD}_G(X^2)\operatorname{SD}_G(Y^2)=\Var_G(X^2).
\end{align*}
Therefore, by~\eqref{eq:covariance:formula:asymptotic:variance}, we have that $\sigma_G^2(M) =\Var_G(X^2)+1-1^2=\Var_G(X^2)$ and $\sigma_G^2(W) =\Var_G(X^2)+1-(-1)^2=\Var_G(X^2)$.

\noindent \ref{item:bounds:asymptotic:variance})
By Cauchy--Schwarz inequality, it holds that $\Cov(X^2,Y^2)\leq \operatorname{SD}(X^2)\operatorname{SD}(Y^2)=\Var(X^2)$.
Since $\Cov(X,Y)^2\geq 0$, the desired inequalities hold by~\eqref{eq:covariance:formula:asymptotic:variance}.
When $C=\frac{M+W}{2}$, we have that
\begin{align*}
(X,Y)\deq B  (G^\i(U),G^\i(U))+ (1-B)(G^\i(U),G^\i(1-U)),
\end{align*}
where $B \sim \Bern(1/2)$ and $U \sim \Unif(0,1)$ independent of $B$.
Therefore,
\begin{align*}
\Cov(X,Y)=\frac{1}{2}\Cov(G^\i(U),G^\i(U)) + \frac{1}{2}\Cov(G^\i(U),G^\i(1-U))=\frac{1}{2}(1)+\frac{1}{2}(-1)=0,
\end{align*}
and $\Cov(X^2,Y^2)=\Cov(G^\i(U)^2,G^\i(U)^2)=\Var_G(X^2)$.
Using~\eqref{eq:covariance:formula:asymptotic:variance}, we have that $\sigma_G\left(\frac{M+W}{2}\right)=\Var_G(X^2)+1-0^2=\Var_G(X^2)+1$.

\noindent \ref{item:boundedness:asymptotic:variance})
Since the fourth moment of $G$ is finite, we have that $\Var_G(X^2)<\infty$.
Therefore $\sigma_G^2(C)<\infty$ by~\ref{item:bounds:asymptotic:variance}).

\noindent \ref{item:reflection:invariance:asymptotic:variance})
The desired result follows from~\eqref{eq:covariance:formula:asymptotic:variance} since radial symmetry of $G \in \mathcal G$ implies that
\begin{align}\label{eq:almost:sure:equalities:radial:symmetry}
(G^\i(U),G^\i(V))&\aseq
(-G^\i(U_{\nu_1}),G^\i(V_{\nu_1}))\aseq
(G^\i(U_{\nu_2}),-G^\i(V_{\nu_2}))
\aseq(-G^\i(U_{\nu_1\circ \nu_2}),-G^\i(V_{\nu_1 \circ \nu_2})).
\end{align}

\end{proof}

\begin{proposition}[Concavity of $\sigma_G^2(C)$]\label{prop:concavity:sigma:G:C:asymptotic:variance}
For $G \in \mathcal G_4$, the map $C \mapsto \sigma_G^2(C)$ is concave with respect to convex combinations of copulas.
In particular, for $k \in [-1,1]$, the map $C \mapsto \sigma_G^2(C)$ is linear on $\mathcal C_G(k)=\{C \in \mathcal C_2: \kappa_G(C)=k\}$.

\end{proposition}

\begin{proof}
For $C, C' \in \mathcal C_2$ and $p \in [0,1]$, define the random vector $(\tilde X,\tilde Y)=(G^\i(\tilde U),G^\i(\tilde V))$ where $(\tilde U,\tilde V)= B (U,V) + (1-B)(U',V')$, $(U,V)\sim C$, $(U',V')\sim C'$ and $B\sim \Bern(p)$ is independent of $(U,V)$ and $(U',V')$.
Then $(\tilde U,\tilde V)\sim\tilde C_p$ where $\tilde C_p=pC + (1-p)C'$.
Moreover, we have that
\begin{align*}
(\tilde X,\tilde Y)=(G^\i(\tilde U),G^\i(\tilde V))=(G^\i(BU+(1-B)U'),G^\i(BV+(1-B)V'))=B(X,Y)+(1-B)(X',Y'),
\end{align*}
where  $(X,Y)=(G^\i(U),G^\i(V))$ and $(X',Y')=(G^\i(U'),G^\i(V'))$.

From this representation, we have that
$\Cov(\tilde X^2,\tilde Y^2)=p\Cov(X^2,Y^2)+(1-p)\Cov({X'}^2,{Y'}^2)$ and $\Cov(\tilde X,\tilde Y)=p\Cov(X,Y)+(1-p)\Cov(X',Y')$.
Therefore, we have that
\begin{align*}
\sigma_G^2(\tilde C_p)& =\Var(\tilde X \tilde Y)= \Cov(\tilde X^2,\tilde Y^2)+ 1 -\Cov(\tilde X,\tilde Y)^2\\
&= p\Cov(X^2,Y^2)+(1-p)\Cov({X'}^2,{Y'}^2) + 1 - (p\Cov(X,Y)+(1-p)\Cov(X',Y'))^2\\
&\geq  p\Cov(X^2,Y^2)+(1-p)\Cov({X'}^2,{Y'}^2) + 1 - p\Cov(X,Y)^2-(1-p)\Cov(X',Y')^2\\
&= p \Var(XY)+(1-p)\Var(X'Y')=p\sigma_G^2(C)+(1-p)\sigma_G^2(C'),
\end{align*}
where the inequality in the third line holds since
\begin{align*}
p\Cov(X,Y)^2+(1-p)\Cov(X',Y')^2 &- (p\Cov(X,Y)+(1-p)\Cov(X',Y'))^2\\
&= p(1-p)(\Cov(X,Y)- \Cov(X',Y'))^2 \geq0.
\end{align*}
Therefore $C \mapsto \sigma_G^2(C)$ is concave.
When $C, C' \in \mathcal C_G(k)$ for some $k \in [-1,1]$, equality holds in the inequality above, and thus  $C \mapsto \sigma_G^2(C)$ is linear.
\end{proof}

By concavity of $C \mapsto \sigma_G^2(C)$, it holds that $\text{ext}(\mathcal D)=\underline C_G(\mathcal D)$ for a closed convex set $\mathcal D\subseteq \mathcal C_2$ of copulas, where $\text{ext}(\mathcal D)$ is the set of extremal points in $\mathcal D$.
Therefore we have the following formula for $\underline \sigma_G^2(\mathcal D)$.

\begin{corollary}[Attaining copulas of $\underline \sigma_G^2(\mathcal D)$]\label{cor:min:lower:asymptotic:variance:extremal:points}

For $L \in \IN$ and $C_1,\dots,C_L \in \mathcal C_2$, let $\mathcal D=\overline{\text{conv}}(C_1,\dots,C_L) \subseteq \mathcal C_2$ be the closed set of convex combinations of $C_1,\dots,C_L$.
Then $\underline C_G(\mathcal D)= \argmin_{l=1,\dots,L}\sigma_G^2(C_l)$ and $\underline \sigma_G^2(\mathcal D)= \min_{l=1,\dots,L}\sigma_G^2(C_l)$.
\end{corollary}

\begin{comment}
The map $C \mapsto \sigma_G^2(C)=\Cov(X^2,Y^2)+1 - \Cov(X,Y)^2$ is in general not linear with respect to convex combinations of copulas since $C \mapsto \Cov(X,Y)^2$ is quadratic.
On the other hand, the map $C \mapsto \Cov(X^2,Y^2)$ is linear by Lemma~\ref{lemma:linearity:copula:X2:Y2},

and thus, $C \mapsto \sigma_G^2(C)$ is linear on a restricted subset of $\mathcal C_2$ as shown by the following proposition.

\begin{proposition}[Linearity of $C \mapsto \sigma_G^2(C)$]\label{prop:linearity:sigma:G:C:asymptotic:variance}
For a continuous concordance-inducing distribution $G \in \mathcal G_4^\text{c}$ and a constant $k \in [-1,1]$, the map $C \mapsto \sigma_G^2(C)$ is linear with respect to convex combinations of copulas on $\mathcal C_G(k)=\{C \in \mathcal C_2: \kappa_G(C)=k\}$.

\end{proposition}

\begin{proof}
For $\tilde C_p=pC + (1-p)C'$, $C, C' \in \mathcal C_2$ and $(\tilde U,\tilde V)\sim \tilde C_p$, we have that $(\tilde X,\tilde Y)=(G^\i(\tilde U),G^\i(\tilde V))=(G^\i(BU+(1-B)U'),G^\i(BV+(1-B)V'))=B(X,Y)+(1-B)(X',Y')$
where  $(X,Y)=(G^\i(U),G^\i(V))$ and $(X',Y')=(G^\i(U'),G^\i(V'))$ with $(U,V)\sim C$, $(U',V')\sim C'$ and $B\sim \Bern(p)$ being independent.
From this representation, we have that
$\Cov(\tilde X^2,\tilde Y^2)=p\Cov(X^2,Y^2)+(1-p)\Cov({X'}^2,{Y'}^2)$ and $\Cov(\tilde X,\tilde Y)=p\Cov(X,Y)+(1-p)\Cov(X',Y')$.
If $C \in \mathcal C_G(k)$, then $\Cov(X,Y)=\kappa_G(C)=k$ and $\Cov(X',Y')=\kappa_G(C')=k$.
Therefore, we have that $\Cov(\tilde X,\tilde Y)=k$ and thus
\begin{align*}
\sigma_G^2(\tilde C_p)& =\Var(\tilde X \tilde Y)= \Cov(\tilde X^2,\tilde Y^2)+ 1 -\Cov(\tilde X,\tilde Y)^2\\
&= p\Cov(X^2,Y^2)+(1-p)\Cov({X'}^2,{Y'}^2) + 1 - (p\Cov(X,Y)+(1-p)\Cov(X',Y'))^2\\
&= p (\Cov(X^2,Y^2)+1-k^2 ) + (1-p) (\Cov({X'}^2,{Y'}^2)+1-k^2 )\\
&= p \Var(XY)+(1-p)\Var(X'Y')=p\sigma_G^2(C)+(1-p)\sigma_G^2(C'),
\end{align*}
which shows the desired property.
\end{proof}
\end{comment}

Next we study continuity of the map $(G,C)\mapsto \sigma_G^2(C)$, which leads to attainability of the suprema and infima in $\overline \sigma_G^2(\mathcal D)$, $\underline \sigma_G^2(\mathcal D)$, $\overline \sigma_{\ast}^2(\mathcal H, \mathcal D)$ and $\underline \sigma_{\ast}^2(\mathcal H, \mathcal D)$ for closed sets $\mathcal H \subseteq \mathcal G_4$ and $\mathcal D \subseteq \mathcal C_2$.
To this end, let the product space $\mathcal G_4 \times \mathcal C_2$ be metrized by the uniform norm
\begin{align*}
d_{\infty}((G,C),(G',C'))&=d_{\infty}(G,G')\vee d_{\infty}(C,C')\\
&=\sup_{x \in \IR}|G(x)-G'(x)|\vee \sup_{(u,v)\in [0,1]^2} |C(u,v)-C'(u,v)|.
\end{align*}

\begin{proposition}[Continuity of $C \mapsto \sigma_G^2(C)$]\label{prop:continuity:asymptotic:variance:C}
Let $G \in \mathcal G_4$ and $\mathcal D\subseteq \mathcal C_2$ be a closed subset in $(\mathcal C_2,d_{\infty})$.
Then the map $C \mapsto \sigma_G^2(C)$ is continuous on $\mathcal D$, and thus

$\overline C_G(\mathcal D)=\argsup_{C \in \mathcal D}\sigma_G^2(C) $ and $\underline C_G(\mathcal D)=\arginf_{C \in \mathcal D}\sigma_G^2(C)$ are non-empty.

\end{proposition}

\begin{proof}
For $G \in \mathcal G_4$, $C_n \in \mathcal D$, $n=1,2,\dots$ and $C\in \mathcal D$ such that $\lim_{n\rightarrow \infty}C_n =C$ pointwise, let $(U_n,V_n)\sim C_n$, $(U,V)\sim C$, $(X_n,Y_n)=(G^\i(U_n),G^\i(V_n))$, $(X,Y)=(G^\i(U),G^\i(V))$, $H_n(x,y)=\Prob(X_n\leq x,Y_n \leq y)$, $H_n^{[2]}(x,y)=\Prob(X_n^2\leq x,Y_n^2\leq y)$, $H(x,y)=\Prob(X\leq x,Y \leq y)$ and $H^{[2]}(x,y)=\Prob(X^2\leq x,Y^2\leq y)$ for $(x,y)\in\IR^2$.
Then the pointwise convergence of $C_n \rightarrow C$ implies $H_n \rightarrow H$ and $H_n^{[2]} \rightarrow H^{[2]}$ pointwise, since, as $n \rightarrow \infty$, we have that
\begin{align*}
H_n(x,y)=C_n(G(x),G(y)) \rightarrow C(G(x),G(y))=H(x,y),
\quad (x,y) \in \IR^2,
\end{align*}
and that
\begin{align*}
H_n^{[2]}(x,y)&=H_n\left(\sqrt{x},\sqrt{y}\right)-H_n\left(-\sqrt{x}-,\sqrt{y}\right)-H_n\left(\sqrt{x},-\sqrt{y}-\right)+H_n\left(-\sqrt{x}-,-\sqrt{y}-\right)\\
&\rightarrow
H\left(\sqrt{x},\sqrt{y}\right)-H\left(-\sqrt{x}-,\sqrt{y}\right)-H\left(\sqrt{x},-\sqrt{y}-\right)+H\left(-\sqrt{x}-,-\sqrt{y}-\right)\\
&=H^{[2]}(x,y), \quad (x,y) \in \IR_{+}^2,
\end{align*}
where $\IR_{+}^2=\{(x,y)\in \IR^2: x,\, y \geq 0\}$.

Therefore, by the Hoeffding's identity \citep[Lemma~7.27]{mcneil2015quantitative}, we have that
\begin{align*}
\lim_{n \rightarrow \infty}\Cov(X_n^2,Y_n^2) &= \lim_{n \rightarrow \infty} \iint_{\IR^2}\left(H_n^{[2]}(x,y)-G^{[2]}(x)G^{[2]}(y)\right)\,\rd x \rd y \\
&= \iint_{\IR^2}\left(H^{[2]}(x,y)-G^{[2]}(x)G^{[2]}(y)\right)\,\rd x \rd y= \Cov(X^2,Y^2),
\end{align*}
where $G^{[2]}(x)=\Prob(X^2\leq x)$, $x \in \IR$, for $X\sim G$.
Interchanging the limit and the integral above is justified by Lebesgue's dominated convergence theorem since the integrand $H_n^{[2]}(x,y)-G^{[2]}(x)G^{[2]}(y)$ is bounded above by $ M(G^{[2]}(x),G^{[2]}(y))-G^{[2]}(x)G^{[2]}(y)$, which is integrable as
\begin{align*}
\iint_{\IR^2}\left(M(G^{[2]}(x),G^{[2]}(y))-G^{[2]}(x)G^{[2]}(y)\right)\,\rd x \rd y
\leq \Var_G(X^2) < \infty.
\end{align*}

Similarly, it holds that $\lim_{n \rightarrow \infty}\Cov(X_n,Y_n)=\Cov(X,Y)$. Therefore, we have the continuity
\begin{align*}
\lim_{n \rightarrow \infty}\sigma_G^2(C_n)&=\lim_{n \rightarrow \infty}\left(\Cov(X_n^2,Y_n^2)+1 - \Cov(X_n,Y_n)^2\right)

= \Cov(X^2,Y^2)+1 - \Cov(X,Y)^2= \sigma_G^2(C),
\end{align*}
which completes the proof.

\end{proof}

To guarantee the continuity of $G \mapsto \overline \sigma_G^2(\mathcal D)$ and $G \mapsto \underline \sigma_G^2(\mathcal D)$ we require $G \in \mathcal G_4$ to have a bounded support.

\begin{proposition}[Continuity of $G \mapsto \overline \sigma_G^2(\mathcal D)$ and $G \mapsto \underline \sigma_G^2(\mathcal D)$]\label{prop:continuity:asymptotic:variance:C:G}
Let  $\mathcal H \subseteq \mathcal G_4^{\textup{b}}$ and $\mathcal D \subseteq \mathcal C_2$ be closed subsets in $(\mathcal G_4^{\textup{b}},d_{\infty})$ and $(\mathcal C_2,d_{\infty})$, respectively.
Then the maps $G \mapsto \overline \sigma_G^2(\mathcal D)$ and $G \mapsto \underline \sigma_G^2(\mathcal D)$ are continuous on $\mathcal H$, and thus

$\overline G_{\ast}(\mathcal H, \mathcal D)$ and $\underline G_{\ast}(\mathcal D)$ are non-empty.

\end{proposition}

\begin{proof}
To show the continuity of $G \mapsto \overline \sigma_G^2(\mathcal D)$ and $G \mapsto \underline \sigma_G^2(\mathcal D)$, it suffices to show the continuity of the map $(G,C)\mapsto \sigma_G^2(C)$ by Berge's maximum theorem \citep{berge1997topological}.
Since $|\sigma_G^2(C)-\sigma_{G'}^2(C')|\leq |\sigma_G^2(C)-\sigma_{G'}^2(C)|+|\sigma_{G'}^2(C)-\sigma_{G'}^2(C')|$ for $C,C' \in \mathcal D$ and $G,G' \in \mathcal H$, it suffices to show the continuity of $C \mapsto \sigma_G^2(C)$ for a fixed  $G \in \mathcal H$, and of $G \mapsto \sigma_{G}^2(C)$ for a fixed $C \in \mathcal D$.
Continuity of $C \mapsto \sigma_G^2(C)$ follows from Proposition~\ref{prop:continuity:asymptotic:variance:C}.
Therefore, it suffices to show the continuity of $G \mapsto \sigma_{G}^2(C)$.

By \citet[Lemma 2.1.5]{nelsen2006introduction}, we have that
\begin{align*}
|H_n(x,y)-H(x,y)|\leq  |G_n(x)-G(x)|+|G_n(y)-G(y)|, \quad (x,y) \in \IR^2,
\end{align*}
and thus the pointwise convergence of $G_n \rightarrow G$ implies $H_n \rightarrow H$ and $H_n^{[2]} \rightarrow H^{[2]}$ pointwise.
Since $G_n, G \in \mathcal G_4^{\text{b}}$ for $n\in \IN$,
there exists a bounded subset $S \subset \IR_{+}$ such that $\cup_{n \in \IN} \text{\supp}(G_n^{[2]}) \cup \text{\supp}(G^{[2]}) \subseteq S$ where $\supp(G)$ is the support of $G$.
Since
\begin{align*}
H_n^{[2]}(x,y)-G_n^{[2]}(x)G_n^{[2]}(y)
\begin{cases}
=0 & \text{ if } (x,y)\notin \supp(G_n^{[2]})^2,\\
\leq 1 & \text{ otherwise},
\end{cases}
\end{align*}
we have that
\begin{align*}
\lim_{n\rightarrow \infty}\Cov(X_n^2,Y_n^2) &=  \lim_{n\rightarrow \infty}\iint_{\IR^2}\left(H_n^{[2]}(x,y)-G_n^{[2]}(x)G_n^{[2]}(y)\right)\,\rd x \rd y\\
&= \lim_{n\rightarrow \infty}\iint_{S}\left(H_n^{[2]}(x,y)-G_n^{[2]}(x)G_n^{[2]}(y)\right)\,\rd x \rd y
=\iint_{S}\left(H^{[2]}(x,y)-G^{[2]}(x)G^{[2]}(y)\right)\,\rd x \rd y \\
&= \Cov(X^2,Y^2),
\end{align*}
where interchanging the limit and the integral is verified by the bounded convergence theorem.
Similarly we have that $\lim_{n\rightarrow \infty}\Cov(X_n,Y_n)=\Cov(X,Y)$, and thus $\lim_{n\rightarrow \infty}\sigma_{G_n}^2(C)=\sigma_{G}^2(C)$ as desired.
\end{proof}

\subsection{Optimal location shift of $G$}\label{subsec:optimal:location:shift}

Although $\kappa_G$ is invariant under location-scale transforms of $G$ by Proposition~\ref{prop:properties:g:transformed:correlations} Part~\ref{item:kappa:g:invariance:location:scale:transform}, the asymptotic variance $\sigma_G^2(C)$ of its canonical estimator $\hat \kappa_G$ may not be the case.
To see this, let $G_0 \in \mathcal G_4$ be a concordance-inducing distribution with mean zero and variance one, and let $G_{\mu,\sigma}(x)=G_0(\frac{x-\mu}{\sigma})$ be the corresponding concordance-inducing distribution of the same type as $G_0$ but with mean $\mu \in \IR$ and variance $\sigma^2 >0$.
A canonical estimator of $\kappa_{G_{\mu,\sigma}}$ for known $\mu$ and $\sigma$ is then given by
\begin{align*}
\hat \kappa_{G_{\mu,\sigma}} = \frac{1}{n}\sum_{i=1}^n \frac{G_{\mu,\sigma}^\i(U_i)G_{\mu,\sigma}^\i(V_i)}{\sigma^2} - \left(\frac{\mu}{\sigma}\right)^2.
\end{align*}
By the CLT, asymptotic normality follows for $\hat \kappa_{G_{\mu,\sigma}}$ with the asymptotic variance given by
\begin{align*}
\sigma_{G_{\mu,\sigma}}^2(C)=\Var\left(\frac{G_{\mu,\sigma}^\i(U)G_{\mu,\sigma}^\i(V)}{\sigma^2}\right).
\end{align*}
Since $G_{\mu,\sigma}^\i(U)/\sigma=G_{\mu/\sigma,1}^\i(U)$ and $G_{\mu,\sigma}^\i(V)/\sigma=G_{\mu/\sigma,1}^\i(V)$, one can assume that $\sigma=1$ without changing the asymptotic variance $\sigma_{G_{\mu,\sigma}}^2(C)$; in this sense $\sigma_G^2(C)$ is invariant under scale transforms of $G$.
On the other hand, $\sigma_G^2(C)$ changes under location transforms of $G$ since shifting $G^\i$ by $\mu \in \IR$ leads to the asymptotic variance $\Var((X+\mu)(Y+\mu))=\Var(XY+\mu(X+Y))\neq \Var(XY)$ for $X=G^\i(U)$ and $Y=G^\i(V)$.

Since the canonical estimator $\hat \kappa_{G_{\mu,\sigma}}$ estimates the same quantity $\kappa_{G_0}$ regardless of the mean $\mu$ and variance $\sigma^2$ of $G$, a natural choice of $(\mu,\sigma)$ is such that $\sigma =1$ and $\mu$ minimizes the asymptotic variance $\sigma_{G_{\mu,1}}^2(C)$.
For a fixed concordance-inducing distribution $G_0 \in \mathcal G_4$ with mean zero and variance one, denote by $G_\mu(x)=G_0(x-\mu)$ the concordance-inducing distribution of the same type as $G_0$ but with mean $\mu\in \IR$.
For $X= X_0+\mu \sim G_\mu$ and $Y= Y_0+\mu \sim G_\mu$ with $X_0=G_0^\i(U)$ and $Y_0=G_0^\i(V)$, the asymptotic variance is given by
\begin{align*}
\sigma_{G_\mu}^2(C)&=\Var(XY)=\Var((X_0+\mu)(Y_0+\mu))=\Var(X_0Y_0+\mu(X_0+Y_0))\\
&= \Var(X_0Y_0)+ 2\mu \Cov(X_0Y_0,X_0+Y_0)+\mu^2 \Var(X_0+Y_0).
\end{align*}
Therefore the desired $\mu$ is given as follows.

\begin{definition}[Optimal shift of $G_0$]
For $G_0 \in \mathcal G_4$ and $C \in \mathcal C_2$, the minimizer of $\mu \mapsto \sigma_{G_\mu}^2(C)$ is called the \emph{optimal (location) shift} of $G_0$ under $C$, and is given by
\begin{align*}
\mu_{\ast} = \mu_{\ast}(G_0,C)=\begin{cases}
-\frac{\Cov(X_0Y_0, X_0+Y_0)}{\Var(X_0+Y_0)}, & \text{ if } \Var(X_0+Y_0)>0,\\
0, & \text{ if } \Var(X_0+Y_0)=0,
\end{cases}
\end{align*}
where $(U,V)\sim C$ and $(X_0,Y_0)=(G_0^\i(U),G_0^\i(V))$.
The optimal asymptotic variance is then given by
\begin{align*}
\sigma_{G_{\mu_\ast}}^2(C)=
\begin{cases}
\Var(X_0Y_0)-\frac{\Cov(X_0Y_0,X_0+Y_0)^2}{\Var(X_0+Y_0)},& \text{ if } \Var(X_0+Y_0)>0,\\
\Var(X_0Y_0), & \text{ if } \Var(X_0+Y_0)=0.
\end{cases}
\end{align*}

\end{definition}

For $G_0 \in \mathcal G_4$, the degenerate case $\Var(X_0+Y_0)=0$ occurs if and only if $X_0+Y_0\aseq0$.
In this case, we have that

\begin{align*}
\sigma_{G_\mu}^2(C)= \Var(X_0Y_0)+ 2\mu \Cov(X_0Y_0,0)+\mu^2 \Var(0)= \Var(X_0Y_0)=\sigma_{G_0}^2(C).
\end{align*}

The following proposition states that $\mu_{\ast}=0$ for a certain class of copulas.

\begin{proposition}[Sufficient condition for $\mu_{\ast}=0$]\label{prop:sufficient:condition:optimal:shift:zero}
For a copula $C \in \mathcal C_2$ and a concordance-inducing distribution $G_0 \in \mathcal G_4$ with mean zero and variance one, $\mu_{\ast}(G_0,C)=0$ holds if $C$ is radially symmetric $C=C_{\nu_1\circ \nu_2}$, that is,

$(U,V)\deq (1-U,1-V)$ for $(U,V)\sim C$.

\end{proposition}

\begin{proof}
By definition of $\mu_{\ast}$, it suffices to consider the case when $\Var(X_0+Y_0)>0$ for $X_0 = G_0^\i(U)$ and $Y_0 = G_0^\i(V)$ with $(U,V)\sim C$.

Since $\E[X_0+Y_0]=\E[X_0]+\E[Y_0]=0$, we have that $\Cov(X_0Y_0,X_0+Y_0)=\E[X_0Y_0(X_0+Y_0)]-\E[X_0Y_0]\E[X_0+Y_0]=\E[X_0Y_0(X_0+Y_0)]$.
Therefore, it suffices to show that $\E[X_0Y_0(X_0+Y_0)]=0$ when $C$ is radially symmetric.

When $C$ is radially symmetric, we have that $(U,V)\deq (U_{\nu_1\circ \nu_2},$ $V_{\nu_1\circ \nu_2})$ and $(U_{\nu_1},V_{\nu_1})\deq (U_{\nu_2},V_{\nu_2})$ for $(U,V)\sim C$.

Together with the identity
\begin{align*}
1&=\bone_{\{U>1/2,\,V>1/2\}} + \bone_{\{U\leq 1/2,\,V>1/2\}} + \bone_{\{U>1/2,\,V\leq 1/2\}} + \bone_{\{U\leq 1/2,\,V\leq 1/2\}}\\
&= \bone_{\{U>1/2,\,V>1/2\}} + \bone_{\{U_{\nu_1}>1/2,\,V_{\nu_1}>1/2\}} + \bone_{\{U_{\nu_2}>1/2,\,V_{\nu_2}>1/2\}} + \bone_{\{U_{\nu_1\circ \nu_2}>1/2,\,V_{\nu_1\circ \nu_2}>1/2\}},
\end{align*}
we have, by~\eqref{eq:almost:sure:equalities:radial:symmetry}, that
\begin{align*}

\E[X_0Y_0  (X_0+Y_0)]
  &= \sum_{\varphi  \in \{\iota, \nu_1,\nu_2, \nu_1\circ \nu_2\}}\E[\bone_{\{U_{\varphi}>1/2,V_{\varphi}>1/2\}}G_0^\i(U)G_0^\i(V)( G_0^\i(U)+ G_0^\i(V))]\\

&=  \E[\bone_{\{U>1/2,V>1/2\}}G_0^\i(U)G_0^\i(V)( G_0^\i(U)+ G_0^\i(V))]\\
& \hspace{5mm}- \E[\bone_{\{U_{\nu_1\circ \nu_2}>1/2,V_{\nu_1\circ \nu_2}>1/2\}}G_0^\i(U_{\nu_1\circ \nu_2})G_0^\i(V_{\nu_1\circ \nu_2})( G_0^\i(U_{\nu_1\circ \nu_2})+ G_0^\i(V_{\nu_1\circ \nu_2}))]\\
&\hspace{5mm}+ \E[\bone_{\{U_{\nu_1}>1/2,V_{\nu_1}>1/2\}}G_0^\i(U_{\nu_1})G_0^\i(V_{\nu_1})( G_0^\i(U_{\nu_1})-G_0^\i(V_{\nu_1}))]\\
&\hspace{5mm}- \E[\bone_{\{U_{\nu_2}>1/2,V_{\nu_2}>1/2\}}G_0^\i(U_{\nu_2})G_0^\i(V_{\nu_2})( G_0^\i(U_{\nu_2})- G_0^\i(V_{\nu_2}))]\\
&=0,
\end{align*}

which gives the desired result $\mu_{\ast}(G,C)=0$.
\end{proof}

By Proposition~\ref{prop:sufficient:condition:optimal:shift:zero}, a location shift of $G_0$ does not change the asymptotic variance $\sigma_{G_0}^2(C)$ when $C$ is, for example, $M$, $W$, $\Pi$, a Gaussian copula, $t$ copula or their mixtures.

On the other hand, shifting $G_0$ may improve $\sigma_{G_0}^2(C)$ if $C$ is, for example, a Clayton or Gumbel copula since they are not radially symmetric.
The next proposition states that the optimal asymptotic variance can be obtained analytically when $(X,Y)=(G^\i(U),G^\i(V))$ is a normal variance mixture, that is,
\begin{align*}
(X,Y)=(\mu_1,\mu_2)+\sqrt{W}(Z_1,Z_2),
\quad (\mu_1,\mu_2)\in \IR^2,
\quad (Z_1,Z_2)\sim \N_2(\bzero_2,\Sigma),
\end{align*}
where $\Sigma$ is a $2$-dimensional square positive definite matrix and $W \geq 0$ is a non-negative and nondegenerate random variable independent of $(Z_1,Z_2)$.

\begin{proposition}[$\sigma_{G_{\mu_\ast}}^2(C)$ for normal variance mixture]\label{prop:asymptotic:variance:normal:variance:mixture}
For $G \in \mathcal G_4$ and $(U,V)\sim C \in \mathcal C_2$, suppose that $(X,Y)=(G^\i(U),G^\i(V))$ is a normal variance mixture.
Then the optimal asymptotic variance is given by
\begin{align}\label{eq:optimal:asymptotic:variance:normal:variance:mixture}
\sigma_{G_{\mu^\ast}}^2(C)= \left(2\frac{\E[W^2]}{\E[W]^2}  -1\right) \rho^2(X,Y) + \frac{\E[W^2]}{\E[W]^2}.
\end{align}
\end{proposition}

\begin{proof}
Since normal variance mixtures are radially symmetric, the optimal asymptotic variance is obtained when $\mu^{\ast}=0$.
Since $\sigma_{G_{\mu^\ast}}^2(C)$ is invariant under scale transforms of $G_{\mu_{\ast}}$, we standardize $(X,Y)$ to apply the covariance formula~\eqref{eq:covariance:formula:asymptotic:variance}.
Namely, we set $\Sigma=P/\E[W]$ where $P$ is a correlation matrix with off-diagonal entry $\rho=\rho(X,Y)$.
In this setup, we have that
\begin{align*}
\Var(X)=\Var(Y)=\E[WZ_1^2]-\E\left[\sqrt{W}Z_1\right]^2=\E[W]\Var(Z_1)-0=\E[W]\frac{1}{\E[W]}=1.
\end{align*}
Since $\Cov(Z_1^2,Z_2^2)=2\rho^2/\E[W]^2$, the law of total covariance implies that

\begin{align*}
\Cov(X^2,Y^2)&=\E[\Cov(X^2,Y^2)\ |\ W)] + \Cov(\E[X^2\ |\ W],\ \E[Y^2\ |\ W]) \\
&= \E[W^2\Cov(Z_1^2,Z_2^2)] + \Cov(\E[Z_1^2]W,\E[Z_2^2]W)\\
&=\E[W^2]\Cov(Z_1^2,Z_2^2) + \Var(W)\Var(Z_1)\Var(Z_2)\\
&= 2\rho^2\frac{\E[W^2]}{\E[W]^2}+\frac{\Var(W)}{\E[W]^2}.

\end{align*}

Therefore,~\eqref{eq:covariance:formula:asymptotic:variance} yields
\begin{align*}
\sigma_{G_{\mu^\ast}}^2(C)&=\Var(XY)=\Cov(X^2,Y^2)+1-\Cov(X,Y)^2
=2\rho^2\frac{\E[W^2]}{\E[W]^2}+\frac{\Var(W)}{\E[W]^2}+ 1 - \rho^2,
\end{align*}
and thus the desired result follows.
\end{proof}

By~\eqref{eq:optimal:asymptotic:variance:normal:variance:mixture}, the function $\rho \mapsto \sigma_{G_{\mu_\ast}}^2(C)$ is convex since $\E[W^2]\geq \E[W]^2$.
For normal distributions, $W=1$ and thus $\sigma_{G_{\mu_\ast}}^2(C)=\rho^2(X,Y)+1$.
Finally, as we will see in Section~\ref{sec:simulation:study}, the variance $\Var(X^2)$, $X\sim G$, is observed to affect convexity or concavity of $\rho \mapsto \sigma_{G_{\mu_\ast}}^2(C)$.
For the case of normal variance mixtures,

we have that
\begin{align}
\label{eq:optimal:asymptotic:variance:normal:variance:mixture:variance:X:squared}
\sigma_{G_{\mu^\ast}}^2(C)= \left(\Var(X^2)-\frac{\E[W^2]}{\E[W]^2}\right) \rho^2(X,Y) + \frac{\E[W^2]}{\E[W]^2},
\end{align}
since
\begin{align*}
\Var(X^2)&=\E[W^2Z_1^4]-(\E[WZ_1^2])^2
=\E[W^2]\left(\frac{3}{\E[W]^2}\right)-\E[W]^2\left(\frac{1}{\E[W]^2}\right)
=3\frac{\E[W^2]}{\E[W]^2}-1.
\end{align*}

\section{Optimal concordance-inducing distributions}
\label{sec:optimal:concordance:inducing:distributions}

In this section we investigate optimal best and worst asymptotic variances and their attaining concordance-inducing distributions for certain choices of $\mathcal D\subseteq \mathcal C_2$.

\subsection{Asymptotic variance for fundamental and Fr\'echet copulas}\label{subsec:asymptotic:variance:fundamental:copulas}

We consider the case when $\mathcal D \subset \mathcal C_2$ is a set of fundamental copulas $M$, $\Pi$ and $W$, or their mixtures.
By radial symmetry of these copulas, the optimal shift $\mu_\ast$ is zero and thus it suffices to consider standardized concordance-inducing distributions in $\mathcal G_4$.

\begin{definition}[Fr\'echet copula]\label{def:frechet:copulas}
A bivariate \emph{Fr\'echet copula} is defined by
\begin{align*}
C_{\bp}^{\operatorname{F}} =  p_M M + p_{\Pi}\Pi + p_{W} W,\quad  \bp = (p_M,p_{\Pi},p_W)\in \Delta_3,
\end{align*}
where $\Delta_3=\{(p_1,p_2,p_3)\in \IR_{+}^3: p_1+p_2+p_3=1\}$ is the standard unit simplex on $\IR^3$.
The set of all Fr\'echet copulas is denoted by $\mathcal C^{\operatorname{F}}= \{C_{\bp}^{\operatorname{F}}: \bp \in \Delta_3\}$.
\end{definition}

In addition to their financial applications, Fr\'echet copulas can be used to approximate bivariate copulas; see \citet{yang2006bivariate}.
Moreover, for any $G \in \mathcal G_4$, the transformed rank correlation $\kappa_G$ can take any value in $[-1,1]$ since, by Proposition~\ref{prop:properties:g:transformed:correlations} Part~\ref{item:linearity:kappa:g}, it holds that
\begin{align}\label{eq:kappa:g:Frechet:copulas}
\kappa_G(C_{\bp}^{\operatorname{F}})=p_M \kappa_G(M) + p_{\Pi}\kappa_G(\Pi) + p_{W} \kappa_G(W)=p_M-p_W\in [-1,1].
\end{align}

The following proposition is an immediate consequence from Proposition~\ref{prop:basic:properties:asymptotic:variance} Part~\ref{item:fundamental:copulas:asymptotic:variance}.

\begin{proposition}[Optimal asymptotic variances for fundamental copulas]\label{prop:optimal:g:asymptotic:variance:fundamental:copulas}

Let $\mathcal H\subseteq \mathcal G_4$.
\begin{enumerate}
  \item\label{item:optimal:G:independence}

  $\underline\sigma_{\ast}^2(\mathcal H,\{\Pi\})=\overline\sigma_{\ast}^2(\mathcal H,\{\Pi\})=1$
   and
   $\underline G_\ast(\mathcal H,\{\Pi\})=\overline G_\ast(\mathcal H,\{\Pi\})=\mathcal H$.
 \item\label{item:optimal:G:extremal:dependence}
   Suppose $\mathcal D=\{M\}$, $\{W\}$ or $\{M,W\}$. Then
  \begin{align*}
      \underline \sigma_\ast^2(\mathcal H,\mathcal D)=\overline \sigma_\ast^2(\mathcal H,\mathcal D)=\inf_{G \in \mathcal H}\Var_G(X^2)\quad \text{and}\quad
       \underline G_\ast(\mathcal H,\mathcal D)=\overline G_\ast(\mathcal H,\mathcal D)=\arginf_{G \in \mathcal H}\Var_G(X^2).
  \end{align*}
  \item\label{item:optimal:G:fundamental:copulas}

$\underline \sigma_\ast^2(\mathcal H,\{M,\Pi,W\})= 1 \wedge \inf_{G \in \mathcal H}\Var_G(X^2)$ and
$\overline \sigma_\ast^2(\mathcal H,\{M,\Pi,W\})=1 \vee \inf_{G \in \mathcal H}\Var_G(X^2)$.
Moreover,

\begin{align*}

\underline G_{\ast}(\mathcal H, \{M,\Pi,W\})&=
\begin{cases}
\arginf_{G \in \mathcal H}\Var_G(X^2), &  \text{ if } \inf_{G \in \mathcal H}\Var_G(X^2)<1,\\
\mathcal H, & \text{ if }\inf_{G \in \mathcal H}\Var_G(X^2)\geq 1,
\end{cases}\\
\overline G_{\ast}(\mathcal H, \{M,\Pi,W\})&=
\begin{cases}
\mathcal H, &  \text{ if } \inf_{G \in \mathcal H}\Var_G(X^2)<1,\\
\arginf_{G \in \mathcal H}\Var_G(X^2), & \text{ if }\inf_{G \in \mathcal H}\Var_G(X^2)\geq 1.
\end{cases}
\end{align*}
\end{enumerate}
\end{proposition}

The next proposition provides the best and worst asymptotic variances and their attainers when $\mathcal D=\mathcal C^{\operatorname{F}}$.

\begin{proposition}[Best and worst asymptotic variances for Fr\'echet copulas]\label{prop:worst:best:asymptotic:variances:Frechet:copulas}
For a concordance-inducing distribution $G \in \mathcal G_4$, the best and worst asymptotic variances on $\mathcal C^{\operatorname{F}}$ are given by
\begin{align*}
\underline \sigma_{G}^2(\mathcal C^{\operatorname{F}})=1 \wedge \Var_G(X^2)
\quad \text{and}\quad
\overline \sigma_{G}^2(\mathcal C^{\operatorname{F}})=1+\Var_G(X^2)
\end{align*}
with the sets of attaining copulas given by

\begin{align*}
\underline C_G(\mathcal \mathcal C^{\operatorname{F}})

&=
\begin{cases}
\{M, W\}, & \text{ if } 0\leq \Var_G(X^2)<1,\\
\{M, \Pi,W\},& \text{ if }\Var_G(X^2)=1,\\
\{\Pi\},& \text{ if }\Var_G(X^2)>1,
\end{cases}\\
\overline C_G(\mathcal C^{\operatorname{F}})

&=
\begin{cases}
\left\{ \frac{M+W}{2} \right\}, & \text{ if } \Var_G(X^2)>0,\\
\left\{p \frac{M+W}{2} + (1-p)\Pi: p \in [0,1] \right\}, & \text{ if } \Var_G(X^2)=0.
\end{cases}
\end{align*}
\end{proposition}

\begin{proof}
Fix $G \in \mathcal G_4$ and $C_{\bp}^{\operatorname{F}} \in \mathcal C^{\operatorname{F}}$ with $\bp=(p_M,p_{\Pi},p_W) \in \Delta_3$.

For $X=G^\i(U)$ and $Y=G^\i(V)$ with $(U,V)\sim C_{\bp}^{\operatorname{F}}$, we have that $\Cov(X^2,Y^2)=(p_M + p_W)\Var_G(X^2)$ and that $\Cov(X,Y)=p_M-p_W$.
Therefore, by~\eqref{eq:covariance:formula:asymptotic:variance}, it holds that
\begin{align*}
\sigma_G^2(C_{\bp}^{\operatorname{F}})=(p_M + p_W)v + 1 - (p_M-p_W)^2 =:f(p_M,p_W),
\end{align*}
where $v=\Var_G(X^2)$.

Since the Hessian of $f$
\begin{align*}
H(p_M,p_W)=\begin{pmatrix}
\frac{\partial}{\partial p_M^2}f(p_M,p_W)& \frac{\partial}{\partial p_M p_W}f(p_M,p_W)
\\
\frac{\partial}{\partial p_W p_M}f(p_M,p_W) & \frac{\partial}{\partial p_W^2}f(p_M,p_W)\\
\end{pmatrix}
= \begin{pmatrix}
-2& 2\\
2& -2\\
\end{pmatrix},
\end{align*}
is nonpositive definite, $f$ is a concave function.
For $(p_M,p_W) \in \IR^2$ such that $0\leq p_M,p_W$ and $p_M+p_W\leq 1$, consider the reparametrization $(p,0) + r(-1,1)=(p-r,r)$ where $0\leq r \leq p \leq 1$.
Then
\begin{align*}
f(p-r,r)=p v +1-(p-r)^2-r^2+2(p-r)r=-4\left(r-\frac{p}{2}\right)^2 + p v + 1,
\end{align*}
and thus $f$ represents a parabolic cylinder.

For a fixed $p \in [0,1]$, the function $r \mapsto f(p-r,r)$ has a maximum $\overline f(p)=p v+1$ when $r=p/2$, and a minimum $\underline f(p)=-p^2+p v+1$ when $r=0$ or $r=p$.
Since $v\geq 0$, the maximum of $f$ is given by $v+1$ with the maximum attained by $p=1$ when $v>0$, and by any $p\in [0,1]$ when $v=0$.
Therefore, we have that $\overline \sigma_G^2(\mathcal C^{\operatorname{F}})=v+1 = \sigma_G^2(C)$ with $C=\frac{M+W}{2}$ when $v>0$, and with $C=p \left(\frac{M+W}{2}\right) + (1-p)\Pi$ for any $p \in [0,1]$ when $v=0$.

For the minimum of $f$, notice that the function $\underline f(p)=-p^2+p v+1$, $0\leq p \leq 1$, is a concave parabola, and thus the minimum of $\underline f(p)$ is attained at $p=0$ or $p=1$.
With $\underline f(0)=1$ and $\underline f(1)=v$, the minimum of $f$ and its attainers are given by $\underline \sigma_G^2(\mathcal C^{\operatorname{F}})=1\wedge v = \sigma_G^2(C)$ with $C=M$ or $W$ when $0\leq v < 1$, with $C=M$, $\Pi$ or $W$ when $v=1$ and with $C=\Pi$ when $v>1$.

\end{proof}

Note that the result in Proposition~\ref{prop:worst:best:asymptotic:variances:Frechet:copulas} is consistent with Corollary~\ref{cor:min:lower:asymptotic:variance:extremal:points}
since $\text{ext}(\mathcal D^{\text{F}})=\{M,\Pi,W\}$.
In the proof of Proposition~\ref{prop:worst:best:asymptotic:variances:Frechet:copulas}, although $(p_M,p_W)=(1/2,1/2)$ is the unique point attaining the maximum $v+1$ of $f$ when $v>0$, $f$ takes the value $v$ at the points $(p_M,p_W)=(1,0)$ and $(0,1)$, and is greater than $v$ on $\{(p_M,p_W) \in [0,1]^2: p_M+p_W=1\}$.
Therefore, if $v=\Var_G(X^2)$ is sufficiently large, the asymptotic variance $\sigma_G^2(C)$

takes large values in $[\Var_G(X^2),\Var_G(X^2)+1]$ if $C=p M + (1-p)W$ for $p \in [0,1]$.

\begin{remark}[Restrictions of $\mathcal C^{\operatorname{F}}$]
For a concordance-inducing distribution $G \in \mathcal G_4$, consider the set of Fr\'echet copulas such that its transformed rank correlation $\kappa_G$ takes values in $[\underline k,\overline k]$ for $-1 \leq \underline k \leq \overline k \leq 1$, that is,
\begin{align*}
\mathcal C_{\underline k,\overline k}^{\operatorname{F}}(G)=\{C \in \mathcal C^{\operatorname{F}}: \underline k \leq \kappa_G(C) \leq \overline k\}.
\end{align*}
By~\eqref{eq:kappa:g:Frechet:copulas}, the restriction $\underline k \leq \kappa_G(C) \leq \overline k$ reduces to $\underline k \leq p_M-p_W \leq \overline k$ and thus $\mathcal C_{\underline k,\overline k}^{\operatorname{F}}(G)$ does not depend on the choice of $G$.
Consequently, the maximum and minimum of the asymptotic variance $\sigma_G^2(C)$ on $\mathcal C_{\underline k,\overline k}^{\operatorname{F}}(G)$ can be found by calculating $\max f(p_M,p_W)$ and $\min f(p_M,p_W)$ subject to $0\leq p_M,p_W$, $p_M+p_W\leq 1$ and $ \underline k \leq p_M-p_W \leq \overline k$.
This maximum and minimum always exist since $(p_M,p_W)\mapsto f(p_M,p_W)$ is bounded, concave and the feasible set is compact in $\IR^2$.
\end{remark}

Proposition~\ref{prop:worst:best:asymptotic:variances:Frechet:copulas} immediately leads to the optimal best and worst asymptotic variances on $\mathcal D=\mathcal C^{\operatorname{F}}$ by the following corollary.

\begin{corollary}[Optimal best and worst asymptotic variances for Fr\'echet copulas]\label{cor:optimal:asymptotic:variances:Frechet:copulas}
For $\mathcal H \subseteq \mathcal G_4$, the optimal best and worst asymptotic variances are given by
\begin{align*}
\underline \sigma_{\ast}^2(\mathcal H, \mathcal C^{\operatorname{F}})=1 \wedge \inf_{G \in \mathcal H}\Var_G(X^2)
\quad\text{and}\quad
\overline \sigma_{\ast}^2(\mathcal H, \mathcal C^{\operatorname{F}})=1+\inf_{G \in \mathcal H}\Var_G(X^2),
\end{align*}
with the sets of attaining concordance-inducing distributions given, respectively, by
\begin{align*}
\underline G_{\ast}(\mathcal H, \mathcal C^{\operatorname{F}})&=
\begin{cases}
\arginf_{G \in \mathcal H}\Var_G(X^2), &  \text{ if } \inf_{G \in \mathcal H}\Var_G(X^2)<1,\\
\mathcal H, & \text{ if }\inf_{G \in \mathcal H}\Var_G(X^2)\geq 1,
\end{cases}\\
\overline G_{\ast}(\mathcal H, \mathcal C^{\operatorname{F}})&=\arginf_{G \in \mathcal H}\Var_G(X^2).
\end{align*}
\end{corollary}

Compared with the optimal best and worst asymptotic variances from Proposition~\ref{prop:optimal:g:asymptotic:variance:fundamental:copulas} Part~\ref{item:optimal:G:fundamental:copulas}, the lower bound $\underline \sigma_{\ast}^2(\mathcal H,\mathcal D)$ obtained in Proposition~\ref{cor:optimal:asymptotic:variances:Frechet:copulas} remains unchanged but the upper bound $\overline \sigma_{\ast}^2(\mathcal H,\mathcal D)$ increases since the attaining copulas $p\left(\frac{M+W}{2}\right)+(1-p)\Pi$, $p \in [0,1]$, are not included in the set $\mathcal D$ in Proposition~\ref{prop:optimal:g:asymptotic:variance:fundamental:copulas}.
Nevertheless, the best and worst asymptotic variances are the functions of $\Var_G(X^2)$ when $\mathcal D$ is a set of fundamental or Fr\'echet copulas, and thus we have the following result.

\begin{corollary}[Optimal concordance-inducing distributions on fundamental or Fr\'echet copulas]
\label{cor:optimal:concordance:inducing:functions:fundamental:frechet:copulas}
Suppose that $\mathcal D$ is a set of fundamental or Fr\'echet copulas, that is, $\mathcal D=\{\Pi\}$, $\{M\}$, $\{W\}$, $\{M,W\}$, $\{M,\Pi,W\}$ or $\mathcal C^{\text{F}}$. Then
\begin{enumerate}
\item\label{item:total:order:funndamental:frechet:copuas} $\leq_{\mathcal D}$ is a total order,
\item\label{item:preference:order:funndamental:frechet:copuas} $G'\leq_{\mathcal D} G$ if $\Var_{G}(X^2)\leq \Var_{G'}(X^2)$, and
\item\label{item:optimal:concordance:inducing:distributions:funndamental:frechet:copuas} $G_{\ast}(\mathcal H, \mathcal D)=\arginf_{G \in \mathcal H}\Var_G(X^2)$.
\end{enumerate}
\end{corollary}

Corollary~\ref{cor:optimal:concordance:inducing:functions:fundamental:frechet:copulas} states that concordance-inducing distributions having a smaller variance of $X^2$ for $X\sim G$ are more preferable in terms of best and worst asymptotic variances when $\mathcal D$ is a set of fundamental or Fr\'echet copulas.
As a consequence, heavy-tailed concordance-inducing distributions, such as a Student $t$ distribution with degrees of freedom $4 < \nu <\infty$, are not recommendable choices.
In particular, popular measures of concordance introduced in Example~\ref{ex:transformed:rank:correlations} are ordered as follows.

\begin{corollary}[Preference orders for $\rho_{\text{S}}$, $\beta$ and $\zeta$]
\label{cor:preference;orders:popular:mocs}
Suppose that $\mathcal D$ is a set of fundamental or Fr\'echet copulas.
Then $\zeta \leq_{\mathcal D} \rho_{\text{S}}\leq_{\mathcal D} \beta$.

\end{corollary}

\begin{proof}
For $G\in\mathcal G$, denote by $V(G)$ (or $V(\kappa_G)$) the variance of $X^2$ where $X$ follows the standardized distribution of $G$ so that $\E[X]=0$ and $\Var(X)=1$.
Then the concordance-inducing distributions of $\rho_{\text{S}}$, $\beta$ and $\zeta$ are the uniform distribution on $\left(-\sqrt{3},\sqrt{3}\right)$, the symmetric Bernoulli distribution on $\{-1,1\}$ and the standard normal distribution $N(0,1)$.
Since
\begin{align*}
V(\rho_{\text{S}})=V(\U(0,1))=0.8,\quad
V(\beta)=V(\Bern(1/2))=0\quad\text{and}\quad
V(\zeta)=V(\N(0,1))=2,
\end{align*}
the result follows from Corollary~\ref{cor:optimal:concordance:inducing:functions:fundamental:frechet:copulas} Part~\ref{item:preference:order:funndamental:frechet:copuas}.

\end{proof}

\subsection{Optimality of Blomqvist's beta}\label{subsec:optimality:Blomqvist}

In this section, we show that Blomqvist's beta is an optimal $G$-transformed rank correlation under some conditions on $\mathcal D \in \mathcal C_2$.
For $G \in \mathcal G_4$, we identify $G \in \mathcal G_4$ with $\kappa_G$, and thus we allow $\sigma_{G}^2$ to be written as $\sigma_\beta^2$ for the standardized symmetric Bernoulli distribution $G \in \mathcal G_4$.

\begin{definition}[Balancedness of copulas]\label{def:balanced:copulas}
Let $p(C)=C(1/2,1/2)+\bar C(1/2,1/2)$ for $C \in \mathcal C_2$.
A copula $C \in \mathcal C_2$ is called \emph{balanced} if $p(C)=1/2$,
\emph{imbalanced} if $p(C)\neq 1/2$, \emph{totally positively imbalanced (TPI)} if $p(C)=1$ and  \emph{totally negatively imbalanced (TNI)} if $p(C)=0$.
\end{definition}

It is straightforward to check that $\Pi$ and $(M+W)/2$ are balanced, $M$ is TPI and $W$ is TNI.

\begin{proposition}[Asymptotic variance of Blomqvist's beta]
\label{prop:asymptotic:variances:beta}
Let $C \in \mathcal C_2$.
Then the following properties hold for $\beta$.
\begin{enumerate}
\item\label{item:optimal:shift:asymptotic:variances:beta} $\mu_{\ast}(\beta,C)=0$.
\item\label{item:explicit:form:asymptotic:variances:beta} $\sigma_\beta^2(C)=4p(C)(1-p(C))=1-\beta^2(C)$.
\item\label{item:upper:lower:bounds:asymptotic:variances:beta} $0\leq \sigma_\beta^2(C)\leq 1$.
\item\label{item:lower:bound:asymptotic:variances:beta} $\sigma_\beta^2(C)=0$ if and only if $C$ is a TPI or TNI copula.
\item\label{item:upper:bound:asymptotic:variances:beta} $\sigma_\beta^2(C)=1$ if and only if $C$ is balanced.
\end{enumerate}
\end{proposition}

\begin{proof}
Let $G$ be the standardized symmetric Bernoulli distribution.

\noindent \ref{item:optimal:shift:asymptotic:variances:beta})
For $(X,Y)=(G^\i(U),G^\i(V))$ with $(U,V)\sim C$, it suffices to consider the case when $\Var(X+Y)>0$ by definition of $\mu_\ast$.

Since $G^\i(u)=2\bone_{\{u>1/2\}}-1$, $u \in (0,1)$, we have that
\begin{align*}
X+Y&=
\begin{cases}
2, & \text{ if }\  \{U >  1/2,\ V >  1/2\},\\
-2, & \text{ if }\ \{U \leq  1/2,\ V \leq  1/2\},\\
0, & \text{ if }\  \{U > 1/2,\ V \leq 1/2\}\cup \{U \leq  1/2,\ V >  1/2\},
\end{cases}\\
XY&=\begin{cases}
1, & \text{ if }\  \{U \leq 1/2,\ V \leq 1/2\}\cup \{U >  1/2,\ V >  1/2\},\\
-1, & \text{ if }\  \{U > 1/2,\ V \leq 1/2\}\cup \{U \leq  1/2,\ V >  1/2\}.
\end{cases}
\end{align*}
Since $C(1/2)=\bar C(1/2,1/2)$, we have that $\E[X+Y]=2\bar C(1/2,1/2)-2 C(1/2,1/2)=0$, and thus
\begin{align*}
\Cov(X+Y,XY)&=\E[(X+Y)XY] - \E[X+Y]\E[XY] = 2\bar C(1/2,1/2)-2 C(1/2,1/2)-0 = 0,
\end{align*}
which implies that $\mu_\ast(\beta,C)=0$.

\noindent \ref{item:explicit:form:asymptotic:variances:beta})
Using the notation
\begin{align*}
p(C)=\Prob\left(\{U \leq 1/2,\ V \leq  1/2\}\cup \{ U >  1/2,\ V >  1/2\}\right)=C(1/2,1/2)+\bar C(1/2,1/2)=2C(1/2,1/2),
\end{align*}
we have that $\beta(C)=4C(1/2,1/2)-1=2p(C)-1$ and that
\begin{align*}
\sigma_\beta^2(C)=\Var(XY)=4 p(C) (1-p(C))=1-\beta^2(C).
\end{align*}

\noindent \ref{item:upper:lower:bounds:asymptotic:variances:beta})
This immediately comes from \ref{item:explicit:form:asymptotic:variances:beta}) since $0 \leq p(C)\leq 1$.

\noindent \ref{item:lower:bound:asymptotic:variances:beta})
By \ref{item:explicit:form:asymptotic:variances:beta}), $\sigma_\beta^2(C)=0$ if and only if $p(C)=0$ or $1$, that is, $C$ is a TPI or TNI copula.

\noindent \ref{item:upper:bound:asymptotic:variances:beta})
By \ref{item:explicit:form:asymptotic:variances:beta}), $\sigma_\beta^2(C)=1$ if and only if $p(C)=1/2$, that is, $C$ is a balanced copula.

\end{proof}

\begin{remark}[Asymptotic variance of $\beta$ for elliptical copulas]
\label{remark:asymptotic:variances:Blomqvist:beta:elliptical}
Blomqvist's beta admits an explicit form $\beta(C)=\frac{2}{\pi}\arcsin (\rho)$ when $C$ is an \emph{elliptical copula} with correlation parameter $\rho\in[-1,1]$.
Therefore, by Proposition~\ref{prop:asymptotic:variances:beta} Part~\ref{item:explicit:form:asymptotic:variances:beta},
we have that $\sigma_{\beta}^2(C)=1-\left(\frac{2}{\pi}\arcsin (\rho)\right)^2$,
which coincides with the result derived in \citet[Proposition~9]{schmid2007nonparametric}.
Note that not all the asymptotic variances derived in \citet{schmid2007nonparametric} coincide with ours due to the location shift of the estimators of $\kappa_G$.
\end{remark}

Next we prove the optimality of Blomqvist's beta under certain conditions on $\mathcal D$.

\begin{proposition}[Optimality of Blomqvist's beta]\label{prop:optimality:blomqvist:beta}
Let $\mathcal D \subseteq \mathcal C_2$ and $\beta \in \mathcal H\subseteq \mathcal G_4$.
\begin{enumerate}
\item\label{item:optimality:blomqvist:beta:lower}
If $C^\ast \in \mathcal D$ for some TPI or TNI copula $C_\ast$, then $\beta \in \underline G_\ast(\mathcal H,\mathcal D)$.
\item\label{item:optimality:blomqvist:beta:upper}
If $\Pi \in \mathcal D$, then $\beta \in \overline G_\ast(\mathcal H,\mathcal D)$.
\item\label{item:optimality:blomqvist:beta}
If $\Pi,\ C_\ast \in \mathcal D$ for some TPI or TNI copula $C_\ast$, then $\beta \in G_\ast(\mathcal H,\mathcal D)$.
\end{enumerate}
Statements \ref{item:optimality:blomqvist:beta:lower},~\ref{item:optimality:blomqvist:beta:upper} and~\ref{item:optimality:blomqvist:beta} remain valid if $\sigma_G^2(C)$, $G \in \mathcal G_4$, in Definitions~\ref{Def:optimal:best:worst:asymptotic:variances} and~\ref{def:preference:optimal:concordance:inducing:distribution} is replaced by the optimally shifted asymptotic variance $\sigma_{G_{\mu_{\ast}}}^2(C)$.

\end{proposition}

\begin{proof}

\noindent \ref{item:optimality:blomqvist:beta:lower})
By Proposition~\ref{prop:asymptotic:variances:beta} Part~\ref{item:lower:bound:asymptotic:variances:beta}, we have that
\begin{align}\label{eq:key:inequalities:optimality:blomqvist:beta:lower}
\underline \sigma_\beta^2(\mathcal D)=\sigma_\beta^2(C_\ast)=0 \leq \underline \sigma_G^2(\mathcal D)\quad \text{for all } G \in \mathcal G_4,
\end{align}
and thus $\underline \sigma_\ast^2(\mathcal H,\mathcal D)=0$ and $\beta \in \underline G_\ast(\mathcal H,\mathcal D)$.

\noindent \ref{item:optimality:blomqvist:beta:upper})
By Proposition~\ref{prop:basic:properties:asymptotic:variance} Part~\ref{item:fundamental:copulas:asymptotic:variance} and Proposition~\ref{prop:asymptotic:variances:beta} Parts~\ref{item:upper:lower:bounds:asymptotic:variances:beta} and~\ref{item:upper:bound:asymptotic:variances:beta}, we have that
\begin{align}\label{eq:key:inequalities:optimality:blomqvist:beta:upper}
\overline \sigma_\beta^2(\mathcal D)=\sigma_\beta^2(\Pi)=1 = \sigma_G^2(\Pi) \leq \overline \sigma_G^2(\mathcal D)\quad \text{for all } G \in \mathcal G_4,
\end{align}
and thus $\overline \sigma_\ast^2(\mathcal H,\mathcal D)=1$ and $\beta \in \overline G_\ast(\mathcal H,\mathcal D)$.

\noindent \ref{item:optimality:blomqvist:beta})
$\beta \in G_\ast(\mathcal H,\mathcal D)$ immediately follows from \ref{item:optimality:blomqvist:beta:lower}) and \ref{item:optimality:blomqvist:beta:upper}).

Finally, all the relations in~\eqref{eq:key:inequalities:optimality:blomqvist:beta:lower} and~\eqref{eq:key:inequalities:optimality:blomqvist:beta:upper} remain valid even if $\sigma_G^2(C)$, $G \in \mathcal G_4$, is replaced by the optimally shifted asymptotic variance $\sigma_{G_{\mu_{\ast}}}^2(C)$.
Therefore, all the optimality results in
\ref{item:optimality:blomqvist:beta:lower}),
\ref{item:optimality:blomqvist:beta:upper}) and
\ref{item:optimality:blomqvist:beta})
hold for correspondingly modified versions of Definitions~\ref{Def:optimal:best:worst:asymptotic:variances} and~\ref{def:preference:optimal:concordance:inducing:distribution}.

\end{proof}

Proposition~\ref{prop:optimality:blomqvist:beta} states that Blomqvist's beta is an optimal choice of $G$-transformed rank correlation for typical choices of $\mathcal D$, such as $\mathcal C_2$, $\mathcal C_2^{\succeq}= \{C \in \mathcal C_2: C \succeq \Pi\}$ or $\mathcal C_2^{\preceq}= \{C \in \mathcal C_2: C \preceq \Pi\}$.

\subsection{Uniqueness of the optimality of $\beta$}\label{subsec:uniqueness:optimality:beta}

In this section we investigate whether Blomqvist's beta is the unique optimal $G$-transformed rank correlation, that is, whether $G_{\ast}(\mathcal H,\mathcal D)=\{\beta\}$.
The next proposition states that this uniqueness holds under some condition on $\mathcal D$.

\begin{proposition}[Uniqueness of $\beta$ for $\overline G_{\ast}(\mathcal H,\mathcal D)$ and $G_{\ast}(\mathcal H,\mathcal D)$]
\label{prop:uniqueness:beta:worst:G}
Let $\mathcal H \subseteq \mathcal G_4$ and $\mathcal D \subseteq \mathcal C_2$ be such that $\beta \in \mathcal H$, $\Pi \in \mathcal D$ and $\mathcal D$ contains a TPI or TNI copula.
If, in addition, $(M+W)/2 \in \mathcal D$, then $\overline G_{\ast}(\mathcal H,\mathcal D)=\{\beta\}$ and thus $G_{\ast}(\mathcal H,\mathcal D)=\{\beta\}$.

The statement remains valid if $\sigma_G^2(C)$, $G \in \mathcal G_4$, in Definitions~\ref{Def:optimal:best:worst:asymptotic:variances} and~\ref{def:preference:optimal:concordance:inducing:distribution} is replaced by the optimally shifted asymptotic variance $\sigma_{G_{\mu_{\ast}}}^2(C)$.
\end{proposition}

\begin{proof}
By Proposition~\ref{prop:basic:properties:asymptotic:variance} Part~\ref{item:bounds:asymptotic:variance}, we have that
$\overline \sigma_G^2(\mathcal D)=1+\Var_G(X^2)$ for all $G \in \mathcal H$.
This upper bound remains valid if $G \in \mathcal H$ is optimally shifted since $(M+W)/2$ is radially symmetric.
Therefore, regardless of whether the optimal shift is taken into account, the optimal worst asymptotic variance $\overline \sigma_{\ast}^2(\mathcal H,\mathcal D)= 1$ is attained if and only if $G \in \mathcal H$ satisfies $\Var_G(X^2)=0$, that is, $X\sim G$ is the standardized symmetric Bernoulli distribution.
Consequently, we have that $\overline G_{\ast}(\mathcal H,\mathcal D)=\{\beta\}$ and thus $G_{\ast}(\mathcal H,\mathcal D)=\{\beta\}$ as desired.

\end{proof}

Proposition~\ref{prop:uniqueness:beta:worst:G} does not address whether $\underline G_\ast(\mathcal H,\mathcal D)=\{\beta\}$.
This, however, is rarely the case as we will see in what follows.

For given $\mathcal H \subseteq \mathcal G_4$ and $\mathcal D \subseteq \mathcal C_2$, assume that $\beta \in \mathcal H$ and that $\mathcal D$ contains at least one TPI or TNI copula.
Then the following equivalence relations hold:

\begin{align}\label{eq:equivalent:conditions:attaining:lower:bound:asymptotic:variance:bernoulli}
\nonumber G \in \underline G_{\ast}(\mathcal H,\mathcal D) \quad &\Leftrightarrow \quad
\text{there exists } C \in \mathcal D \text{ such that } \sigma_G^2(C)=0\\
& \Leftrightarrow \quad G^\i(U)G^\i(V)\aseq a \text{ for some } a \in \IR \text{ and } (U,V)\sim C \in \mathcal D.
\end{align}
The next proposition provides necessary conditions on $a \in \IR$, $G \in \mathcal H$ and $C \in \mathcal D$ in~\eqref{eq:equivalent:conditions:attaining:lower:bound:asymptotic:variance:bernoulli}.

\begin{proposition}[Necessary conditions on  $G \in \underline G_{\ast}(\mathcal H,\mathcal D)$]\label{prop:necessary:conditions:g:attain:optimal:best:asymptotic:variance}
For $\mathcal H \subseteq \mathcal G_4$ and $\mathcal D \subseteq \mathcal C_2$, suppose that $\beta \in \mathcal H$ and that $\mathcal D$ contains at least one TPI or TNI copula.
If $G \in \underline G_{\ast}(\mathcal H,\mathcal D)$, then $C \in \mathcal D$ and $a \in \IR$ in~\eqref{eq:equivalent:conditions:attaining:lower:bound:asymptotic:variance:bernoulli} satisfy the following conditions.
\begin{enumerate}
\item\label{item:necessary:conditions:point:mass:case} If $\Prob(X=0)>0$ for $X\sim G$, then $a=0$ and $\Prob(X=0)\geq 1/2$.
\item\label{item:necessary:conditions:no:point:mass:case} If $\Prob(X=0)=0$, then $a \neq 0$ and the copula $C$ is either TPI or TNI with $0<a\leq 1$ if $C$ is TPI and $-1\leq a<0$ if $C$ is TNI.
Moreover, the distribution function $G_{+}(x)=2G(x)-1$, $x> 0$, satisfies
\begin{align}\label{eq:G:plus:identity}
\E_{G_{+}}[Z]\geq |a|^{1/2},\quad Z\sim G_{+}\quad \text{and}\quad
G_{+}(x)=1-G_{+}\left(\frac{|a|}{x}-\right),\quad x> 0.
\end{align}
In particular, it holds that $\Prob(Z>|a|^{1/2})=\Prob(Z<|a|^{1/2})$ for $Z \sim G_{+}$.
\end{enumerate}
\end{proposition}

\begin{proof}
For $G \in \mathcal H$ and $C \in \mathcal D$ in~\eqref{eq:equivalent:conditions:attaining:lower:bound:asymptotic:variance:bernoulli}, write $(X,Y)=(G^\i(U),G^\i(V))$.

\noindent \ref{item:necessary:conditions:point:mass:case})
In this case, we have that $\Prob(XY=0)>0$ and thus $a \in \IR$ in~\eqref{eq:equivalent:conditions:attaining:lower:bound:asymptotic:variance:bernoulli} necessarily has to be $a=0$.
If $XY\aseq 0$ holds, then $X\neq 0$ implies that $Y=0$.
Together with $X\deq Y$, we have that

$\Prob(X\neq 0)\leq \Prob(Y=0)=\Prob(X=0)$,

which leads to the condition $\Prob(X=0)\geq 1/2$.

\noindent \ref{item:necessary:conditions:no:point:mass:case})
In this case, we have that $\Prob(XY=0)=0$ since
\begin{align*}
XY  \begin{cases}
 >0, & \text{ if } \{U\leq 1/2,\ V \leq 1/2\}\cup \{U> 1/2,\ V > 1/2\},\\
 <0,  & \text{ if } \{U\leq 1/2,\ V > 1/2\}\cup \{U> 1/2,\ V \leq  1/2\}.\\
\end{cases}
\end{align*}
Therefore, $a\in \IR$ in~\eqref{eq:equivalent:conditions:attaining:lower:bound:asymptotic:variance:bernoulli} necessarily has to be unequal to $0$.
Since $\Prob(XY>0)=p(\{U\leq 1/2,\ V \leq 1/2\}\cup \{U> 1/2,\ V > 1/2\})=p(C)$ and $\Prob(XY<0)=p(\{U\leq 1/2,\ V > 1/2\}\cup \{U> 1/2,\ V \leq  1/2\})=1-p(C)$, the random variable $XY$ can never be almost surely a constant if $0<p(C)<1$.
Therefore, it holds that $p(C)=0$ or $1$, and thus $C$ is either TPI or TNI.

Assume that $C$ is TPI.
Then $a>0$ since $\Prob(XY>0)=1$.
By the TPI assumption of $C$, we have that
 \begin{align*}
 X_{+}&=X\ | \ \{U>1/2,V>1/2\}=X\ | \ \{U>1/2\} \sim G_{+},\\
 Y_{+}&=Y\ | \ \{U>1/2,V>1/2\}=Y\ | \ \{V>1/2\} \sim G_{+},\\
 X_{-}&=X\ | \ \{U\leq 1/2,V\leq 1/2\}=X\ | \ \{U\leq 1/2\} \sim G_{-},\\
 Y_{-}&=X\ | \ \{U\leq 1/2,V\leq 1/2\}=Y\ | \ \{V\leq 1/2\} \sim G_{-},
 \end{align*}
 where
\begin{align*}
G_{+}(x)=\begin{cases}
2G(x)-1, & \text{ if } x > 0,\\
0, & \text{ if } x \leq  0,\\
\end{cases}
\quad \text{and}\quad
G_{-}(x)=\begin{cases}
1, & \text{ if } x > 0,\\
2G(x), & \text{ if } x \leq  0.\\
\end{cases}
\end{align*}
In addition to the equalities $X_{+}\deq Y_{+}$ and $X_{-}\deq Y_{-}$, we have that $X_{+}\deq - X_{-}$ and $Y_{+}\deq - Y_{-}$ since
\begin{align*}
\Prob(-X_{-}\leq x)&= \Prob(X_{-}\geq -x)=1-G_{-}((-x)-)\\
&=\begin{cases}
1-1 = 0, & \text{ if } x < 0,\\
1-2G((-x)-)=1-2(1-G(x))=2G(x)-1,& \text{ if } x \geq 0,
\end{cases}\\&= G_{+}(x)
\end{align*}
by radial symmetry of $G$ and the condition that $\Prob(X=0)=0$.
 Moreover, since $XY\aseq a$, it holds that
 \begin{align*}
 X_{+}Y_{+}=XY\ | \ \{U>1/2,V>1/2\}\aseq a \quad\text{and}\quad
 X_{-}Y_{-}=XY\ | \ \{U\leq1/2,V\leq 1/2\}\aseq a.
 \end{align*}

Since $X_{+}Y_{+}\aseq a$ and $Y_{+}>0$ a.s., we have that $X_{+} \aseq a/Y_{+}$.
Therefore, Jensen's inequality implies that
\begin{align*}
\E[X_{+}]=\E\left[\frac{a}{Y_{+}}\right]=a \E\left[\frac{1}{Y_{+}}\right]\geq \frac{a}{\E[Y_{+}]}=\frac{a}{\E[X_{+}]},
\end{align*}
which yields the mean condition $\E[X_{+}]\geq \sqrt{a}$.

Since $X_{+}\deq - X_{-}$ and $\Var(X)=\E[X^2]=1$, we have that
\begin{align*}
1&=\E[X^2]=\Prob\left(U>\frac{1}{2}\right)\E\left[X^2\ \biggl|\ U>\frac{1}{2}\right] +\Prob\left(U\leq \frac{1}{2}\right)\E\left[X^2\ \biggl|\ U\leq \frac{1}{2}\right]
=\frac{1}{2}\E[X_{+}^2] + \frac{1}{2}\E[X_{-}^2] =\E[X_{+}^2],
\end{align*}
and thus $\E[X_{+}^2]=1$.
Using $X_{+}^2 \aseq (a/Y_{+})^2>0$ a.s.\ and Jensen's inequality, we have that
\begin{align*}
1=\E[X_{+}^2] =\E\left[\left(\frac{a}{Y_{+}}\right)^2\right]\geq \frac{a^2}{\E[Y_{+}^2]}=\frac{a^2}{\E[X_{+}^2]},
\end{align*}
which yields $-1\leq a\leq 1$.
Together with $a>0$, we have the inequalities $0<a\leq 1$.

Moreover, $X_{+} \aseq a/Y_{+}$ implies that
\begin{align*}
G_{+}(x)=\Prob(X_{+}\leq x)=\Prob\left(\frac{a}{Y_{+}}\leq x\right)=1-\Prob\left(Y_{+}<\frac{a}{x}\right)=1-G_{+}\left(\frac{a}{x}-\right),
\quad x>0,
\end{align*}
which leads to the identity~\eqref{eq:G:plus:identity}.
The symmetry $\Prob(Z>a^{1/2})=\Prob(Z<a^{1/2})$ for $Z \sim G_{+}$ is obtained as a special case by taking $x=\sqrt{a}>0$ in~\eqref{eq:G:plus:identity}.

Next assume that $C$ is TNI.
Then $a<0$ since $\Prob(XY<0)=1$.
By the TNI assumption, we have that
 \begin{align*}
 X_{+}&=X\ | \ \{U>1/2,V \leq 1/2\}=X\ | \ \{U>1/2\} \sim G_{+},\\
 Y_{+}&=Y\ | \ \{U\leq 1/2,V>1/2\}=Y\ | \ \{V>1/2\} \sim G_{+},\\
 X_{-}&=X\ | \ \{U\leq 1/2,V> 1/2\}=X\ | \ \{U\leq 1/2\} \sim G_{-},\\
 Y_{-}&=X\ | \ \{U > 1/2,V\leq 1/2\}=Y\ | \ \{V\leq 1/2\} \sim G_{-}.
 \end{align*}
 As in the TPI case, it holds that $X_{+}\deq Y_{+}$, $X_{-}\deq Y_{-}$, $X_{+}\deq - X_{+}$ and $Y_{+}\deq - Y_{+}$.
 Moreover, $XY\aseq a$ implies that
 \begin{align*}
 X_{+}Y_{-}=XY\ | \ \{U>1/2,V\leq 1/2\}\aseq a\quad\text{and}\quad
 X_{-}Y_{+}=XY\ | \ \{U\leq1/2,V >  1/2\}\aseq a.
 \end{align*}
 From these equalities, all the necessary conditions derived in the TPI case hold with $-Y_{-},\ -X_{-} \sim G_{+}$ and $-a>0$ since $X_{+}(-Y_{-})\aseq (-X_{-})Y_{+} \aseq  -a$.
\end{proof}

\begin{comment}
\begin{proposition}[Characterization of $\underline G_{\ast}(\mathcal G_4,\mathcal D)$]\label{prop:characterization:optimal:best:asymptotic:variance}
A given concordance-inducing distribution $G \in \mathcal G_4$ satisfies $G \in \underline G_{\ast}(\mathcal G_4,\mathcal D)$ with $\underline\sigma_{\ast}^2(\mathcal G_4,\mathcal D)=0$ if and only if either of the following conditions holds:
\begin{enumerate}[label=(C\arabic*)]
\item\label{item:condition:neither:tpi:tni:case} There exists $C \in \mathcal D$ which is neither TPI nor TNI such that $|G^\i(U)||G^\i(V)|\aseq0$ for $(U,V)\sim C$. In this case the distribution function $G$ is required to have a point mass $\Prob(X=0)\geq 1/2$ for $X \sim G$.
\item\label{item:condition:tpi:tni:case} There exists a TPI or TNI copula $C \in \mathcal D$ such that $|G^\i(U)||G^\i(V)|\aseq a$ for $(U,V)\sim C$ and some non-negative constant $a\geq 0$.
When $a=0$, the distribution function $G$ is required to have a point mass $\Prob(X=0)\geq 1/2$ for $X \sim G$.
When $a>0$, $G$ is required to satisfy $\Prob(X=0)=0$ and the identity
\begin{align*}
G_{+}(x)+G_{+}(a/x-)=1\quad \text{for all }x>0,
\end{align*}
where $G_{+}$ is the distribution function of $|X|$.
\end{enumerate}
\end{proposition}

\begin{proof}
For any concordance-inducing distribution $G \in \mathcal G_4$, it holds that $G(u)\geq 0$ if $u\geq 1/2$ and $G(u)\leq 0$ if $u\leq 1/2$ since $G$ is radially symmetric with mean zero.
Therefore, for any copula $C \in \mathcal C_2$, we have that

A given concordance-inducing distribution $G \in \mathcal G_4$ satisfies $G\in \underline G_\ast(\mathcal G_4,\mathcal D)$ with $\underline\sigma_{\ast}^2(\mathcal G_4,\mathcal D)=0$ if and only if $\underline \sigma_{G}^2(\mathcal D)=\underline\sigma_{\ast}^2(\mathcal G_4,\mathcal D)=0$, that is, there exists $C \in \mathcal D$ such that $\sigma_G^2(C)=\Var(X Y)=0$ for $X = G^\i(U)$ and $Y = G^\i(V)$ with $(U,V)\sim C$.
Equivalently, $X Y$ is almost surely a constant.
If Condition~\ref{item:condition:neither:tpi:tni:case} holds, then $0<p(C)<1$ and $XY\aseq0$, which implies that $\sigma_G^2(C)=0$.
If Condition~\ref{item:condition:tpi:tni:case} holds, then $p(C)=1$ or $0$.
In either case, $|X||Y|\aseq a$ for some $a\geq 0$, which implies that $\sigma_G^2(C)=0$.
Therefore, both of~\ref{item:condition:neither:tpi:tni:case} and~\ref{item:condition:tpi:tni:case} imply that $G\in \underline G_\ast(\mathcal G_4,\mathcal D)$ with $\underline\sigma_{\ast}^2(\mathcal G_4,\mathcal D)=0$.

We now show the converse.
Let $C \in \mathcal D$ be a copula such that $\sigma_G^2(C)=\Var(X Y)=0$ for $X = G^\i(U)$ and $Y = G^\i(V)$ with $(U,V)\sim C$.
The copula $C$ is either TPI, TNP or neither of the two.

If $C$ is neither TPI nor TNI, then $XY$ equals $|X||Y|$ and $-|X||Y|$ with positive probabilities.
Therefore, $XY$ is almost surely a constant if and only if $|X||Y|=-|X||Y|=0$ almost surely.
$|X|$ and $|Y|$ are not identically $0$ since $G$ is nondegenerate.
Therefore,  $G$ is required to have a positive point mass at $0$ so that $|X||Y|=0$.

Next, if $C$ is TPI or TNP, then $XY=|X||Y|$ or $XY\aseq-|X||Y|$.
Therefore, $XY$ is a constant if and only if $|X||Y|\aseq a$ for some $a \geq 0$.
If $a=0$, then $G$ is required to have a point mass $\Prob(X=0)\geq 1/2$ by the same discussion as in the previous case.
Suppose $a>0$. If $\Prob(X=0)>0$, then $\Prob(|X||Y|=0)>0$, which violates the identity $|X||Y|\aseq a>0$.
Therefore, $G$ is required not to have a positive point mass at $0$.
Since $|Y|\neq 0$ a.s., we have $|X|\aseq a/|Y|$.
Consequently,
\begin{align*}
G_{+}(x)=\Prob(|X|\leq x)=\Prob(a/|Y|\leq x)=1-\Prob(|Y|<a/x)=G_{+}(a/x-),
\end{align*}
which completes the proof.
\end{proof}
\end{comment}

By Proposition~\ref{prop:necessary:conditions:g:attain:optimal:best:asymptotic:variance},
not all $G \in \mathcal G_4$ and $C \in \mathcal C_2$ can attain the optimal best asymptotic variance $\sigma_G^2(C)=0$.
The following examples show non-Bernoulli concordance-inducing distributions attaining this lower bound.
Let $M(n,\{J_i\},\pi,w)$ denote a shuffle-of-$M$ with $n$ being the number of connected components in its support, $\{J_i\}=\{J_1,\dots,J_n\}$ being a finite partition of $[0,1]$ into $n$ closed subintervals, $\pi$ being a permutation of $\{1,\dots,n\}$ and $w:\{1,\dots,n\}\rightarrow \{-1,1\}^n$ being a function indicating whether the strip $J_i\times J_{\pi(i)}$ is flipped ($w(i)=1$) or not ($w(i)=-1$); see \citet[Section~3.2.3]{nelsen2006introduction}.

\begin{example}[Non-Bernoulli concordance-inducing distributions in $\underline G_\ast(\mathcal G_4,\mathcal D)$]\label{example:non:bernoulli:optimal:best:g:functions}
\hspace{2mm}

\begin{enumerate}
\item \emph{The case when $\Prob(X=0)>0$}: Let $X\sim G$ be an equally weighted mixture of $0$ and $\U(-\sqrt{6},\sqrt{6})$. Then $\E[X]=0$, $\Var(X)=1$ and $\E[X^4]<\infty$, and thus $G \in \mathcal G_4$.
This is Case~\ref{item:necessary:conditions:point:mass:case} of Proposition~\ref{prop:necessary:conditions:g:attain:optimal:best:asymptotic:variance} since $\Prob(X=0)=1/2$.
Consider $C_1=M(4,\cup_{i=1}^4[(i-1)/4,i/4],\{2,1,4,3\},\bone_4)$, $C_2=M(4,\cup_{i=1}^4[(i-1)/4,i/4],\{3,4,1,2\},\bone_4)$ and $C_3=M(4,\cup_{i=1}^4[(i-1)/4,i/4],\{2,4,1,3\},\bone_4)$, where $\bone_d=(1,\dots,1)\in\IR^d$.
Then $C_1$ is TPI, $C_2$ is TNI and $C_3$ is neither TPI nor TNI.
Moreover, they satisfy $\sigma_G^2(C_k)=0$, $k=1,2,3$, since $G^\i(U)G^\i(V)\aseq 0$ with $(U,V)\sim C_k$ for $k=1,2,3$.

\item \emph{The case when $\Prob(X=0)=0$}:
Let $X\sim G$ be a discrete uniform distribution on the four points $\{-a/b,-b,b,a/b\}$ where $a=1/\sqrt{2}$ and $b=\sqrt{1-\sqrt{2}/2}$ with $b \approx 0.541$ and $a/b \approx 1.307$.

Then it is straightforward to check that $G \in \mathcal G_4$.
Define $(X,Y)=(G^\i(U),G^\i(V))$ with $(U,V)\sim C_4= M(4,\cup_{i=1}^4[(i-1)/4,i/4],\{2,1,4,3\},-\bone_4)$.
Then $(X,Y)=(-a/b,-b)$, $(-b,-a/b)$, $(b,a/b)$ and $(a/b,b)$ are equiprobable, and thus  $\sigma_G^2(C_4)=0$ since $XY\aseq a$.

This case belongs to Case~\ref{item:necessary:conditions:no:point:mass:case} of Proposition~\ref{prop:necessary:conditions:g:attain:optimal:best:asymptotic:variance} since $C_4$ is TPI, $0< a \leq 1$ and $\E_{G_{+}}[Z]\approx 0.924 > 0.841 \approx \sqrt{a}$.
\end{enumerate}
\end{example}

\section{Comparison of $\kappa_G$ and Kendall's tau}\label{sec:comaprison:kendall:tau}

In Section~\ref{subsec:optimality:Blomqvist}, we showed that Blomqvist's beta is an optimal $G$-transformed rank correlation under some conditions on $\mathcal D\subseteq \mathcal C_2$.
In this section, we show that Kendall's tau, although it is not a transformed rank correlation, attains the same optimal best and worst asymptotic variances that Blomqvist's beta does.

Kendall's tau $\tau:\mathcal C_2 \rightarrow \IR$ is defined by
\begin{align}\label{eq:kendall:tau}
\tau(C)=4\int_{[0,1]^2}C(u,v)\,\rd C(u,v)-1,
\end{align}
and is a measure of concordance; see \citet{scarsini1984}.
Moreover, it is not a $G$-transformed rank correlation since $\tau$ is not linear with respect to a mixture of copulas; see \citet[Remark~2]{hofert2019compatibility}.
Since $\tau(C)=\rho(\bone_{\{U> \tilde U\}},\bone_{\{V> \tilde V\}})$ where $(U,V)\sim C$ and $(\tilde U,\tilde V)\sim C$ is an independent copy of $(U,V)$, $\tau$ can also be written as
\begin{align}\label{eq:kendall:representation:correlation}
\tau(C) = \rho(g(U,\tilde U),g(V,\tilde V)),\quad \text{where}\quad
g(l,m)=\begin{cases}
1 & \text{ if }l> m ,\\
-1&  \text{ if }l \leq  m,\\
\end{cases}
\end{align}
by invariance of $\rho$ under location-scale transforms.
According to~\eqref{eq:kendall:representation:correlation}, we consider the following estimator of $\tau(C)$ and its asymptotic variance.

\begin{definition}[Canonical estimator of $\tau$]
\label{def:canonical:estimator:tau}
For $C \in \mathcal C_2$, the \emph{canonical estimator} of $\tau$ is given by
\begin{align*}
\hat \tau=\hat \tau(C;n)=\frac{1}{n}\sum_{i=1}^n   g(U_i,\tilde U_i)g(V_i,\tilde V_i),\quad\text{where}\quad
(U_1,V_1),\dots,(U_n,V_n),
(\tilde U_1,\tilde V_1),\dots, (\tilde U_n,\tilde V_n) \iidsim C,\quad n\in \IN.
\end{align*}
Denote by $\sigma_\tau^2(C)=\Var(g(U,\tilde U)g(V,\tilde V))$ the asymptotic variance of $\hat \tau$ such that
\begin{align*}
\sqrt{n}\left\{\hat \tau - \tau(C)\right\} \darrow \N(0,\sigma_\tau^2(C)),
\quad \text{as}\ n \rightarrow \infty.
\end{align*}
\end{definition}

For $(U,V),\  (\tilde U,\tilde V)\iidsim C$, write $X=g(U,\tilde U)$ and $Y=g(V,\tilde V)$.
As discussed in Section~\ref{subsec:optimal:location:shift}, location shift of $X$ and $Y$ does not change $\tau(C)=\rho(X,Y)$ but may affect $\sigma_\tau^2(C)$.
Thus we define the optimal location shift as follows.

\begin{definition}[Optimal shift of $g$]
For $C\in \mathcal C_2$, the \emph{optimal (location) shift} of $g$ under $C$ is given by
\begin{align*}
\mu_{\ast} = \mu_{\ast}(\tau,C)=\begin{cases}
-\frac{\Cov(XY, X+Y)}{\Var(X+Y)}, & \text{ if } \Var(X+Y)>0,\\
0, & \text{ if } \Var(X+Y)=0.
\end{cases}
\end{align*}
where $X=g(U,\tilde U)$ and $Y=g(V,\tilde V)$ for $(U,V),\  (\tilde U,\tilde V)\iidsim C$.
\end{definition}

\begin{proposition}[Basic properties of $\sigma_\tau^2(C)$]
\label{prop:basic:properties:tau:asymptotic:variance}
Let $C \in \mathcal C_2$.
Then the following properties hold for $\tau$.

\begin{enumerate}
\item\label{item:optimal:shift:tau:asymptotic:variance}
$\mu_\ast(\tau,C)=0$.
\item\label{item:explicit:form:tau:asymptotic:variance}
$\sigma_{\tau}^2(C)=4p_\tau(C)(1-p_\tau(C))=1-\tau^2(C)$
where
$p_\tau(C)=2\int_{[0,1]^2}C(u,v)\,\rd C(u,v)$.
\item\label{item:bounds:kendall:asymptotic:variance}
$0\leq \sigma_\tau^2(C) \leq 1$.

\item\label{item:lower:bound:kendall:asymptotic:variance}
$\sigma_{\tau}^2(C)=0$ if and only if $\tau(C)=1$ or $-1$, that is, $C=M$ or $W$, respectively.

\item\label{item:upper:bound:kendall:asymptotic:variance}
$\sigma_{\tau}^2(C)=1$ if and only if $\tau(C)=0$,

which holds when $C=C_{\nu_1}$ or $C=C_{\nu_2}$.

\end{enumerate}
\end{proposition}

\begin{proof}
For $(U,V),\  (\tilde U,\tilde V)\iidsim C$, write $X=g(U,\tilde U)$ and $Y=g(V,\tilde V)$.

\noindent \ref{item:optimal:shift:tau:asymptotic:variance})
The statement holds when $\Var(X+Y)=0$.
Assume that $\Var(X+Y)>0$.
Then we have that
\begin{align*}
X+Y&=
\begin{cases}
2, & \text{ if }\  \{U >  \tilde U,\ V >  \tilde V\},\\
-2, & \text{ if }\ \{U \leq  \tilde U,\ V \leq  \tilde V\},\\
0, & \text{ if }\  \{U > \tilde U,\ V \leq \tilde V\}\cup \{U \leq  \tilde U,\ V >  \tilde V\},
\end{cases}\\
XY&=\begin{cases}
1, & \text{ if }\  \{U \leq \tilde U,\ V \leq \tilde V\}\cup \{U >  \tilde U,\ V >  \tilde V\},\\
-1, & \text{ if }\  \{U > \tilde U,\ V \leq \tilde V\}\cup \{U \leq  \tilde U,\ V >  \tilde V\}.
\end{cases}
\end{align*}
Then $\E[X+Y]=0$ and $\Cov(X+Y,XY)=0$ by calculation since
\begin{align*}
\Prob(U >  \tilde U,\ V >  \tilde V)=\Prob(U\leq  \tilde U,\ V \leq \tilde V)=\int_{[0,1]^2}C(u,v)\,\rd C(u,v).
\end{align*}
Therefore, we have that $\mu_\ast(\tau,C)=0$ as desired.

\noindent \ref{item:explicit:form:tau:asymptotic:variance})
By definition of Kendall's tau~\eqref{eq:kendall:tau}, we have that
\begin{align*}
\Prob(\{U\leq \tilde U,\ V\leq \tilde V\}\cup \{U> \tilde U,\ V> \tilde V\})

= 2\int_{[0,1]^2}C(u,v)\,\rd C(u,v)=p_\tau(C)=\frac{\tau(C)+1}{2},
\end{align*}
and thus $\sigma_{\tau}^2(C)=\Var(XY)=4p_\tau(C)(1-p_\tau(C))=1-\tau^2(C)$.

\noindent \ref{item:bounds:kendall:asymptotic:variance})
This is an immediate consequence from \ref{item:explicit:form:tau:asymptotic:variance}) and $0\leq p_{\tau}(C)\leq 1$.

\noindent \ref{item:lower:bound:kendall:asymptotic:variance})
$\sigma_{\tau}^2(C) = 0$ is attained if and only if $p_\tau(C)=1$ or $0$, that is, $\tau(C)=1$ or $-1$, respectively.

By \citet[Theorem~3]{embrechtsmcneilstraumann2002}, $\tau(C)=1$ or $-1$ if and only of $C=M$ or $W$, respectively.

\noindent \ref{item:upper:bound:kendall:asymptotic:variance})
$\sigma_{\tau}^2(C) = 1$ is attained if and only if $p_\tau(C)=1/2$, that is, $\tau(C)=0$.
When $C=C_{\nu_1}$ or $C=C_{\nu_2}$, the change of sign axiom of measures of concordance in Definition~\ref{def:axioms:MOC} implies that $\tau(C)=\tau(C_{\nu_1})=-\tau(C)$ or $\tau(C)=\tau(C_{\nu_2})=-\tau(C)$, either of which yields $\tau(C)=0$.

\end{proof}

\begin{remark}[Asymptotic variance of $\tau$ for elliptical copulas]\label{remark:explicit:forms:asymptotic:variances:Kendall:tau}
When $C$ is an elliptical copula with correlation parameter $\rho\in[-1,1]$, we have that $\tau(C)=\frac{2}{\pi}\arcsin (\rho)$; see \citet{hult2002multivariate}.
Therefore, by Proposition~\ref{prop:asymptotic:variance:kendall:tau} Part~\ref{item:explicit:form:tau:asymptotic:variance}, we have that
$\sigma_{\tau}^2(C)=1-\left(\frac{2}{\pi}\arcsin (\rho)\right)^2$, which also equals $\sigma_{\beta}^2(C)$ as derived in Remark~\ref{remark:asymptotic:variances:Blomqvist:beta:elliptical}.

\end{remark}

Similar to the case of $G$-transformed rank correlations, we consider the following best and worst asymptotic variances.

\begin{definition}[Optimal best and worst asymptotic variances for $\tau$]
For $\mathcal D \subseteq C_2$, the \emph{best and worst asymptotic variances} and their \emph{attaining copulas} on $\mathcal D$ are defined, respectively, by
\begin{align*}
\underline \sigma_{\tau}^2(\mathcal D)=\inf_{C \in \mathcal D}\sigma_{\tau}^2(C),
\quad
\overline \sigma_{\tau}^2(\mathcal D)=\sup_{C \in \mathcal D}\sigma_{\tau}^2(C),
\quad
\underline C_{\tau}(\mathcal D)=\arginf_{C \in \mathcal D}\sigma_{\tau}^2(C),
\quad
\overline C_{\tau}(\mathcal D)=\argsup_{C \in \mathcal D}\sigma_{\tau}^2(C).
\end{align*}
\end{definition}

Properties of the best and worst asymptotic variances of $\tau$ are summarized as follows.

\begin{proposition}[Best and worst asymptotic variances of Kendall's tau]\label{prop:asymptotic:variance:kendall:tau}\hspace{2mm}
Let $\mathcal D \subseteq C_2$ and $\beta \in \mathcal H \subseteq \mathcal G_4$.
\begin{enumerate}
\item\label{item:attain:lower:bound:kendall:asymptotic:variance}
$\underline \sigma_{\tau}^2(\mathcal D)=\underline\sigma_{\ast}^2(\mathcal H,\mathcal D)=0$ and $\underline C_{\tau}(\mathcal D)=\{M,W\}\cap \mathcal D$ if $M \in \mathcal D$ or $W \in \mathcal D$.

\item\label{item:attain:upper:bound:kendall:asymptotic:variance}
$\overline \sigma_{\tau}^2(\mathcal D)=\overline\sigma_{\ast}^2(\mathcal H,\mathcal D)=1$ and $\Pi \in \overline C_{\tau}(\mathcal D)$ if $\Pi \in \mathcal D$.

\item \label{itemattain:upper:bound:kendall:asymptotic:variance:Frechet:copulas}
$\overline \sigma_{\tau}^2(\mathcal C^{\text{F}})=1$ and
$\overline C_{\tau}(\mathcal C^{\text{F}})= \left\{ p \frac{M+W}{2} + (1-p)\Pi: p \in [0,1]\right\}$.
\end{enumerate}

\end{proposition}

\begin{proof}
\ref{item:attain:lower:bound:kendall:asymptotic:variance}) and \ref{item:attain:upper:bound:kendall:asymptotic:variance}) immediately follow from Proposition~\ref{prop:basic:properties:tau:asymptotic:variance} Parts~\ref{item:bounds:kendall:asymptotic:variance},~\ref{item:lower:bound:kendall:asymptotic:variance} and~\ref{item:upper:bound:kendall:asymptotic:variance}, and Proposition~\ref{prop:optimality:blomqvist:beta}.

\noindent \ref{itemattain:upper:bound:kendall:asymptotic:variance:Frechet:copulas})
By Proposition~\ref{prop:basic:properties:tau:asymptotic:variance} Part~\ref{item:upper:bound:kendall:asymptotic:variance}, it holds that $\sigma_\tau^2(C)=1$ for $C\in\mathcal C_2$ if and only if $\tau(C)=0$.

For a Fr\'echet copula, we have that $\tau(\mathcal C_{(p_M,p_{\Pi},p_W)}^{\text{F}})=(p_M-p_W)(p_M+p_W+2)/3$; see \citet[Example~5.3]{nelsen2006introduction}.
Therefore, $\tau(\mathcal C^{\text{F}})=0$ holds if and only if $p_M=p_W$, and thus the desired result follows.
\end{proof}

Proposition~\ref{prop:asymptotic:variance:kendall:tau}

states that Kendall's tau attains the optimal best and worst asymptotic variances of transformed rank correlations, which are also attained by Blomqvist's beta.

Taking into account the drawback of Blomqvist's beta that it depends only on the local value $C(1/2,1/2)$ of a copula $C$, Kendall's tau can be a good alternative of Blomqvist's beta in terms of best and worst asymptotic variances.

\begin{remark}[Comparability of $\kappa_G$ and $\tau$]
\label{remark:comparability:kappa:g:tau}

Since Representation~\eqref{eq:kendall:representation:correlation} of Kendall's tau in terms of Pearson's correlation coefficient depends on two independent copies $(U,V)\sim C$ and $(\tilde U,\tilde V)\sim C$, the canonical estimator $\hat \tau$ requires twice more samples from $C$ than $\hat \kappa$ does.
Therefore, if the estimators $\hat \tau$ and $\hat \kappa_G$ are compared based on their actual variances (instead of their asymptotic variances), then $\Var(\hat \tau)=\sigma_{\tau}^2(C)/n$ should be multiplied by $2$ to be compared with $\Var(\hat \kappa_G)=\sigma_G^2(C)/n$.
With this modification, Kendall's tau still attains the optimal best asymptotic variance $\underline\sigma_{\ast}^2(\mathcal G_4,\mathcal D)=0$ since $2 \underline \sigma_{\tau}^2(\mathcal D)=0$.

On the other hand, Kendall's tau fails to attain the optimal worst asymptotic variance $\overline\sigma_{\ast}^2(\mathcal G_4,\mathcal D)=1$ since
$2 \overline \sigma_{\tau}^2(\mathcal D)=2>1$.
\end{remark}

\begin{remark}[Alternative estimators of Kendall's tau]\label{remark:alternative:estimator:kendall:tau}
Instead of the canonical estimator $\hat \kappa$ in Definition~\ref{def:canonical:estimator:tau}, one could compare $\hat \kappa_G$ with other estimators of Kendall's tau, such as
\begin{align*}
{\hat \tau}^{\maltese}(C;n)= \frac{1}{n}\sum_{i=1}^n g(U_i,U_{i+1})g(V_i,V_{i+1}),\quad
(U_1,V_1),\dots,(U_{n+1},V_{n+1}) \iidsim C,
\end{align*}
by taking the required sample size into account.
For $X_i = g(U_i,U_{i+1})$ and $Y_i=g(V_i,V_{i+1})$, the two random variables $Z_i=X_iY_i$ and $Z_j=X_jY_j$ are independent whenever $|i-j|\geq 2$.
Therefore, $Z_1,Z_2,\dots,$ are weakly dependent, and thus the CLT holds with the asymptotic variance given by

\begin{align*}

\sigma_{\tau}^{2\maltese}(C)=\Var(X_1Y_1)+2\Cov(X_1Y_1,X_2Y_2).
\end{align*}

Since $n+1$ samples are required to construct the estimator $\hat \tau^{\maltese}$, the modification factor $(n+1)/n$ is asymptotically $1$ as $n \rightarrow \infty$, and thus $\sigma_\tau^{2\maltese}(C)$ can be directly compared with the asymptotic variance $\sigma_G^2(C)$ of $\hat \kappa_G$.

\begin{comment}One can show that the covariance term in~\eqref{eq:asymptotic:variance:kendall:sophisticated} equals zero when $C=M$ or $W$, and thus $\underline \sigma_{\tau}^{2\maltese}(\mathcal D)=0={\underline \sigma}_{\ast}^2(\mathcal G_4,\mathcal D)$ if $M \in \mathcal D$ or $W \in \mathcal D$.
Therefore, the conclusion that Kendall's tau attains the best asymptotic variance remains valid for the estimator ${\hat \tau}^{\maltese}$.
Although the upper bound ${\overline \sigma}_{\tau}^{2\maltese}(\mathcal D)$ is not known, $\hat \tau$ cannot be more preferable than $\hat \tau^{\maltese}$ since
\begin{align*}
\sigma_{\tau}^{2\maltese}(C)\leq \Var(g(U_1,U_2)g(V_1,V_2)) + \Var(g(U_1,U_2)g(V_1,V_2))\leq 1+1=2 = {\overline \sigma}^{2\star}_\tau(\mathcal D).
\end{align*}

\end{comment}
\end{remark}

\section{Simulation study}\label{sec:simulation:study}

In this section, we conduct a simulation study to compare the asymptotic variances $\sigma_G^2(C)$ for various copulas $C \in \mathcal C_2$ and concordance-inducing distributions $G\in \mathcal G_4$.
Not only Spearman's rho $\rho_{\text{S}}$, Blomqvist's beta $\beta$ and van der Waerden's coefficient $\zeta$,we also consider $G$-transformed rank correlations with $G$ given by a Student $t$ distribution $t(\nu)$ with $\nu=10$ degrees of freedom and a Beta distribution with shape parameters $(0.5,0.5)$; note that both are radially symmetric, have finite fourth moments, and thus belong to $\mathcal G_4$ after standardization (mean zero and variance one).
The Beta$(0.5,0.5)$ concordance-inducing distribution has a different shape from the others since it puts an increasing probability mass as locations farther away from the center.
Kendall's tau is also considered for comparison.
Besides standardized concordance-inducing distributions, we also consider optimally shifted ones as introduced in Section~\ref{subsec:optimal:location:shift}.
As underlying copulas, we consider Gaussian $C_{\rho}^{\text{Ga}}$, Student $t$ $C_{\rho,\nu}^{t}$ and Clayton copulas $C_{\theta}^{\text{Cl}}$ where $\rho \in [-1,1]$ is a correlation parameter, $\nu>0$ is a degrees of freedom and $\theta\geq -1$ is a shape parameter.
The experiment consists of the following three steps.

\begin{enumerate}\setlength{\itemsep}{-0mm}
\item\label{item:step:set:parameters:copulas} Set $\rho = -0.99 +1.98 k / 49$ for $k=0,1,\dots,49$, $\nu=5$ and $\theta = 2\rho/(1-\rho)$ (which yields $\tau(C_{\theta}^{\text{Cl}})=\rho$) in $C=C_{\rho}^{\text{Ga}}$, $C_{\rho,\nu}^{t}$ and $C_{\theta}^{\text{Cl}}$.
\item\label{item:step:simulate:sample:C} For each copula $C$ in Step~\ref{item:step:set:parameters:copulas}, simulate $(U_1,V_1),\dots,(U_n,V_n)\iidsim C$ with $n=10^5$.
\item\label{item:step:compute:estimates:asymptotic:variances} Based on the samples generated in Step~\ref{item:step:simulate:sample:C}, estimate $\sigma_G^2(C)$ and $\sigma_{\tau}^2(C)$ by the sample variances of $G^\i(U_i)G^\i(V_i)$, $i=1,\dots,n$, and of $g(U_i,U_{i+n/2})g(V_i,V_{i+n/2})$, $i=1,\dots,n/2$, where $G$ is a standardized, and optimally shifted uniform, Beta$(0.5,0.5)$, normal, $t(10)$ and symmetric Bernoulli distribution function, and $g$ is as defined in~\eqref{eq:kendall:representation:correlation}.
\end{enumerate}

\begin{figure}[h]
  \centering
  \vspace{0mm}
  \includegraphics[width=12 cm]{MainPlotAsymVar.pdf}
  \vspace{0mm}
  \caption{Estimates of asymptotic variances $\sigma_G^2(C)$ and $\sigma_{\tau}^2(C)$ against correlation parameters $\rho \in [-0.99,0.99]$ of $C=C_{\rho}^{\text{Ga}}$ (red), $C_{\rho,\nu}^{t}$ (blue) with $\nu=5$ and $C_{\theta}^{\text{Cl}}$ (green) with $\theta = 2\rho/(1-\rho)$ for $G$-transformed rank correlation coefficients $\kappa_G$ (all except bottom-right) and Kendall's tau $\tau$ (bottom-right).
  The concordance-inducing distribution $G$ is set to be standardized (solid lines) and optimally shifted (dashed lines) uniform, Beta$(0.5,0.5)$, normal, $t(10)$ and symmetric Bernoulli distribution.

  The black dotted lines represent $y=1$, $V(G)$ and $1+V(G)$ with $V(\beta)=0$, $V(\rho_{\text{S}})=0.8$, $V(\zeta)=2$, $V(t(10))=3$, $V(\text{Beta(0.5,0.5)})=0.5$ and $V(\tau)=0$.}
  \label{fig:asymptotic:variances}
\end{figure}

The estimates of $\sigma_G^2(C)$ and $\sigma_{\tau}^2(C)$ computed in Step~\ref{item:step:compute:estimates:asymptotic:variances} are plotted in Figure~\ref{fig:asymptotic:variances}.
Observations from these plots are summarized as follows.
Recall that, for $G\in\mathcal G$, we denote by $V(G)$ (or $V(\kappa_G)$) the variance of $X^2$ where $X$ follows the standarzied distribution of $G$ so that $\E[X]=0$ and $\Var(X)=1$.
In addition, we write $V(\tau)=\Var(g(U,\tilde U))$ for $U,\, \tilde U \iidsim \Unif(0,1)$.

\noindent
{\bf Symmetry}:
For all copulas $C$, the curves of $\sigma_G^2(C)$ and $\sigma_{\tau}^2(C)$ against the correlation parameter $\rho$ were almost symmetric around $\rho=0$.
For $C=C_{\rho}^{\text{Ga}}$ and $C_{\rho,\nu}^{t}$, the symmetry is a consequence from Proposition~\ref{prop:basic:properties:asymptotic:variance} Part~\ref{item:reflection:invariance:asymptotic:variance} since

$C_{-\rho}^{\text{Ga}}=\nu_1(C_{\rho}^{\text{Ga}})$ and $C_{-\rho,\nu}^{t}=(\nu_1(C_{\rho,\nu}^{t})$.

This argument does not apply to Clayton copulas, and thus the curves $\rho \mapsto \sigma_G^2(C_{2\rho/(1-\rho)}^{\text{Cl}})$ and $\rho \mapsto \sigma_{\tau}^2(C_{2\rho/(1-\rho)}^{\text{Cl}})$ are not precisely symmetric.

\noindent
{\bf Convexity and concavity}:
The curves of $\sigma_G^2(C)$ and $\sigma_{\tau}^2(C)$ are convex when $V(G)>1$ (which holds if $G$ is normal or $t(10)$), and concave when $V(G)<1$ (which holds if $G$ is Bernoulli, uniform and Beta$(0.5,0.5)$, and if Kendall's tau is considered).
This observation is consistent with~\eqref{eq:optimal:asymptotic:variance:normal:variance:mixture:variance:X:squared}, and with the asymptotic variances obtained in Remarks~\ref{remark:asymptotic:variances:Blomqvist:beta:elliptical} and~\ref{remark:explicit:forms:asymptotic:variances:Kendall:tau}.

\noindent
{\bf Best and worst asymptotic variances}:

For all cases of $C=C_{\rho}^{\text{Ga}}$, $C_{\rho,\nu}^{t}$ and $C_{\theta}^{\text{Cl}}$, the best and worst asymptotic variances were approximately $1 \wedge V(G)$ and $1\vee V(G)$, respectively.

When $V(G)>1$ (normal or $t(10)$), the lower bound $\sigma_G^2(C)=1$ was attained at $\rho=0$ and the upper bound $\sigma_G^2(C)=V(G)$ was attained at $\rho=\pm 1$.
When $V(G)<1$ (Bernoulli, uniform, Beta$(0.5,0.5)$ and Kendall), the lower bound $\sigma_G^2(C)=V(G)$ was attained at $\rho=\pm 1$ and the upper bound $\sigma_G^2(C)=1$ was attained at $\rho=0$.
Note that, for $C=C_{\rho,\nu}^{t}$, the curve was slightly above these lower and upper bounds at $\rho=0$ since $C_{0,\nu}^{t}\neq \Pi$.

Since we only consider specific classes of copulas, the global upper bound $1+V(G)$ (see Proposition~\ref{prop:basic:properties:asymptotic:variance} Part~\ref{item:bounds:asymptotic:variance}) was not attained except in the cases of Blomqvist's beta and Kendall's tau where $V(\beta)=V(\tau)=0$.

\noindent
{\bf Choice of $G$; normal or Student, and uniform or Beta}:
As seen for the best and worst asymptotic variances, the variance $V(G)$ is an important quantity determining the best and worst asymptotic variances.
As theoretically indicated, concordance-inducing distributions with smaller $V(G)$ are more preferable.
Therefore, the normal concordance-inducing distribution is more preferable than $t(10)$ since $V(\zeta)=2 < 3 = V(t(10))$.
In fact, for all copulas considered, $\zeta$ had a smaller asymptotic variance than $\kappa_{t(10)}$.

Interestingly, $\kappa_{\text{Beta}(0.5,0.5)}$ typically had smaller asymptotic variance than $\rho_{\text{S}}$ since $V(\text{Beta})=0.5 < 0.8 = V(\rho_{\text{S}})$.

\noindent
{\bf Blomqvist's beta and Kendall's tau}:
As indicated in Remark~\ref{remark:explicit:forms:asymptotic:variances:Kendall:tau}, the curves of $\sigma_\beta^2(C)$ and $\sigma_\tau^2(C)$ seem to coincide when $C=C_{\rho}^{\text{Ga}}$ and $C_{\rho,\nu}^{t}$.

On the other hand, $\sigma_{\beta}^2(C_{\theta}^{\text{Cl}})$ and $\sigma_{\tau}^2(C_{\theta}^{\text{Cl}})$ are in general different since
$\sigma_{\beta}^2(C_{\theta}^{\text{Cl}})=1-\beta^2(C_{\theta}^{\text{Cl}})$ and $\sigma_{\tau}^2(C_{\theta}^{\text{Cl}})=1-\tau^2(C_{\theta}^{\text{Cl}})$ by Proposition~\ref{prop:asymptotic:variances:beta} Part~\ref{item:explicit:form:asymptotic:variances:beta} and Proposition~\ref{prop:basic:properties:tau:asymptotic:variance} Part~\ref{item:explicit:form:tau:asymptotic:variance}, but $\beta(C_{\theta}^{\text{Cl}})=4(2^{\theta+1}-1)^{-1/\theta}-1$ and $\tau(C_{\theta}^{\text{Cl}})=\theta/(\theta+2)$.

\noindent
{\bf Strength and the model of dependence}:
Compared with the choice of concordance-inducing distribution, the strength of dependence $\rho$ and the choice of $C$ seem to be less influencial on the asymptotic variance $\sigma_G^2(C)$.
Furthermore, for any concordance-inducing distribution, the difference of $\sigma_G^2(C)$ among different copulas $C=C_{\rho}^{\text{Ga}}$, $C_{\rho,\nu}^{t}$ and $C_{\theta}^{\text{Cl}}$ was typically smaller than the difference of $\sigma_G^2(C)$ among different levels of dependence $\rho$.

\noindent
{\bf Effect of optimal shifts}:
As theoretically indicated in Proposition~\ref{prop:sufficient:condition:optimal:shift:zero}, Proposition~\ref{prop:asymptotic:variances:beta} Part~\ref{item:optimal:shift:asymptotic:variances:beta} and Proposition~\ref{prop:basic:properties:tau:asymptotic:variance} Part~\ref{item:optimal:shift:tau:asymptotic:variance},
the solid and dotted curves of asymptotic variances overlapped when $C=C_{\rho}^{\text{Ga}}$ or $C_{\rho,\nu}^{t}$, and when $\beta$ and $\tau$ are considered.
For other cases, the optimal shift reduced the asymptotic variance.
However, even when the copula is $C_{\theta}^{\text{Cl}}$, only a small reduction was observed by the optimal shift.

\section{Concluding remarks}\label{sec:concluding:remark}

To answer the question which measure of concordance is best to be used, we proposed to compare $G$-transformed rank correlations $\kappa_G$ in terms of their best and worst asymptotic variances $\sigma_G^2(C)$ on a set of copulas $\mathcal D$.
When $\mathcal D$ is a set of fundamental copulas or Fr\'echet copulas, we showed that the best and worst asymptotic variances are increasing functions of $\Var_G(X^2)$, $X\sim G$, which leads to the comparison criterion that concordance-inducing distributions $G$ having smaller $\Var_G(X^2)$ are more preferable.
Since $\Var_G(X^2)$ attains its minimum $\Var_G(X^2)=0$ if and only if $G$ is a standardized symmetric Bernoulli distribution,

we proved that Blomqvist's beta $\beta$ uniquely attains the optimal best and worst asymptotic variances among all transformed rank correlations under certain conditions on $\mathcal D$.

Considering the drawback of $\beta$ that it depends only on the local value $C(1/2,1/2)$ of a copula $C$, we also compared $\kappa_G$ with Kendall's tau $\tau$.
Based on the representation of $\tau$ in terms of Pearson's linear correlation coefficient, we found that $\tau$ also attains the optimal best and worst asymptotic variances that $\beta$ does, although $\tau$ is not a transformed rank correlation.

Since the estimator of $\tau$ requires twice more samples than that of $\kappa_G$ does, the optimality of $\tau$ is violated if this required sample size is taken into account.

In a simulation study, we investigated the curve of the asymptotic variance of $\kappa_G$ and $\tau$ against the strength of dependence of the underlying copula.
We observed that the curve of $\sigma_G^2(C)$ was typically symmetric and convex or concave with the best and worst asymptotic variances give by $1\wedge \Var_G(X^2)$ and $1\vee \Var_G(X^2)$ depending on $\Var_G(X^2)>1$ or $\Var_G(X^2)<1$.

These observations support the criterion that concordance-inducing distributions $G$ with smaller $\Var_G(X^2)$ are more preferable.
Consequently, heavy-tailed concordance-inducing distributions, such as Student $t$ distributions with small degrees of freedom, are not recommended in comparison to the normal distribution, which leads to van der Waerden's coefficient $\zeta$.
In addition, we found that Beta distribution-based transformed rank correlations can be good alternatives to Spearman's rho $\rho_{\text{S}}$.

Other than Kendall's tau, there are still important measures of concordance which are not included in the class of transformed rank correlations, such as Gini's gamma.
Studying a broader framework of comparing these measures of concordance is part of future research.
Further investigation is also required for parametric classes of concordance-inducing distributions such as Beta$(\alpha,\beta)$ with $\alpha=\beta>0$.
Given the limitations of Fr\'echet copulas in practice,

another direction of future work is to investgate optimal concordance-inducing distributions under more practical choices of sets of underlying copulas, such as a set of parametric copulas or a ball of copulas around a given reference copula.
A comparison of multivariate measures of concordance, and of matrices of pairwise bivariate measures of concordance are also interesting directions for future research.
Finally, it is of interest whether and how the results in this paper change if
all measures of concordance are compared in terms of their asymptotic variance without assuming that marginal distributions are known, and thus only pseudo-samples from the underlying copulas are available.

\section*{Acknowledgments}
The authors thank Alexander Schied and Ruodu Wang at University of Waterloo for their constructive comments.

\appendix

\begin{comment}
\section{Proof of Theorem~\ref{thm:characterization:correlation:based:mocs}}\label{app:proof:theorem}

In this section we prove Theorem~\ref{thm:characterization:correlation:based:mocs}.
We first show the following lemma.

\begin{lemma}[Discontinuity points of left-continuous functions]\label{lemma:discontinuity:points:left:cont:functions}
Let $g:(0,1)\rightarrow \IR$ be a left-continuous function.
Then $g$ has at most countably many discontinuity points.
\end{lemma}

\begin{proof}
For a left-continuous function $g: (0,1)\rightarrow \IR$, denote by $\mathfrak D(g)$ the set of discontinuity points.
For $n \in \IN$, let
\begin{align*}
\mathfrak D_n(g)=\left\{u \in (0,1): \max\left(g(u)-\liminf_{v \downarrow u}g(v),\ \limsup_{v \downarrow u}g(v) - g(u)\right) > \frac{1}{n}\right\}.
\end{align*}
Then $\mathfrak D(g) = \cup_{n \in \IN} \mathfrak D_n(g)$.
Suppose, by way of contradiction, that $\mathfrak D(g)$ is uncountable.
Then there exists $n_0 \in \IN$ such that $\mathfrak D_{n_0}(g)$ is uncountable.
Define
\begin{align*}
\mathfrak D_{n_0}'(g)=\{u \in (0,1): \text{ there exists $\epsilon>0$ such that } (u-\epsilon,u)\cap \mathfrak D_{n_0}(g) = \emptyset\}.
\end{align*}
Then $\mathfrak D_{n_0}'(g)$ is at most countable since each $u \in \mathfrak D_{n_0}'(g)$ can be associated with rational numbers in the open interval $(u-\epsilon,u)$.
Since $\mathfrak D_{n_0}(g)$ is uncountable, there exists $u^{\ast} \in \mathfrak D_{n_0}(g)\backslash \mathfrak D_{n_0}'(g)$.
Since $u^{\ast} \notin \mathfrak D_{n_0}'(g)$, it holds that $(u^{\ast} -\epsilon,u^{\ast})\cap \mathfrak D_{n_0}(g) \neq \emptyset$ for all $\epsilon>0$.
This contradicts the left-continuity of $g$ since, for any increasing sequence $\{u_m^{\ast}\}_{m \in \IN}\subset (0,1)$ such that $u_m^{\ast} \uparrow u^{\ast} \in \mathfrak D_{n_0}(g)$, it holds that $|g(u^\ast)-g(u_m^{\ast})|>1/n_0$ for sufficiently large $m \in \IN$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:characterization:correlation:based:mocs}]
A map of Form~\eqref{eq:def:g:transformed:rank:correlation} is a measure of concordance by \citet[Theorem~1]{hofert2019compatibility}.
We show the converse, that is, if  $\kappa_{g_1,g_2}$ is a measure of concordance, it can be of Form~\eqref{eq:def:g:transformed:rank:correlation}.
By \citet[Proposition~1]{hofert2019compatibility}, $g_1$ and $g_2$ satisfy
\begin{align}\label{eq:monotonicity:g1:g2}
(g_1(u')-g_1(u))(g_2(v')-g_2(v))\geq 0
\end{align}
for any continuity points $0<u<u' < 1$ of $g_1$ and $0<v<v'<1$ of $g_2$.
By Lemma~\ref{lemma:discontinuity:points:left:cont:functions}, discontinuity points of $g_1$ and $g_2$ are at most countable, and thus~\eqref{eq:monotonicity:g1:g2} also holds for countably many discontinuity points of $g_1$ and $g_2$ by considering increasing and decreasing sequences of continuity points of $g_1$ and $g_2$ approaching their discontinuity points.
Therefore, $g_1$ and $g_2$ are monotone in the sense that $(g_1(u')-g_1(u))(g_2(v')-g_2(v))\geq 0
$ for any $0<u<u' < 1$ and $0<v<v'<1$.

\TODO{The following statements on constructing $G_1$ and $G_2$ should be correct but can be stated more clearly. The problems are only at the flat parts and jump parts of $g$.
Flat parts of $g$ leads to the jump of $G$. By right-continuity of $G$, this part is uniquely determined.
On this part, $G^\i$ is flat, and thus $g=G^\i$ there.
Jump part (at most countably many) of $g$ leads to the flat part of $G$ where no probability mass is assigned on this interval.
Therefore, $G^\i$ also has a jump there, but by left-continuity of $G^\i$, this part is uniquely determined and $g=G^\i$.
In conclusion it should hold that $g=G^\i$ anywhere on $(0,1)$.
}
When $g_1$ and $g_2$ are both non-decreasing, let $G_1$ and $G_2$ be distribution functions of $g_1(U)$ and $g_2(U)$ for $U \sim \U(0,1)$.
Since $g_1$ and $g_2$ are both non-decreasing and left-continuous, it holds that $g_1(u)= G_1^\i(u)$ and $g_2(u)=G_2^\i(u)$ for all $u \in (0,1)$.
The same statement holds when $g_1$ and $g_2$ are both non-inccreasing by replacing $g_1$ and $g_2$ with $-g_1$ and $-g_2$, respectively.
Therefore, if $g_1$ and $g_2$ are monotone left-continuous functions, it holds that $\rho(g_1(U),g_2(V))=\rho(G_1^\i(U),G_2^\i(V))$ for distribution functions $G_1$ and $G_2$ defined above.
Finally, these distribution functions must be of the same type as some non-degenerate radially symmetric distribution function $G$ with finite second moment by \citet[Proposition~1]{hofert2019compatibility}, which completes the proof.
\end{proof}

\end{comment}

\section{A class of discrete concordance-inducing distributions}\label{app:sec:class:discrete:concordance:inducing:functions}

As claimed by \citet{schmid2007nonparametric}, one of the advantages of Blomqvist's beta is that it admits an explicit form whenever the copula is given analytically.
This advantage can be extended to a wider class of discrete concordance-inducing distributions.
For $m\in \IN$, $\bz=(z_1,\dots,z_m) \in \IR_{+}^m$ and $\bp=(p_0,p_1,\dots,p_m) \in \IR_{+}^{m+1}$ such that $0<z_1<\cdots<z_m$, $p_0 + 2\sum_{i=1}^m p_i=1$ and $\sum_{i=1}^m p_i z_i^2=1/2$, consider a discrete distribution $G_{m,\bz,\bp}$ supported on $-z_m,\dots,-z_1,0,z_1,\dots,z_m$ with corresponding probabilities $p_m,\dots,p_1,p_0,p_1,\dots,p_m$.
Then $G_{m,\bz,\bp}$ is a concordance-inducing distribution with mean zero and variance one.
As a special case, Blomqvist's beta arises when $m=1$, $z_1=1$ and $(p_0,p_1)=(0,1/2)$.
Let $p_{+}=p_1+\cdots + p_m$, $I_{-i}=[p_{+}-\sum_{j=1}^{i} p_j,\ p_{+}-\sum_{j=1}^{i-1} p_j]$, $I_0=[p_{+},\ p_{+}+p_0]$ and $I_i=[p_{+}+p_0+\sum_{j=1}^{i-1} p_j,\ p_{+}+p_0+\sum_{j=1}^i p_j]$ for $i=1,\dots,m$.
Then
\begin{align*}
\kappa_{G_{m,\bz,\bp}}(C)=\E[G_{m,\bz,\bp}^\i(U)G_{m,\bz,\bp}^\i(V)]=\sum_{(i,j) \in \{-m,\dots,m\}}z_i z_j V_C(I_i\times I_j),
\end{align*}
and
\begin{align*}
\sigma_{G_{m,\bz,\bp}}^2(C) &=\Var(XY)=\E[(XY)^2]-(\E[XY])^2\\
&=  \sum_{(i,j) \in \{-m,\dots,m\}}z_i^2 z_j^2 V_C(I_i\times I_j)- \left(\sum_{(i,j) \in \{-m,\dots,m\}}z_i z_j V_C(I_i\times I_j)\right)^2,
\end{align*}
where $z_{-i}=-z_i$ for $i=1,\dots,m$ and $V_C(A)$, $A \subseteq [0,1]^2$, is a volume of $A$ measured by $C$.
Therefore, $\kappa_{G_{m,\bz,\bp}}(C)$ and $\sigma_{G_{m,\bz,\bp}}^2(C)$ admit explicit forms if $V_C(I_i\times I_j)$ can be written explicitly for all $(i,j) \in \{-m,\dots,m\}^2$.

\section{Joint distribution of $(X^2,Y^2)$}\label{app:sec:joint:distributions:X2:Y2}

For $G \in \mathcal G_4$ and $C \in \mathcal C_2$, the asymptotic variance $\sigma_G^2(C)$ depends on the joint distribution of $(X^2,Y^2)$, where $(X,Y)=(G^\i(U),G^\i(V))$, through the covariance formula~\eqref{eq:covariance:formula:asymptotic:variance}.

By radial symmetry of $G$, the marginal distribution of $X^2$ (and that of $Y^2$) is given by
\begin{align*}
G^{[2]}(x)&=\Prob(X^2 \leq x)=\Prob\left(-\sqrt{x}\leq X \leq \sqrt{x}\right)=G\left(\sqrt{x}\right)-G\left(-\sqrt{x}-\right)
= 2 G(\sqrt{x})-1,\quad x \geq 0.
\end{align*}
The following proposition describes the copula of $(X^2,Y^2)$ when $G$ is continuous.

\begin{proposition}[Copula of $(X^2,Y^2)$]\label{prop:copula:x2:y2}
Let $G \in \mathcal G_4^\text{c}$ be a continuous concordance-inducing distribution.
For a copula $C \in \mathcal C_2$ and $(X,Y)=(G^\i(U),G^\i(V))$ with $(U,V)\sim C$, the copula of $(X^2,Y^2)$ is given by
\begin{align*}
C^{[2]}(u,v)=\sum_{\varphi \in \{ \iota,\nu_1,\nu_2,\nu_1\circ\nu_2\}}\bar C_{\varphi}\left(\frac{1}{2},\frac{1}{2}\right) C_{\varphi,(1/2,1/2)}\left(\frac{u+1}{2},\frac{v+1}{2}\right),
\end{align*}
where $(U_\varphi, V_\varphi)\sim C_\varphi=\varphi(C)$ and
\begin{align*}
C_{\varphi,(1/2,1/2)}(u,v)=\Prob\left(U_\varphi\leq u,\ V_\varphi \leq v\ | \ U_{\varphi}>1/2,\ V_{\varphi} > 1/2 \right).
\end{align*}
\end{proposition}

\begin{proof}
By continuity of $G$, we have that $X>0$ if $U>1/2$ and $X\leq 0$ if $U\leq 1/2$.
Therefore,
\begin{align*}
G^{[2]}(X^2)&=2G\left(\sqrt{X^2}\right)-1=2G(|X|)-1
=\begin{cases}
2G(X)-1 = 2U-1, & \text{ if } U > 1/2,\\
2G(-X)-1 = 2(1-G(X))-1=1-2U, & \text{ if } U \leq 1/2.
\end{cases}
\end{align*}
Using this relationship, we have that
\begin{align*}
C^{[2]}(u,v)& =\Prob(G^{[2]}(X^2)\leq u,G^{[2]}(Y^2)\leq v)\\
&= \Prob\left(U>\frac{1}{2},\ V>\frac{1}{2}\right)\Prob\left(2U-1\leq u,\ 2V-1\leq v\ \biggl| \ U>\frac{1}{2},\ V>\frac{1}{2}\right)\\
& \hspace{5mm} + \Prob\left(U\leq \frac{1}{2},\ V>\frac{1}{2}\right)\Prob\left(1-2U\leq u,\ 2V-1\leq v\ \biggl| \ U\leq \frac{1}{2},\ V>\frac{1}{2}\right)\\
& \hspace{5mm} + \Prob\left(U>\frac{1}{2},\ V\leq \frac{1}{2}\right)\Prob\left(2U-1\leq u,\ 1-2V\leq v\ \biggl| \ U>\frac{1}{2},\ V\leq \frac{1}{2}\right)\\
& \hspace{5mm} + \Prob\left(U\leq \frac{1}{2},\ V\leq \frac{1}{2}\right)\Prob\left(1-2U\leq u,\ 1-2U\leq v\ \biggl| \ U\leq \frac{1}{2},\ V\leq \frac{1}{2}\right)\\
&= \Prob\left(U>\frac{1}{2},\ V>\frac{1}{2}\right)\Prob\left(U\leq \frac{u+1}{2},\ V\leq \frac{v+1}{2}\ \biggl| \ U>\frac{1}{2},\ V>\frac{1}{2}\right)\\
& \hspace{5mm} + \Prob\left(1-U> \frac{1}{2},\ V>\frac{1}{2}\right)\Prob\left(1-U\leq \frac{u+1}{2},\ V\leq \frac{v+1}{2}\ \biggl| \ 1-U> \frac{1}{2},\ V>\frac{1}{2}\right)\\
& \hspace{5mm} + \Prob\left(U>\frac{1}{2},\ 1-V>\frac{1}{2}\right)\Prob\left(U\leq \frac{u+1}{2},\ 1-V\leq \frac{v+1}{2}\ \biggl| \ U>\frac{1}{2},\ 1-V >  \frac{1}{2}\right)\\
& \hspace{5mm} + \Prob\left(1-U >  \frac{1}{2},\ 1-V > \frac{1}{2}\right)\Prob\left(1-U\leq \frac{u+1}{2},\ 1-V\leq \frac{v+1}{2}\ \biggl| \ 1-U >  \frac{1}{2},\ 1-V >  \frac{1}{2}\right),\\
&=\sum_{\varphi \in \{ \iota,\nu_1,\nu_2,\nu_1\circ\nu_2\}}\bar C_{\varphi}\left(\frac{1}{2},\frac{1}{2}\right) C_{\varphi,(1/2,1/2)}\left(\frac{u+1}{2},\frac{v+1}{2}\right).
\end{align*}
\end{proof}

As an application of Proposition~\ref{prop:copula:x2:y2}, we have that
\begin{align*}
C^{[2]}=\begin{cases}
\frac{1}{4}\Pi +\frac{1}{4}\Pi +\frac{1}{4}\Pi +\frac{1}{4}\Pi =\Pi, & \text{ if } C=\Pi,\\
\frac{1}{2} M +  0 + 0 +  \frac{1}{2}M  =M, & \text{ if } C=M,\\
0 + \frac{1}{2} M + \frac{1}{2} M + 0 =M, & \text{ if } C=W.\\
\end{cases}
\end{align*}

Next we study the copula $C^{[2]}$ when $C$ is a convex combination of copulas.

\begin{lemma}[Convex combination of $C^{[2]}$]\label{lemma:linearity:copula:X2:Y2}
Suppose that the concordance-inducing distribution $G \in \mathcal G_4^{\textup{c}}$ is continuous.
For a convex combination ${\tilde C}_p = pC + (1-p) C'$ of $C$ and $C'$ where $p \in [0,1]$ and $C, C' \in \mathcal C_2$, we have that
\begin{align*}
{\tilde C}_p^{[2]} = pC^{[2]} + (1-p) {C'}^{[2]}.
\end{align*}
\end{lemma}

\begin{proof}
Consider a random vector $(\tilde U,\tilde V)= B (U,V)+ (1-B)(U',V') \sim \tilde C_p$ where $(U,V)\sim C$, $(U',V')\sim C'$ and $B\sim \Bern(p)$ are independent of each other.
For $(X,Y)=(G^\i(U),G^\i(V))$, $(X',Y')=(G^\i(U'),G^\i(V'))$ and $(\tilde X,\tilde Y)=(G^\i(\tilde U),G^\i(\tilde V))$, we have that
\begin{align*}
(\tilde X,\tilde Y)&=(G^\i(BU+(1-B)U'),G^\i(BV+(1-B)V'))

=B(X,Y)+(1-B)(X',Y'),
\end{align*}
and thus $(\tilde X^2,\tilde Y^2)=B(X^2,Y^2) + (1-B)({X'}^2,{Y'}^2)$.
Since $\tilde X^2 \deq \tilde Y^2 \sim G^{[2]}$, the copula of $(\tilde X^2,\tilde Y^2)$ is given by
\begin{align*}

  {\tilde C}_p^{[2]}(u,v)
   &=\Prob(G^{[2]}(BX^2 + (1-B){X'}^2)\leq u,\ G^{[2]}(BY ^2 + (1-B){Y'}^2)\leq v)\\
&= \Prob(B=1) \Prob(G^{[2]}(X^2)\leq u,\ G^{[2]}(Y^2)\leq u)+ \Prob(B=0)\Prob(G^{[2]}({X'}^2)\leq u,\ G^{[2]}({Y'}^2)\leq u)\\
&= pC^{[2]}(u,v)+(1-p){C'}^{[2]}(u,v).
\end{align*}
\end{proof}

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