text:
\begin{document}
\maketitle

\begin{abstract}
Statistical inference based on divergence measures have
a long history.

Recently, Maji, Ghosh and Basu (2014)
have introduced a general family of divergences
called the

logarithmic super divergence (LSD) family. This family acts
as a superfamily for both of the logarithmic power
divergence (LPD) family (eg. Renyi, 1961) and the logarithmic density
power divergence (LDPD) family introduced by
Jones et al.~(2001).
In this paper we describe the asymptotic properties
of the inference procedures resulting from this
divergence in discrete models. The properties are well supported
by real data examples.

\end{abstract}
\vspace{2 mm}

{\em \bf Key words}: asymptotic properties, logarithmic density power divergence,
logarithmic power divergence, logarithmic super divergence, $S$-divergence,
statistical inference.
\section{Introduction}
\label{Sec_Introduction}

The

density-based minimum divergence approach
has long been an important parametric inference tool.

In this approach the

closeness between the data and the model is measured by

a density-based divergence between the data density and the
model density, such as
a $\chi^2$ type divergence or a $\phi$-divergence
(Csisz\'{a}r 1963, 1967a,b; Ali and Silvey 1966)
 or a Bregman divergence (Bregman, 1967).
Apart from their natural appeal, most

of these
methods are very useful for their inherent robustness properties.

A prominent member of the class of density-based divergences
is the Pearson's $\chi^2$ (Pearson, 1900)
which started its journey
from the very early days

of formal research
in statistics.
From the robustness perspective, however, Beran's 1977
work is the first useful reference in the literature
of density-based minimum divergence inference.

In the present paper
we focus on some variants of the power divergence (PD) measure of
Cressie and Read (1984) and the density power divergence (DPD) of Basu et al.~(1998)
and discuss various properties related to statistical inference based on a
generalized superfamilies of these measures.
The primary divergence class
of logarithmic super divergences (LSDs)
which is of interest to us in this paper
has been proposed recently by Maji, Ghosh and Basu (2014).
In the present paper we establish the theoretical asymptotic
properties of the resulting statistical procedures.
\paragraph{}
The rest of the paper is organized as follows. Section \ref{SEC:LPD_LDPD}
describes the logarithmic power divergence family and the
logarithmic density power divergence family. Section \ref{SEC:lsd_defn}
gives the form and the estimating equation of the
logarithmic super divergence (LSD) family whereas Section \ref{SEC:lsd_properties}
establishes the asymptotic distribution for LSD estimator. Section \ref{SEC:est_example}
gives some illustration of the proposed estimation procedure with real data.

Section \ref{SEC:testing} gives the asymptotic
distribution of the test statistic using LSD measure for both one sample
and two sample problem and Section \ref{SEC:testing:example}
provides a hypothesis testing example.

Concluding remarks are in Section \ref{Sec_Conclusion}.

\section{The Logarithmic Power Divergence (LPD) and the Logarithmic
Density Power Divergence (LDPD) Families}\label{SEC:LPD_LDPD}
Jones et al.~(2001) described
a class of divergence measures

which

do not require
any nonparametric smoothing techniques for their construction.

This

family of divergences is given by
\begin{equation}\label{EQ:LDPD}
{\rm LDPD}_\beta(g,f) = \displaystyle \log \int  f^{1+\beta} -
\left(1 + \frac{1}{\beta}\right)
\log \int f^\beta g + \frac{1}{\beta} \log \int g^{1+\beta},
\end{equation}
$\beta\geq0$,
where LDPD stands for logarithmic density power divergence.

For the case $\beta=0$ we define
$${\rm LDPD}_0(g,f)=
\lim_{\beta\rightarrow0} {\rm LDPD}_\beta(g,f)=
\lim_{\lambda\rightarrow0} {\rm PD}_\lambda(g,f)={\rm PD}_0(g,f),$$
where the power divergence (PD) measure has the form

\begin{equation}\label{EQ:cressie-read}
{\rm PD}_\lambda(g,f)=\frac{1}
{\lambda(\lambda + 1)} \int g \left[\left(\frac{g}{f}\right)^\lambda-1\right],
~~~~ -\infty<\lambda<\infty.

\end{equation}

The family of divergences in (\ref{EQ:LDPD})

is similar to the density power divergence family given by

 \begin{equation}\label{EQ:dpd}
    {\rm DPD}_\alpha(g,f) =

\int  \left[f^{1+\alpha} - \left(1 +
\frac{1}{\alpha}\right)  f^\alpha g + \frac{1}{\alpha} g^{1+\alpha}\right] ~

{\rm for} ~\alpha \geq 0.

\end{equation}
The LDPD family may be recovered from the DPD by replacing the
identity function with the logarithm function.
In spite of the similarity between the forms of the
DPD and the LDPD families, Jones et al.~(2001) had originally
developed the latter as a special case of an estimation
method proposed by Windham (1995).

Following the connection shown
between PD and DPD by Patra et al.~(2013),

Maji, Chakraborty and Basu (2014) have recently shown that the same sort of connection exists
between LDPD and the logarithmic power divergence (LPD) family.
To show this connection, the LDPD measure can be written as
\begin{eqnarray}\label{EQ:LDPD_log}
{\rm LDPD}_\beta(g,f) &=& \displaystyle \log \int  f^{1+\beta} - \left(1 + \frac{1}{\beta}\right)
\log \int \left(\frac{g}{f}\right) f^{1+\beta} \\ \nonumber
& & + \frac{1}{\beta} \log \int \left(\frac{g}{f}\right)^{1+\beta} f^{1+\beta}.
\end{eqnarray}
Replacing the $f^{1+\beta}$ term with $f$
in each of the three terms on the right hand side of (\ref{EQ:LDPD_log})
leads
to the density-based divergence $\frac{1}{\beta} \log \int \frac{g^{1+\beta}}{f^\beta}.$
After standardizing this further we express this divergence as
 $\frac{1}{\beta(\beta+1)} \log \int \frac{g^{1+\beta}}{f^\beta}.$ This family of
divergences will be called the logarithmic power divergence family
by Maji, Chakraborty and Basu (2014). Using a
different symbol for the tuning parameter, this
family has the form
\begin{equation}
\label{EQ:LPD}
{\rm LPD}_{\gamma}(g,f)=\frac{1}{\gamma(\gamma+1)} \log \int
\frac{g^{1+\gamma}}{f^\gamma} , \gamma \in \mathbb{R}.
\end{equation}
The limiting forms as $\gamma \rightarrow 0$ and $\gamma \rightarrow -1$
generate, respectively, the likelihood disparity LD (or PD$_{0}$) and the
Kullback-Leibler divergence KLD (or PD$_{-1}$).
These divergences have the form
\begin{equation}
 {\rm LD}(g,f)=  \int g \log \left(\frac{g}{f}\right),
\end{equation}
\begin{equation}
 {\rm KLD}(g,f)=  \int f \log \left(\frac{f}{g}\right).
\end{equation}
For any other value
of $\gamma$ the LPD measure can be seen to be a function of the PD
measure at the same value of $\gamma$. Specifically,
\begin{equation}
 \label{EQ:pd_lpd}
{\rm LPD}_\gamma(g,f)=\frac{1}{\gamma(\gamma+1)} \log
\left[\gamma(\gamma+1){\rm PD}_\gamma(g,f)+1\right].
\end{equation}
Apart from being
briefly considered by

Renyi as a measure of the amount of information

(Renyi, 1961),
the LPD family

is a member
of the ($h,\phi$) divergence family (eg. Pardo, $2006$),
where
$h(x)=\frac{1}{r(r+1)} [r(r+1)\log x+1]$.

\section{The Logarithmic Super Divergence}
\label{SEC:lsd_defn}

We now define the Logarithmic Super Divergence
(or Logarithmic \\ $S$-Divergence)
introduced in Maji, Ghosh and Basu (2014)
and establish
the asymptotic properties of the procedures resulting from it.

The

Logarithmic $S$-Divergence (LSD) is defined (Maji, Ghosh and Basu, 2014) as
\begin{eqnarray}
\label{lsd_form}
		{\rm LSD}_{\beta, \gamma}(g,f)

 &=&  \frac{1}{A} ~ \log \int f^{1+\beta}  -   \frac{1+\beta}{A B} ~
\log \int f^{B} g^{A} \nonumber \\
& & + \frac{1}{B} ~ \log \int g^{1+\beta},
~~~~ \beta > 0, ~-\infty < \gamma < \infty,
\end{eqnarray}
where
\begin{eqnarray}
    A &=& 1+\gamma (1-\beta) \nonumber \\
\mbox{           and               }
    B &=& \beta - \gamma (1-\beta). \nonumber
\end{eqnarray}

For $\beta=0~(A=1+\gamma,B=-\gamma),$ this family
coincides with the

logarithmic power divergence family of (\ref{EQ:LPD}) with parameter $\gamma$,
while $\gamma=0$ gives the logarithmic density power divergence family
in (\ref{EQ:LDPD}) with parameter $\beta$.
\paragraph{}
Replacing the logarithmic function with the identity function in (\ref{lsd_form})
generates the divergence

\begin{eqnarray}
		S_{\alpha, \lambda}(g,f)

 &=&  \frac{1}{A} ~ \int ~ f^{1+\alpha}  -   \frac{1+\alpha}{A B}
~ \int ~~ f^{B} g^{A}  + \frac{1}{B} ~ \int ~~ g^{1+\alpha},
\label{EQ:general_form}
\end{eqnarray}
with $\beta$ and $\gamma$ being replaced by $\alpha$ and $\lambda$
respectively.
This divergence is known as the $S$-divergence and has been introduced by Ghosh et al.~(2013).

\subsection{Estimating Equation}
\label{SEC:lsd_est_eqn}
Consider a parametric class of model densities
$\{ f_\theta : \theta \in \Theta \subseteq {\mathbb{R}}^p \}$
and
suppose that our interest is in estimating

$\theta$. Let $G$ denote the distribution function corresponding
to the true density $g$. The minimum
LSD

functional
$T_{\beta,\gamma}(G)$ at $G$ is defined as
\begin{equation}
 {\rm LSD}_{\beta, \gamma}\left(g,f_{T_{\beta,\gamma}(G)}\right)
= \min \limits_{\theta \in \Theta } {\rm LSD}_{\beta, \gamma}(g,f_\theta).
\end{equation}
It takes the value $\theta$ when the true density $g = f_\theta$ is in the model; when
it does not, $\theta_{\beta,\gamma}^g = T_{\beta,\gamma}(G)$ represents
the best fitting parameter, and $f_{\theta^g}$ is the model element closest
to $g$ in terms of logarithmic super divergence. For simplicity of
 notation, we suppress the subscript $\beta,\gamma$ in $\theta_{\beta,\gamma}^g$.

A simple differentiation gives us the estimating equation
for $\theta$, which is

\begin{equation}
\label{EQ:lsd_est_eqn}
 \frac{\int f_\theta^{1+\beta}  u_\theta}{\int f_\theta^{1+\beta}} =
\frac{\int f_\theta^{B} g^A  u_\theta}{\int f_\theta^{B} g^A }.
\end{equation}

For $\beta=0~(A=1+\gamma,B=-\gamma)$, the equation thus becomes the same as the estimating
equation of the logarithmic power divergence family with parameter $\gamma$.
For $\gamma=0~(A=1, B=\beta)$, on the other hand, it is the estimating
equation for the LDPD measure.
Here $u_\theta(x)=\frac{\partial}{\partial \theta} \log f_\theta(x)$
is the likelihood score function.

\section{Asymptotic Properties of the Minimum LSD Estimators in
Discrete Models}
\label{SEC:lsd_properties}
Under the parametric set-up of Section {\ref{SEC:lsd_est_eqn}},
consider a discrete family of distributions. We will use
the term ``density function'' generally for the sake of a
unified notation, irrespective of whether the distribution is discrete
or continuous. Let $X_1,\ldots,X_n$ be a random sample from the
true distribution having density function $g$. Representing
the logarithmic $S$-divergence in terms of the parameter $\beta$ and $\gamma$
(as given in Section \ref{SEC:lsd_defn}), let
$\hat{\theta}_{\beta,\gamma}$ be the estimator obtained by minimizing
${\rm LSD}_{\beta,\gamma}(\hat{g},f_\theta)$ over $\theta \in \Theta$, where
$\hat{g}$ is a suitable nonparametric density estimate of $g$;
in the discrete case the vector of relative frequencies
based on the sample data is the canonical choice for $\hat{g}$.
For similar theoretical properties of the minimum $S$-divergence estimator, see Ghosh (2014).

\subsection{Asymptotic Properties}
	Suppose $X_1,X_2,\ldots,X_n$ are $n$ independent
and identically distributed observations from a discrete
distribution $G$ modeled by
$\mathcal{F} = \{ F_{\theta} : \theta \in \Theta \subseteq \mathbb{R}^p \}$ and
let the distribution have support $\mathcal{X} = \{ 0,1,2,\ldots \}$ without loss
of generality. Denote the
relative frequency at $x$ as obtained from the data by $r_n(x) = \frac{1}{n}
\sum \limits_{i=1}^n \chi(X_i = x)$ where $\chi(A)$ is the
indicator of $A$.

The minimum LSD estimator
is then obtained

by minimizing the LSD
measure between the data
density $r_n$ and the model density $f_\theta$
with respect to $\theta$.

Minimizing
${\rm LSD}_{\beta,\gamma}(r_n,f_{\theta})$
with respect to $\theta$ is equivalent to
minimizing

$H_n(\theta)$ where
\begin{eqnarray}
    H_n(\theta) = \frac{1}{1+\beta} \left[ \frac{1}{A} \log \sum_x f_{\theta}^{1+\beta}(x)
- \frac{1+\beta}{AB} \log \sum_x f_{\theta}^B(x) r_n^A(x) \right].
\end{eqnarray}
	Now,
\begin{eqnarray}
    \nabla H_n(\theta) &=& \frac{1}{A}
\left[\frac{\sum_x f_{\theta}^{1+\beta}(x) u_\theta(x)}{\sum_x f_{\theta}^{1+\beta}(x)} -
\frac{\sum_x f_{\theta}^B(x) r_n^A(x) u_\theta(x)}{\sum_x f_{\theta}^B(x) r_n^A(x)}\right],

\end{eqnarray}
where $\nabla$ is

the gradient with respect to $\theta$.
Equating
the above

to zero, the

estimating equation becomes
\begin{eqnarray}
    \sum_x K(\delta_n(x)) f_{\theta}^{1+\beta}(x) w_{\theta}(x) = 0,
\end{eqnarray}
where $\delta_n(x) = \frac{r_n(x)}{f_{\theta}(x)},~~
\delta_g(x) = \frac{g(x)}{f_{\theta}(x)},~~
w_{\theta}(x)=[B(\theta)u_\theta(x)-A(\theta)],~~
K(\delta)=\delta^A-1,~~ A(\theta)=\sum_x
f_{\theta}^{1+\beta}(x)u_{\theta}(x) \\ \mbox{ and }
B(\theta)=\sum_x
f_{\theta}^{1+\beta}(x).$

	Define

\begin{eqnarray}
J_g &=& E_g \left[ w_{\theta^g}(X)u_{\theta^g}^T(X) K'(\delta_g^g(X))
 f_{\theta^g}^{\beta}(X) \right] \nonumber \\
 & & -  \sum_x  K(\delta_g^g(x))
f_{\theta}^{1+\beta}(x) \nabla w_{\theta^g}(x) \nonumber \\
& & - (1+\beta) \sum_x  K(\delta_g^g(X))
f_{\theta}^{1+\beta}(x) w_{\theta^g}(x) u_{\theta^g}(x) . \\
    V_g &=& V_g \left[ K'(\delta_g^g(X))
f_{\theta^g}^\beta(X) w_{\theta^g}(X) \right].
\end{eqnarray}

We now present the necessary assumptions for our asymptotic results.

\begin{enumerate}
    \item The model family ${F_\theta}$ is identifiable which
means that different values of the parameter must generate
different probability distributions of the observable variables.
	 \item The probability density function $f_\theta$ of
the model distribution have common support so that the
set $\mathcal{X} = \{ x : f_{\theta}(x) >0 \}$ is independent of $\theta$.
			Also the true distribution $g$ is
compatible with the model family.
	 \item There exists an open subset $\omega \subset \Theta$
 for which the best fitting parameter $\theta^g$ is an interior
point and for almost all $x$, the density $f_{\theta}(x)$ admits
all third derivatives of the type $\nabla_{jkl} f_{\theta}(x)$   $\mbox{ for all } \theta \in \omega$.
Here the subscripts $j,k,l$ of $\nabla$ represent the indicated
partial derivatives.
		\item The matrix $\frac{1+\beta}{A} J_g$ is positive definite.
		\item The quantities

$\sum_x g^{1/2}(x)
f_{\theta}^{\beta}(x) |u_{j\theta}(x)|$,
$\sum_x g^{1/2}(x)
f_{\theta}^{\beta}(x) |w_{j\theta}(x)|$, \\
$\sum_x g^{1/2}(x)
f_{\theta}^{\beta}(x) |u_{j\theta}(x)| |u_{k\theta}(x)|$ and

$\sum_x g^{1/2}(x) f_{\theta}^{\beta}(x) |u_{jk\theta}(x)|$
are bounded $\mbox{ for all } j, k$ and $\mbox{ for all } \theta \in \omega$.

		\item For almost all $x$, there exists functions
$M_{jkl}(x)$, $M_{jk,l}(x)$, $M_{j,k,l}(x)$,
possibly depending on $\beta$ that dominate, in absolute
value,

$f_{\theta}^{\beta}(x) u_{jkl\theta}(x)$, $f_{\theta}^{\beta}(x)
u_{jk\theta}(x) u_{l\theta}(x)$ and $f_{\theta}^{\beta}(x)
u_{j\theta}(x) u_{k\theta}(x) u_{l\theta}(x)$ respectively $ \mbox{ for all } j, k, l$ and
that are uniformly bounded in expectation with respect to $g$ and $f_{\theta}
~~ \mbox{for all } \theta \in \omega$, where
the subscripts of $u$ denote the indicated partial
derivatives of the score function.
		\item The function $\left( \frac{g(x)}{f_{\theta}(x)} \right)^{A-1}$ is
uniformly bounded (by, say, $C$) $\mbox{ for all } \theta \in \omega$.

\end{enumerate}
	To prove the consistency and asymptotic normality of
the minimum LSD estimator, we will assume, for the rest of the paper, that the
seven conditions stated above are satisfied.

\paragraph{}
Based on the above assumptions we now
start the proofs of the required results.
The proofs are primarily along the
lines of Lindsay (1994) and Basu et al.~(2011).
Define $ \eta_n(x) = \sqrt n (\sqrt
\delta_n-\sqrt\delta_g)^2$.

We then have the following lemmas.
\begin{lemma}
\label{lem:lamma1}
  	 For any $k \in [0,2]$, we have
\begin{enumerate}
    \item $E[\eta_n^k(x)] \le n^{\frac{k}{2}}E[|\delta_n(X) -
\delta_g(X)|]^k \le \left[ \frac{g(x)(1-g(x))}{f_{\theta}^2(x)}\right]^{\frac{k}{2}}.$

	 \item   $E[|\delta_n(X) - \delta_g(X)|] \le \frac{2g(x)(1-g(x))}{f_{\theta}(x)}.$
\end{enumerate}
\begin{proof}

For $a,b \ge 0$, we have the inequality $(\sqrt a - \sqrt b)^2 \le |a-b|$. So we get
\begin{eqnarray}
    E[\eta_n^k(x)] &=& n^{\frac{k}{2}} E[(\sqrt
\delta_n-\sqrt\delta_g)^2]^k \nonumber \\
		& \le & n^{\frac{k}{2}} E[|
\delta_n-\delta_g |]^k. \nonumber
\end{eqnarray}

	For the next part see that, $ n r_n(x) \sim Bin( n, g(x) ) ~~ \mbox{ for all } x$. Now,
for any  $k \in [0,2]$, we get by the Lyapunov's inequality that

\begin{eqnarray}
    E[|\delta_n(X) - \delta_g(X)|]^k &\le & \left[ E(\delta_n(X) - \delta_g(X))^2 \right]^{\frac{k}{2}} \nonumber \\					&=& \frac{1}{f_{\theta}^k(x)} \left[ E(r_n(X) - g(X))^2 \right]^{\frac{k}{2}} \nonumber
\\		&=& \frac{1}{f_{\theta}^k(x)} \left[ \frac{g(x)(1-g(x))}{n}\right]^{\frac{k}{2}}. \nonumber
\\ \nonumber
\end{eqnarray}

	For the second part, note that
\begin{eqnarray}
    E[|\delta_n(X) - \delta_g(X)|] &=& \frac{1}{f_{\theta}^k(x)} \left[ E|r_n(X) - g(X)| \right]^{\frac{k}{2}} \nonumber
\\		&\le& \frac{2g(x)(1-g(x))}{f_{\theta}(x)}. \nonumber
\\ \nonumber
\end{eqnarray}
where the last inequality follows from the result about the mean-deviation of a Binomial random variable.
\end{proof}

\end{lemma}

\begin{lemma}
\label{lem:lamma2}
	$E[\eta_n^k(x)] \rightarrow 0$, as $n \rightarrow \infty$,  for $k \in [0,2).$
\begin{proof}
  This follows from Theorem 4.5.2 of Chung (1974) by noting that \\
$n^{1/4} (r_n^{1/2}(x) - g^{1/2}(x)) \rightarrow 0$
with probability one for each $x \in \mathcal{X}$ and by the Lemma \ref{lem:lamma1}(1), $sup_n E[\eta_n^k(x)] $ is bounded.
\end{proof}

\end{lemma}

	Let us now define, $a_n(x) = K(\delta_n(x)) - K(\delta_g(x))$ and
$b_n(x) = (\delta_n(x)-\delta_g(x))K'(\delta_g(x))$.

 We will need the limiting distributions of

$$S_{1n} = \sqrt n \sum_x a_n(x)f_{\theta}^{1+\beta}(x)w_{\theta}(x)$$
and $$S_{2n} = \sqrt n \sum_x
b_n(x)f_{\theta}^{1+\beta}(x)w_{\theta}(x).$$

 Define $\tau_n(x) = \sqrt n |a_n(x) - b_n(x)|$.

\begin{lemma}
\label{lem:lamma3}
	Suppose Assumption 5 holds. Then $E|S_{1n}-S_{2n}| \rightarrow 0 $ as $n \rightarrow \infty$,

and hence $S_{1n}-S_{2n} \xrightarrow{P} 0 $ as $n \rightarrow \infty.$
\begin{proof}
 	By Lemma 2.15 of Basu et ~al.~(2011) [or, Lindsay(1994), Lemma 25], there
exists some positive constant $\beta$ such that
$$
\tau_n(x) \le \beta \sqrt n (\sqrt\delta_n-\sqrt\delta_g)^2 = \beta \eta_n(x).
$$
Also, by Lemma \ref{lem:lamma1}, $ E[\tau_n(x)] \le \beta \frac{g^{1/2}(x)}{f_{\theta}(x)}$.\\
And by Lemma \ref{lem:lamma2}, $E[\tau_n(x)] = \beta E[\eta_n(x)] \rightarrow 0$ as $n \rightarrow \infty$. Thus we get,
\begin{eqnarray}
    E|S_{1n}-S_{2n}| &\le& \sum_x E[\tau_n(x)] f_{\theta}^{1+\beta}(x)|w_{\theta}(x)| \nonumber \\
                     &\le&  \beta \sum_x g^{1/2}(x) f_{\theta}^{\beta}(x)|w_{\theta}(x)| \nonumber \\
							& < &
\infty ~~~~~~ \mbox{ (by Assumption $5$)}. \nonumber \\ \nonumber
\end{eqnarray}
So, by the Dominated Convergence Theorem (DCT),  $E|S_{1n}-S_{2n}| \rightarrow 0 $ as $n \rightarrow \infty$.

Hence, by Markov's inequality, $S_{1n}-S_{2n} \xrightarrow{P} 0 $ as $n \rightarrow \infty$.
\end{proof}

\end{lemma}

\begin{lemma}
\label{lem:lamma4}
	Suppose $V_g =  V_g \left[ K'(\delta_g(X)) f_{\theta}^\beta(X) w_{\theta}(X) \right] $ is finite. Then
$$
S_{1n} \rightarrow N(0, V_g).
$$
\begin{proof}
 	By Lemma \ref{lem:lamma3}, the asymptotic distribution of
$S_{1n}$ and $S_{2n}$ are the same. Now, we have
\begin{eqnarray}
    S_{2n} &=& \sqrt n \sum_x (\delta_n(x) - \delta_g(x))
K'(\delta_g(x)) f_{\theta}^{1+\beta}(x) w_{\theta}(x) \nonumber \\
				&=&  \sqrt n \sum_x (r_n(x) -
g(x)) K'(\delta_g(x)) f_{\theta}^{\beta}(x) w_{\theta}(x) \nonumber \\
			&=&  \sqrt n \left( \frac{1}{n}
\sum_{i=1}^n \left[ K'(\delta_g(X_i)) f_{\theta}^{\beta}(X_i) w_{\theta}(X_i) -
E_g\{ K'(\delta_g(X))f_{\theta}^{\beta}(X)w_{\theta}(X)\} \right]\right) \nonumber \\
			&\rightarrow & N( 0, V_g) ~~~~~ \mbox{(by the
Central Limit Theorem)}. \nonumber \\ \nonumber
\end{eqnarray}

\end{proof}

\end{lemma}

	We will now consider the final theorem of this section
about the consistency and asymptotic normality of the minimum LSD estimator.

\begin{theorem}
\label{th:est_th}
	Under Assumptions $1-7$, there exists a consistent
sequence $\theta_n$ of roots to the minimum LSD estimating equation (\ref{EQ:lsd_est_eqn}).

	Also, the asymptotic distribution of
$\sqrt n (\theta_n - \theta_g)$ is $p-$dimensional normal
with mean $0$ and variance $J_g^{-1}V_g J_g^{-1}$.
\begin{proof}
Because of the lengthy and somewhat messy calculations, the
proof of consistency has been put in Appendix.
\paragraph{}
{\bf Proof of the asymptotic Normality :}		For the Asymptotic normality, we expand
$$\sum_x K(\delta_n(x)) f_{\theta}^{1+\beta}(x)w_{\theta}(x)$$
 in Taylor series about $\theta = \theta^g$ to get
\begin{eqnarray}
\label{EQ:kfw_exapnd}
    & & \sum_x K(\delta_n(x)) f_{\theta}^{1+\beta}(x)w_{\theta}(x)  \nonumber \\
& = & \sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x)w_{\theta^g}(x)
+ \sum_k (\theta_k - \theta_k^g) \nabla_k \left( \sum_x K(\delta_n(x))
f_{\theta}^{1+\beta}(x)w_{\theta}(x) \right)|_{\theta = \theta^g} \nonumber \\
 & & ~~~~~~~~~ + \frac{1}{2} \sum_{k,l} (\theta_k - \theta_k^g)
(\theta_l - \theta_l^g)\nabla_{kl} \left(\sum_x K(\delta_n(x))
f_{\theta}^{1+\beta}(x)w_{\theta}(x)\right)|_{\theta = \theta '} \\ \nonumber
\end{eqnarray}
where, $\theta '$ lies in between $\theta$ and $\theta^g$.

	Now, let $\theta_n$ be the solution of the minimum
LSD estimating equation, which can be assumed to be consistent.
Replace $\theta$ by $\theta_n$ in above (\ref{EQ:kfw_exapnd})
so that the LHS of the equation becomes zero and hence we get
\begin{eqnarray}
\label{EQ:kfw_times}
  &  & - \sqrt n \sum_x K(\delta_n^g(x)) f_{\theta^g}^{1+\beta}(x)
w_{\theta^g}(x)  =  \sqrt n \sum_k (\theta_{nk} -
\theta_k^g) \times \nonumber \\
 & & ~~~~~~~~~~~~ \{ \nabla_k \left( \sum_x K(\delta_n(x))
f_{\theta}^{1+\beta}(x)w_{\theta}(x) \right)|_{\theta = \theta^g}
+ \nonumber \\
& & ~~~~~~~~~~~~~~~~~~~~~~~\frac{1}{2} \sum_{l}
(\theta_{nl} - \theta_l^g)\nabla_{kl} \left(\sum_x K(\delta_n(x))
f_{\theta}^{1+\beta}(x)w_{\theta}(x)\right)|_{\theta = \theta '}
\}. \nonumber \\
\end{eqnarray}
	Note that, the first term within the bracketed quantity
in the RHS of above (\ref{EQ:kfw_times}) converges to $J_g$ with probability
tending to one, while the second bracketed term is an
$o_p(1)$ term (as proved in the proof of consistency part). Also,
by using the Lemma \ref{lem:lamma4}, we get that
\begin{eqnarray}
   & & \sqrt n \sum_x K(\delta_n^g(x))
f_{\theta^g}^{1+\beta}(x)w_{\theta^g}(x) \nonumber \\
 ~~~~~~~~~~~ & = & \sqrt n \sum_x  [K(\delta_n^g(x)) -
K(\delta_g^g(x))]  f_{\theta^g}^{1+\beta}(x)w_{\theta^g}(x)  (\mbox{Using estimating equation)} \nonumber \\
 ~~~~~~~~~~~ & = & S_{1n}|_{\theta=\theta^g}
\rightarrow N_p(0, V_g). \nonumber \\
\end{eqnarray}
Therefore, by Lehmann(1983, Lemma 4.1),

$\sqrt n(\theta_n - \theta^g)$ has asymptotic
distribution as $ N_p(0,J_g^{-1} V_g J_g^{-1})$.
\end{proof}
\end{theorem}

A very interesting observation that follows from the
asymptotic distribution just established is that the
asymptotic distribution of the estimator is independent
of the parameter $\gamma$. Yet the behavior of the
estimator varies widely with $\gamma$ under the presence of
outliers. Here we briefly report the findings
reported by Maji, Ghosh and Basu (2014) in this connection,
which is at least partially indicated by the results
of the current and the subsequent section. Maji, Ghosh and Basu (2014)
have observed that the first order influence function of the minimum
LSD estimator is independent of $\gamma$, predicting that the
robustness properties of the minimum LSD estimators are
similar for each value of $\gamma$. This is immediately
contradicted by the other results of Maji, Ghosh and Basu (2014)
as well as the next section of the current article.
Further exploration by Maji, Ghosh and Basu (2014) indicate
that the second order influence function

gives a much more accurate picture of the
robustness of these estimators. This clearly indicates the
limitation of the first order influence function in quantifying
the robustness of the estimators in this case. In fact
the second order influence analysis (see Maji, Ghosh and Basu, 2014 for details)
shows that the limitation of the first order influence function
can go both ways -- it can fail to indicate the stability of a robust
estimator, and can also describe a highly unstable estimator as
a robust one.

\section{Examples}

\label{SEC:est_example}
\subsection{An Example with a Geometric Model}
The data set given in Basu et al.~(2011),
Table $2.4$, represent the cases of peritonitis
for $390$ kidney patients.

Following Basu et al.~(2011) we
use a

geometric model with

parameter $\theta$ (success probability)

as our target distribution.
A quick look at the data reveals that a geometric model
with $\theta=0.5$ may fit the data well.
We fit a geometric model with parameter $\theta$ using the
LSD measure for various combinations of $\beta$ and $\gamma$.

We can treat the two observations at 10
and 12 as mild to moderate outliers. We have
evaluated the minimum

LSD estimator
in this case for the full data as well as the outlier
deleted data.

The estimates are
presented
shown in
Tables \ref{tab:kidney_w_out} and \ref{tab:kidney_wo_out} respectively.
The estimates highlight an interesting point;

for
$\gamma<0$ or for larger values of $\beta$ with $\gamma\geq0$ the
parameter estimates are close for full data and outlier deleted data.
However for

$\gamma>0$ and

$\beta$ small, the presence or absence
of the outliers do not lead to a substantially
larger difference.
This gives a clear indication about which
combinations of the $(\beta,\gamma)$ values keep
the estimators stable and which are the ones that are easily
affected.

 \begin{table}

	\centering
	\caption{The estimates of the parameter
of the geometric model for

different values of $\gamma$ and $\beta$ for the
Peritonitis Incidence Data with outlier
}
		\begin{tabular}{c c c c c c c } \hline
$\gamma$ \textdownarrow $\beta$ $\rightarrow$

& 0 & 0.2  & 0.4  &  0.6 & 0.8 & 1 \\
\hline

$-1$ & -- & 0.518 & 0.511 & 0.511 & 0.513 & 0.515 \\

$-0.7$ & 0.519 & 0.51 & 0.509 & 0.51 & 0.512 & 0.515 \\

$-0.5$ & 0.510 & 0.508 & 0.508 & 0.51 & 0.512 & 0.515 \\

$-0.3$ & 0.504 & 0.505 & 0.507 & 0.51 & 0.512 & 0.515 \\

$-0.1$ & 0.499 & 0.503 & 0.506 & 0.509 & 0.512 & 0.515 \\
0 & 0.496 & 0.502 & 0.506 & 0.509 & 0.512 & 0.515 \\

0.5 & 0.48 & 0.496 & 0.504 & 0.508 & 0.512 & 0.515 \\

1 & 0.461 & 0.486 & 0.501 & 0.507 & 0.512 & 0.515 \\

1.3 & 0.45 & 0.479 & 0.499 & 0.507 & 0.511 & 0.515 \\

1.7 & 0.438 & 0.469 & 0.495 & 0.506 & 0.511 & 0.515 \\

2 & 0.43 & 0.461 & 0.491 & 0.505 & 0.511 & 0.515 \\
\hline
		\end{tabular}
\label{tab:kidney_w_out}

\end{table}
 \begin{table}

	\centering
	\caption{The estimates of the parameter
of the geometric model for

different values of $\gamma$ and $\beta$ for the
Peritonitis Incidence Data without the two outliers
}
		\begin{tabular}{c c c c c c c } \hline
$\gamma$ \textdownarrow $\beta$ $\rightarrow$

& 0 & 0.2  & 0.4  &  0.6 & 0.8 & 1 \\
\hline

$-1$ & -- & 0.521 & 0.512 & 0.511 & 0.513 & 0.515 \\

$-0.7$ & 0.526 & 0.513 & 0.51 & 0.511 & 0.513 & 0.515 \\

$-0.5$ & 0.518 & 0.511 & 0.509 & 0.51 & 0.512 & 0.515 \\

$-0.3$ & 0.513 & 0.509 & 0.508 & 0.51 & 0.512 & 0.515 \\

$-0.1$ & 0.510 & 0.508 & 0.508 & 0.51 & 0.512 & 0.515 \\
0 & 0.509 & 0.507 & 0.508 & 0.51 & 0.512 & 0.515 \\

0.5 & 0.505 & 0.505 & 0.506 & 0.509 & 0.512 & 0.515 \\

1 & 0.501 & 0.503 & 0.505 & 0.508 & 0.512 & 0.515 \\

1.3 & 0.5 & 0.501 & 0.504 & 0.508 & 0.512 & 0.515 \\

1.7 & 0.498 & 0.5 & 0.504 & 0.508 & 0.511 & 0.515 \\

2 & 0.496 & 0.499 & 0.503 & 0.507 & 0.511 & 0.515 \\
\hline
		\end{tabular}
\label{tab:kidney_wo_out}

\end{table}
\subsection{An Example with a Poisson Model}

This example

gives us the observed frequencies and
corresponding estimated frequencies (Table
\ref{droso_table1}) for several
minimum LSD
estimators

under the Poisson model
for a sex linked recessive lethal test in drosophila (fruit flies)
exposed to a certain chemical. For each of several male flies
one samples about $100$ daughter flies, and
then determines the frequency of the number of
daughter flies having a recessive lethal mutation
in its $X$-chromosome. The data represent a frequency
of frequencies; refer to

Woodruff et al.~(1984) for details. There is a possible case
of outliers corresponding to the observations at $x=3,4$.
\paragraph{}
Table \ref{droso_table1} provides the estimators and the predicted
frequencies for a small number of $\gamma,\beta$ combinations,
together with the fits of the maximum likelihood estimator
(denoted by ML) and the outlier deleted maximum likelihood
estimator obtained by removing the two outliers (denoted
by ML$+$D). Clearly the estimators (and the estimated frequencies)
are substantially different for the ML
and ML$+$D cases, demonstrating that the maximum
likelihood estimator is significantly affected by the presence
of these outliers. Also apart from the ML, the ($\gamma=1,\beta=0.1$)
combination leads to highly unstable estimators.

While Table \ref{droso_table1} provides a small number of ($\gamma,\beta$)
combinations, a large selection is presented in Table \ref{droso_estimate},
where the salient features may be described as follows:
\begin{enumerate}[(a)]
 \item All the estimators corresponding to large negative values of
$\gamma$ and/or values of $\beta$ close to $1$ generate outlier
resistant methods,
\item Estimators corresponding to large positive values of $\gamma$
are relatively poor in terms of robustness, especially for small $\beta$
\item All estimators for $\beta=1$ are identical,
irrespective of the value of $\gamma$.
\end{enumerate}
Once again this example shows that large positive values of $\gamma$
with $\beta$ close to zero are the more unstable distances
within the LSD class.

\begin{table}
\caption{

Fits of the Poisson model to the Drosophila Data using several estimation methods
}
\vspace{2 mm}
\centering
  \begin{tabular}{l c c c c c c c}
\hline
& \multicolumn{7}{c}{Recessive lethal count}\\\cline{2-8}
& 0 & 1 & 2 & 3 & 4 & $\geq 5$
& $\hat{\theta}$
\\
Observed & 23 & 3 & 0 & 1& 1& 0\\
ML & 19.59 & 7.00 & 1.25 & 0.15 & 0.01 & -- & 0.3571\\
ML+D & 24.95 & 2.88 & 0.17 & 0.01 & -- & -- & 0.1154\\
{\rm LSD}$_{\gamma = 1~~\&~~\beta = 0.1}$ & 14.90 & 9.40 & 2.97 & 0.62 & 0.10 & 0.01 & 0.6311\\
{\rm LSD}$_{\gamma = -1~~\&~~\beta = 0.1}$ & 25.95 & 1.98 & 0.07 & -- & -- & -- & 0.0762\\
{\rm LSD}$_{\gamma = 1~~\&~~\beta = 1}$ & 24.59 & 3.19 & 0.21 & 0.01 & -- & -- & 0.1297\\
{\rm LSD}$_{\gamma =-1~~\&~~\beta = 1}$ & 24.59 & 3.19 & 0.21 & 0.01 & -- & -- & 0.1297\\
\hline

\end{tabular}

\label{droso_table1}
 \end{table}
 \begin{table}
	\centering
	\caption{Estimates of the parameter for drosophila data in Table {\ref{droso_table1}}}
		\begin{tabular}{c c c c c c c c c c c c} \hline
$\gamma$ \textdownarrow $\beta$ $\rightarrow$

& 0 & 0.2  & 0.4  &  0.6 & 0.8 & 0.9 & 1 \\

\hline

$-0.8$ & 0.088 & 0.113 & 0.123 & 0.127 & 0.129 & 0.129 & 0.13 \\
$-0.7$ & 0.101 & 0.117 & 0.124 & 0.127 & 0.129 & 0.129 & 0.13 \\
$-0.6$ & 0.112 & 0.121 & 0.126 & 0.128 & 0.129 & 0.129 & 0.13 \\
$-0.5$ & 0.125 & 0.126 & 0.127 & 0.128 & 0.129 & 0.13 & 0.13 \\
$-0.4$ & 0.145 & 0.132 & 0.129 & 0.129 & 0.129 & 0.13 & 0.13 \\
$-0.3$ & 0.177 & 0.139 & 0.131 & 0.129 & 0.129 & 0.13 & 0.13 \\
$-0.2$ & 0.227 & 0.151 & 0.134 & 0.13 & 0.13 & 0.13 & 0.13 \\
$-0.1$ & 0.291 & 0.169 & 0.137 & 0.131 & 0.13 & 0.13 & 0.13 \\
0 & 0.357 & 0.194 & 0.142 & 0.132 & 0.13 & 0.13 & 0.13 \\
0.1 & 0.417 & 0.228 & 0.148 & 0.133 & 0.13 & 0.13 & 0.13 \\
0.2 & 0.47 & 0.269 & 0.157 & 0.134 & 0.13 & 0.13 & 0.13 \\
0.3 & 0.514 & 0.311 & 0.169 & 0.135 & 0.13 & 0.13 & 0.13 \\
0.4 & 0.553 & 0.353 & 0.184 & 0.137 & 0.131 & 0.13 & 0.13 \\
0.5 & 0.586 & 0.393 & 0.204 & 0.139 & 0.131 & 0.13 & 0.13 \\
0.6 & 0.615 & 0.43 & 0.226 & 0.142 & 0.131 & 0.13 & 0.13 \\
0.7 & 0.641 & 0.463 & 0.252 & 0.145 & 0.131 & 0.13 & 0.13 \\
0.8 & 0.663 & 0.494 & 0.278 & 0.149 & 0.131 & 0.13 & 0.13 \\

\hline
		\end{tabular}
\label{droso_estimate}
\end{table}

\section{Testing Parametric Hypothesis using the LSD Measures}
\label{SEC:testing}

Now we focus our attention on hypothesis testing,
the other important \\ paradigm of statistical inference.
\subsection{One Sample problem}
	We consider a parametric family of densities
$\mathcal{F} = \{f_\theta : \theta \in \Theta \subseteq \mathbb{R}^p\}$
as above. Suppose we are given a random sample $X_1, \ldots, X_n$ of
size $n$ from the population. Based on this sample, we want to test the hypothesis
$$
 H_0 : \theta = \theta_0 ~~~ \mbox{against} ~~~~ H_1 : \theta \ne \theta_0.
$$

When the model is correctly
specified and the null hypothesis is correct, $f_{\theta_0}$ is the data
generating density.

We consider the
test statistics based on the LSD with parameter

$\beta$ and $\gamma$
as follows:
\begin{equation}
 W_{\beta,\gamma}(\hat{\theta}_{\beta,\gamma},\theta_0)
=2n~{\rm LSD}_{\beta,\gamma}(f_{\hat{\theta}_{\beta,\gamma}},f_{\theta_0}),

\end{equation}
where ${\rm LSD}_{\beta,\gamma}(f_{\hat{\theta}_{\beta,\gamma}},f_{\theta_0})$ has the form
given in (\ref{lsd_form}).
\begin{theorem}

 The
asymptotic distribution of the test statistic 	$W_{\beta,\gamma}
(f_{{\hat{\theta}}_{\beta,\gamma}},f_{{\theta_0}})$, under the null hypothesis
$H_0 : \theta = \theta_0$, coincides with the distribution of
$$
\sum_{i=1}^r ~  \zeta_i^{\beta}(\theta_0)Z_i^2
$$
where $Z_1, \ldots,Z_r$ are independent standard normal variables,
$\zeta_1^{\beta}(\theta_0), \ldots, \\ \zeta_r^{\beta}
(\theta_0)$  are the nonzero eigenvalues of $A_{\beta}(\theta_0)
J_{\beta}^{-1}(\theta_0) K_{\beta}(\theta_0) J_{\beta}^{-1}(\theta_0)$,
with $J_{\beta}(\cdot)$ and $K_{\beta}(\cdot)$ as defined in Theorem
\ref{th:est_th}

and the matrix $A_{\beta}(\theta_0)$ is defined as
$$
A_{\beta}(\theta_0) = \nabla [  \nabla {\rm LSD}_{\beta,\gamma}(f_\theta, f_{\theta_0}) ]
|_{\theta = \theta_0}

$$
 and
$$
r = rank\left( J_{\beta}^{-1}(\theta_0)
K_{\beta}(\theta_0) J_{\beta}^{-1}(\theta_0)
A_{\beta}(\theta_0) J_{\beta}^{-1}(\theta_0)
K_{\beta}(\theta_0) J_{\beta}^{-1}(\theta_0)\right).
$$

\begin{proof}
  We consider the second order Taylor series expansion
of ${\rm LSD}_{\beta,\gamma}(f_\theta, f_{\theta_0})$ around
$\theta = \theta_0$ at $\theta = \hat{\theta}_{\beta}$ as,
\begin{eqnarray}
\label{EQ:lsd_expansion}
    {\rm LSD}_{\beta,\gamma}\left(f_{\hat{\theta}_{\beta}},f_{\theta_0}\right)
&=& {\rm LSD}_{\beta,\gamma}(f_{\theta_0}, f_{\theta_0}) + \sum_{i=1}^p
 \nabla_i {\rm LSD}_{\beta,\gamma}(f_\theta,
f_{\theta_0})|_{\theta = \theta_0} (\hat{\theta}_{\beta}^i - \theta_0^i) \nonumber
\\
& & + \frac{1}{2} \sum_{i,j} \nabla_{ij} {\rm LSD}_{\beta,\gamma}
(f_\theta, f_{\theta_0})|_{\theta = \theta_0}
(\hat{\theta}_{\beta}^i - \theta_0^i) (\hat{\theta}_{\beta}^j -
\theta_0^j) \nonumber \\
& & + o(||\hat{\theta}_{\beta} - \theta_0||^2),
\end{eqnarray}
where
$\nabla_i$ and $\nabla_{ij}$ represent the indicated partial
derivatives with respect to the components of $\theta$.

Now
we have
$$ {\rm LSD}_{\beta,\gamma}(f_{\theta_0}, f_{\theta_0}) = 0 $$
and $$\nabla_i {\rm LSD}_{\beta,\gamma}(f_\theta,
f_{\theta_0})|_{\theta = \theta_0} =0. $$

Note that the above second order partial derivative of
${\rm LSD}_{\beta,\gamma}(f_\theta, f_{\theta_0})$ at
$\theta = \theta_0 $ is independent of $\gamma$ and
so we will denote that as function of $\beta$ only.
We will denote the second order partial derivatives of
${\rm LSD}_{\beta,\gamma}(f_\theta, f_{\theta_0})$  in
(\ref{EQ:lsd_expansion})
by $a_{ij}^{\beta} (\theta_0)$.

Also denote
$A_{\beta}(\theta_0) = \left( a_{ij}^{\beta} (\theta_0)
\right)_{i,j=1,\ldots,p}$.
Now from the above Taylor series expansion it is clear
that the random variables
\begin{eqnarray}
     W_{\beta,\gamma}({\hat{\theta}_{\beta,\gamma}},{\theta_0}) =
2 n {\rm LSD}_{\beta,\gamma}(f_{\hat{\theta}_{\beta,\gamma}},f_{\theta_0}) \nonumber \\
~~~~ \mbox{and} ~~~~ \sqrt{n} (\hat{\theta}_{\beta,\gamma} - \theta_0)^T
A_{\beta} (\theta_0) \sqrt{n} (\hat{\theta}_{\beta,\gamma} - \theta_0) \nonumber
\end{eqnarray}

have the same asymptotic distribution. Now we know from
the previous section

that the
asymptotic distribution of $\sqrt{n} (\hat{\theta}_{\beta,\gamma}-\theta_0)$
is normal with mean zero and variance $J_{\beta}^{-1}(\theta_0)
K_{\beta}(\theta_0) J_{\beta}^{-1}(\theta_0)$.
   Further we know that for $X \sim N_q(0,\Sigma)$, and a $q-$dimensional
real symmetric matrix $A$, the distribution of the quadratic form
$X^T A X$ is the same as that of
$\sum \limits_{i=1}^r \zeta_i^{\beta} Z_i^2$, where $Z_1, \ldots ,Z_r$ are independent
standard normal variables, $r = rank(\Sigma A \Sigma)$, $r \ge 1$ and
$\zeta_1^{\beta}, \ldots , \zeta_r^{\beta}$ are the nonzero eigenvalues of $A \Sigma$
(Dik and Gunst, 1985, Corollary 2.1). Applying this result with
$ X =  \sqrt{n} (\hat{\theta}_{\beta,\gamma} - \theta_0)$

the theorem is established. It

is evident that the asymptotic distribution of the statistic depends on $\beta$ only
and is independent of $\gamma$.
\end{proof}
\end{theorem}

\begin{theorem}

An approximation to the power
function of the test statistic $W_{\beta,\gamma}( {\hat{\theta}_{\beta,\gamma}}
, {\theta_0})$ for testing $H_0 : \theta =\theta_0$ against
$H_1: \theta \ne \theta_0$ at the significance level $\alpha$ is given by
\begin{eqnarray}
    \pi_{n,\alpha}^{\beta,\gamma} (\theta^*) =
1 - \Phi \left( \frac{\sqrt{n}}{\sigma_{\beta,\gamma
}(\theta^*)} \left( \frac{t_\alpha^{\beta,\gamma}}{2 n}
- {\rm LSD}_{\beta,\gamma}(f_{\theta^*}, f_{\theta_0}) \right) \right), ~~~ \theta^* \ne \theta_0
\end{eqnarray}
where $t_\alpha^{\beta,\gamma}$ is the $(1-\alpha)^{th}$
quantile of the asymptotic distribution of \\ $W_{\beta,\gamma}
( {\hat{\theta}_{\beta,\gamma}} , {\theta_0})$, and $\sigma_{\beta,\gamma}(\theta^*)$ is defined as
\begin{eqnarray}
\label{EQ:sigma_defn}
    \sigma_{\beta,\gamma}^2(\theta) =
M_{\beta,\gamma}(\theta)^T J_\beta^{-1}(\theta)
K_\beta(\theta) J_\beta^{-1}(\theta) M_{\beta,\gamma}(\theta)
\end{eqnarray}
with
$$
M_{\beta,\gamma}(\theta) = \nabla {\rm LSD}_{\beta,\gamma}
( f_\theta , f_{\theta_0}).

$$
\begin{proof}

Fix some $\theta^* \ne \theta_0$. Consider
the first order Taylor series expansion of
${\rm LSD}_{\beta,\gamma}( f_{\hat{\theta}_{\beta,\gamma}} ,
f_{\theta_0})$ under $f_{\theta^*}$ as
$$
{\rm LSD}_{\beta,\gamma}( f_{\hat{\theta}_{\beta,\gamma}} ,
f_{\theta_0}) = {\rm LSD}_{\beta,\gamma}( f_{\theta^*} ,
f_{\theta_0}) + M_{\beta,\gamma}(\theta^*)^T
(\hat{\theta}_{\beta,\gamma} - \theta^*) + o(||\hat{\theta}_{\beta,\gamma} - \theta^*||)
$$
where $M_{\beta,\gamma}$ is as defined in the theorem. Now
we know that, under $\theta^*$,
$$
\sqrt{n} (\hat{\theta}_{\beta,\gamma} - \theta^*) \rightarrow
N(0, J_\beta^{-1}(\theta^*) K_\beta(\theta^*) J_\beta^{-1}(\theta^*) )
 ~~~as ~ n \rightarrow \infty
$$
and $ \sqrt{n} o(||\hat{\theta}_{\beta,\gamma} - \theta^*||) = o_p(1)$. Thus
we get that the random variables
$$
\sqrt{n} \left[ {\rm LSD}_{\beta,\gamma}( f_{\hat{\theta}_{\beta,\gamma}} ,
f_{\theta_0}) - {\rm LSD}_{\beta,\gamma}( f_{\theta^*} , f_{\theta_0})
\right] ~~~~~\mbox{ and } ~~~~~ M_{\beta,\gamma}(\theta^*)^T
\sqrt{n}(\hat{\theta}_{\beta,\gamma} - \theta^*)
$$
have the same asymtotic distribution . Therefore, we have
$$
\sqrt{n} \left[ {\rm LSD}_{\beta,\gamma}( f_{\hat{\theta}_{\beta,\gamma}} ,
f_{\theta_0}) - {\rm LSD}_{\beta,\gamma}( f_{\theta^*} , f_{\theta_0})
\right] \rightarrow N( 0, \sigma_{\beta,\gamma}(\theta^*) )
$$
where $\sigma_{\beta,\gamma}(\theta^*)$ is as given in
(\ref{EQ:sigma_defn}) above. Hence the desired approximation to the power function
follows from the above asymptotic distribution.
\end{proof}

\end{theorem}

\subsection{Two-Sample Problem}
	Again consider a parametric family of densities
$\{f_\theta : \theta \in \Theta \subseteq \mathbb{R}^p\}$ as above in
one sample problem, but here we are given two random
samples $X_1, \ldots, X_n$ of size $n$ and $Y_1, \ldots, Y_m$ of
size $m$ from two

distributions within the model
having parameters $\theta_1$ and
$\theta_2$ respectively and based on these two samples, we want to
test for the homogeneity of the two samples, i.e. to test the hypothesis
$$
 H_0 : \theta_1 = \theta_2 ~~~ \mbox{against} ~~~~ H_1 : \theta_1 \ne \theta_2.
$$
 We will consider the estimator $^{(1)}\hat{\theta}_{\beta,\gamma}$  and
$^{(2)}\hat{\theta}_{\beta,\gamma}$ of $\theta_1$ and $\theta_2$ respectively,
obtained by minimizing the LSD having parameters ${\beta,\gamma}$ and,

as before,

will
consider the test statistic based on the LSD with parameter $\beta$
and $\gamma$ as
given by

\begin{eqnarray}
    S_{\beta,\gamma}\left( ^{(1)}\hat{\theta}_{\beta,\gamma} ,
^{(2)}\hat{\theta}_{\beta,\gamma}\right)
= \frac{2nm}{n+m} ~  {\rm LSD}_{\beta,\gamma}\left( f_{^{(1)}\hat{\theta}_{\beta,\gamma}} ,
f_{^{(2)}\hat{\theta}_{\beta,\gamma}}\right).
\end{eqnarray}

Now, first let us
consider the asymptotic distribution of the test statistic $S_{\beta,\gamma}
\left( ^{(1)}\hat{\theta}_{\beta,\gamma}, ^{(2)}\hat{\theta}_{\beta,\gamma}\right)$ under $H_0$
in the following theorem.
Assume that $\frac{m}{m+n}\rightarrow \omega, (0<\omega<1), \mbox{ as } m\rightarrow \infty
\mbox{ and } n\rightarrow\infty$.
\begin{theorem}
\label{th_two}

The asymptotic distribution of the test
statistic \\	$S_{\beta,\gamma}\left( ^{(1)}\hat{\theta}_{\beta,\gamma} ,
^{(2)}\hat{\theta}_{\beta,\gamma}\right)$, under the null hypothesis $H_0 : \theta_1 = \theta_2$,
coincides with the distribution of
$$
\sum_{i=1}^r ~  \zeta_i^{\beta}(\theta_1)Z_i^2
$$
where $Z_1, \ldots,Z_r$ are independent standard normal variables,
$\zeta_1^{\beta}(\theta_1), \ldots, \\ \zeta_r^{\beta}(\theta_1)$
are the nonzero eigenvalues of $A_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1)
K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1)$, with \\
$J_{\beta}(\cdot)$, $K_{\beta}(\cdot)$
and $A_{\beta}(\cdot)$ as defined in the previous section and
$$
r = rank\left( J_{\beta}^{-1}(\theta_1)
K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1)
A_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1)
K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1)\right).
$$
\begin{proof}
 We have
$$
\sqrt{n} \left( ^{(1)}\hat{\theta}_{\beta,\gamma} -
\theta_1 \right) \rightarrow N( 0, J_{\beta}^{-1}
(\theta_1) K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1) )
$$
and
$$
\sqrt{m} \left( ^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_2 \right)
\rightarrow N( 0, J_{\beta}^{-1}(\theta_2) K_{\beta}(\theta_2)
J_{\beta}^{-1}(\theta_2) ).
$$
Let $\frac{m}{m+n} \rightarrow \omega \in (0,1)$ as $m,n \rightarrow
\infty $. Then we have
$$
\sqrt{\frac{mn}{m+n}} \left( ^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1
\right) \rightarrow N( 0, \omega J_{\beta}^{-1}(\theta_1)
K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1) )
$$
and
$$
\sqrt{\frac{mn}{m+n}} \left( ^{(2)}\hat{\theta}_{\beta,\gamma} -
\theta_2 \right) \rightarrow N( 0, (1-\omega) J_{\beta}^{-1}
(\theta_2) K_{\beta}(\theta_2) J_{\beta}^{-1}(\theta_2) ).
$$
Now, under $H_0 : \theta_1 = \theta_2$, we get that
$$
\sqrt{\frac{mn}{m+n}} \left( ^{(1)}\hat{\theta}_{\beta,\gamma} -
^{(2)}\hat{\theta}_{\beta,\gamma} \right) \rightarrow N( 0,
J_{\beta}^{-1}(\theta_1) K_{\beta}(\theta_1) J_{\beta}^{-1}(\theta_1) ).
$$
Next consider the second order Taylor series expansion of
${\rm LSD}_{\beta,\gamma}\left( f_{\theta_1} , f_{\theta_2}\right)$ around
$\theta_1 = \theta_2$ at $\left( ^{(1)}\hat{\theta}_{\beta,\gamma} ,
^{(2)}\hat{\theta}_{\beta,\gamma} \right)$ as follows
\begin{eqnarray}
    {\rm LSD}_{\beta,\gamma}\left( f_{^{(1)}\hat{\theta}_{\beta,\gamma}} ,
f_{^{(2)}\hat{\theta}_{\beta,\gamma}}\right) & = & \frac{1}{2} \sum_{i,j=1}^p
\left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}(f_{\theta_1},
f_{\theta_2})}{\partial \theta_{1i} \partial \theta_{1j}}\right)_{\theta_1
= \theta_2} \nonumber \\
& & \left(\hat{\theta}_{\beta,\gamma}^{1i} - \theta_{1i}\right)
\left(\hat{\theta}_{\beta,\gamma}^{1j} - \theta_{1j}\right) \nonumber
\\
& & +  \sum_{i,j=1}^p \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}
(f_{\theta_1}, f_{\theta_2})}{\partial \theta_{1i} \partial
\theta_{2j}}\right)_{\theta_1 = \theta_2}\nonumber \\
& &  \left(\hat{\theta}_{\beta,\gamma}^{1i}
- \theta_{1i}\right) \left(\hat{\theta}_{\beta,\gamma}^{2j} - \theta_{2j}\right) \nonumber
\\
& & +  \frac{1}{2} \sum_{i,j=1}^p \left( \frac{\partial^2
{\rm LSD}_{\beta,\gamma}(f_{\theta_1}, f_{\theta_2})}{\partial
\theta_{2i} \partial \theta_{2j}}\right)_{\theta_1 = \theta_2} \nonumber \\
& &
\left(\hat{\theta}_{\beta,\gamma}^{2i} - \theta_{2i}\right) \left(\hat{\theta}_{\beta,\gamma}^{2j} -
\theta_{2j}\right) \nonumber
\\
& & + o\left( ||^{(1)}\hat{\theta}_{\beta,\gamma} -
\theta_1 ||^2 \right) + o\left( ||^{(2)}\hat{\theta}_{\beta,\gamma} -
\theta_2 ||^2 \right). \nonumber
\end{eqnarray}
But for $ i = 1, \ldots, p$, we have
$$
\frac{\partial  {\rm LSD}_{\beta,\gamma}(f_{\theta_1},
f_{\theta_2})}{\partial \theta_{1i}} = \frac{1+\beta}{B}
\left[\frac{\sum f_{\theta_{1i}}^{1+\beta}u_{\theta_{1i}}}
{\sum f_{\theta_{1i}}^{1+\beta}}
- \frac{\sum f_{\theta_{1i}}^A f_{\theta_2}^B u_{\theta_{1i}}}
{\sum f_{\theta_{1i}}^A f_{\theta_2}^B }\right].

$$
where $B = \beta - \gamma(1-\beta)$ and $A = 1+
\gamma(1-\beta)$ and hence
\begin{eqnarray}
   \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}(f_{\theta_1},
f_{\theta_2})}{\partial \theta_{1i} \partial \theta_{1j}}\right)_{\theta_1 = \theta_2}
&=&

a_{ij}^{\beta}(\theta_1), \nonumber \\
 \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}
(f_{\theta_1}, f_{\theta_2})}{\partial \theta_{1i}
\partial \theta_{2j}}\right)_{\theta_1 = \theta_2}
&=&

- a_{ij}^{\beta}(\theta_1), \nonumber \\
 \left( \frac{\partial^2  {\rm LSD}_{\beta,\gamma}(f_{\theta_1},
f_{\theta_2})}{\partial \theta_{2i} \partial \theta_{2j}}\right)_{\theta_1 = \theta_2}
& =&

a_{ij}^{\beta}(\theta_1). \nonumber
\end{eqnarray}
As in the one sample case, here also the second order partial
derivatives depend on $\beta$ only.
	Therefore, we get
\begin{eqnarray}
    2~{\rm LSD}_{\beta,\gamma}\left( f_{^{(1)}\hat{\theta}_{\beta,\gamma}} ,
f_{^{(2)}\hat{\theta}_{\beta,\gamma}}\right)
 &=& (^{(1)}\hat{\theta}_{\beta,\gamma} -
\theta_1)^T A_{\beta}(\theta_1)(^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1) \nonumber \\
& & -
2 (^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1)^T A_{\beta}(\theta_1)
(^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_1) \nonumber \\
& & + (^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_1)^T A_{\beta}(\theta_1)
(^{(2)}\hat{\theta}_{\beta,\gamma} - \theta_1) \nonumber \\
& & + o\left( ||^{(1)}
\hat{\theta}_{\beta,\gamma} - \theta_1 ||^2 \right)
+ o\left( ||^{(2)}
\hat{\theta}_{\beta,\gamma} - \theta_2 ||^2 \right) \nonumber \\
&=&  (^{(1)}\hat{\theta}_{\beta,\gamma} - ^{(2)}\hat{\theta}_{\beta,\gamma})^T
A_{\beta}(\theta_1)(^{(1)}\hat{\theta}_{\beta,\gamma} -
^{(2)}\hat{\theta}_{\beta,\gamma}) \nonumber \\
& & + o\left( ||^{(1)}\hat{\theta}_{\beta,\gamma}
- \theta_1 ||^2 \right)
+ o\left( ||^{(2)}\hat{\theta}_{\beta,\gamma} \nonumber
- \theta_2 ||^2 \right), \nonumber
\end{eqnarray}
with

$$
o\left( ||^{(1)}\hat{\theta}_{\beta,\gamma} - \theta_1 ||^2 \right) =
o_p\left(\frac{1}{n}\right) ~~~~ \mbox{and} ~~~~  o\left( ||^{(2)}\hat{\theta}_{\beta,\gamma} -
\theta_2 ||^2 \right) = o_p\left(\frac{1}{m}\right).
$$
Thus the asymptotic distribution of
$$
S_{\beta,\gamma}\left( ^{(1)}\hat{\theta}_{\beta,\gamma} , ^{(2)}\hat{\theta}_{\beta,\gamma}\right)
= \frac{2nm}{n+m} ~  {\rm LSD}_{\beta,\gamma}\left( f_{^{(1)}\hat{\theta}_{\beta,\gamma}}
, f_{^{(2)}\hat{\theta}_{\beta,\gamma}}\right)
$$
coincides with the distribution of the random variable
$\sum \limits_{i=1}^r ~  \zeta_i^{{\beta}}(\theta_1)Z_i^2$.
Like the one sample case the asymptotic distribution depends
on $\beta$ only.

\end{proof}
\end{theorem}
We have noted that the asymptotic distribution of the LSD based test statistics
under the simple null hypothesis is independent of the parameter $\gamma$.
Maji, Ghosh and Basu (2014) have also reported a similar observation for the robustness of
the corresponding test statistics. They have shown that the first order influence function
of the test statistics is always zero at the simple null and its second order influence function under null,
being a quadratic form in the first order influence function of the minimum LSD estimator used,
is independent of the parameter $\gamma$. However, the numerical illustrations reported in
their paper and in the next section of present paper, this independence is not true for samples
with moderate size. Therefore, as in the case of estimation, the robustness of the LSD based
test of simple null hypothesis can not e indicated in terms of the influence function analysis
even if we even go up to second order. However, Maji, Ghosh and Basu (2014) showed
that the robustness of the minimum LSD estimators can be measured quite accurately
in terms of the secord order influence function of the estimator.
Extending the same idea in case of testing, it is a routine exercise to
see that the third order influence function of the test statistics at the null,
being a function of the second order influence function of the corresponding estimator,
can serve a better measure of robustness in this case.

In this article we have restricted ourselves
to the simple null case. However the results may be extended
to the case involving nuisance
parameters following the same general approach.
\section{A Two-Sample Example}

\label{SEC:testing:example}
Here we will discuss a two sample real data example which
is known to give rise to occasional spurious counts. This experiment
is available in  Woodruff et al.~(1984) and
has been analyzed previously by
Simpson (1989).
This is a sex-linked recessive lethal experiment in drosophila (fruit flies)
to test chemical mutagenicity. Male flies were exposed either to
$2000~\mu g$ butyraldehyde or to control conditions. The
responses are the numbers of recessive lethal
mutations observed among daughters of these flies.
The data are given in Table \ref{tab:twoDdata}.
We will use a Poisson model in this experiment
where the control responses
are supposed to follow Poisson distribution with mean $\theta_0$
and and the treated responses follows a Poisson distribution with mean $\theta_1$.
The two large counts for the treated group appears to be
possible outliers.
 We want to test
$H_0:\theta_0\geq\theta_1$ against $H_1:\theta_0<\theta_1.$
The test statistic for testing this hypothesis is given by,
$$
^*S_{\beta,\gamma}\left( ^{(1)}\hat{\theta}_{\beta,\gamma},
^{(2)}\hat{\theta}_{\beta,\gamma}\right) = \frac{1}{\zeta(^{(0)}\hat{\theta}_{\beta,\gamma})}
 \frac{2nm}{n+m} ~  {\rm LSD}_{\beta,\gamma}\left( p({^{(1)}\hat{\theta}_{\beta,\gamma}})
, p({^{(2)}\hat{\theta}_{\beta,\gamma}})\right),
$$
where
\begin{eqnarray}
    \zeta(^{(0)}\hat{\theta}_{\beta,\gamma})
&=& \frac{A_{\beta,\gamma}(^{(0)}\hat{\theta}_{\beta,\gamma})
K_{\beta,\gamma}(^{(0)}\hat{\theta}_{\beta,\gamma})}
{J_{\beta,\gamma}^2(^{(0)}\hat{\theta}_{\beta,\gamma})}. \nonumber \\

\end{eqnarray}
The asymptotic distribution of the statistic
$^*S_{\gamma,\lambda}( ^{(1)}\hat{\theta}_\beta
, ^{(2)}\hat{\theta}_\beta)$ is chi-square with one degree of freedom
and the corresponding $p$-values are calculated and shown in the tables.
The statistic for $\beta = \gamma = 0$ case is not same as
the likelihood ratio statistic but they are
asymptotically same. Though the tests are different
but the non-robust nature of the likelihood test can be seen
under this set-up also.
The results are shown in
Tables \ref{tab:est2} and \ref{signed_test}.

From the results it is evident that for $\beta \geq 0.6$ and irrespective of $\gamma$
the presence and the absence of outliers
has little impact on $p$-values.

For $\gamma<0$, lower values of $\beta$ also give
close $p$-values but for $\gamma>0$, lower values of $\beta$ the method does not perform well.
It is clear that for large values of $\beta~(\mbox{say}~\geq 0.5)$,
the full data and the outlier deleted data basically lead to  the same conclusion
and almost identical $p$-values irrespective of the value of $\gamma$.
The situation changes when $\beta$ is a small positive value close to $0$.
In this case the role of $\gamma$ becomes decisive. Large positive values
of $\gamma$ and small values of $\beta$ lead to a highly unstable results.
The outlier deleted $p$-values and full data $p$-values are far from
close in these cases. However the negative values of $\gamma$ lead to
stable inference even when $\beta=0$ or in its neighborhood.
On the whole it appears that the two large counts in the treated
group indicate a false significance for the likelihood ratio
test and some other members of our class, but the
more robust members clearly recognize the significance
to be false.

\begin{table}
	\centering
	\caption{Frequencies of the number of recessive lethal daughters for drosophila data}
		\begin{tabular}{c c c c c c c c c } \hline
		$x$      &0 & 1  &2  &3  &4  &5  &6 & 7 \\ \hline
Observed (Control) & 159 & 15 & 3 & 0 & 0 & 0 & 0 & 0 \\
Observed (Treated) & 110 & 11 & 5 & 0 & 0 & 0 & 1 & 1 \\
 \hline
		\end{tabular}
	 \label{tab:twoDdata}
\end{table}
\begin{table}
	\centering
	\caption{Estimated Poisson parameters for the two-sample drosophila example;
the numbers within the parentheses

show the corresponding estimates for the treated case after deleting
the two outliers.
The parameter $\gamma$ is held at $0$  for the first part
and the parameter $\beta$ is held at $0$  for the second part.

}
	\scalebox{0.85} {	\begin{tabular}

{c c c c c c c c c c c }
\hline
	$\beta$

&    $0.2$

&    $ 0.4 $

&    $0.6 $

&    $ 0.8 $

& $1$ \\ \hline

$^{(1)}\hat{\theta}_\beta$   & 0.1091 & 0.1027 & 0.099 & 0.0969 & 0.0957 \\
$^{(2)}\hat{\theta}_\beta$   & 0.153 & 0.1266 & 0.1143 & 0.1077 & 0.1042 \\
 & (0.1432) & (0.1255) & (0.1141) & (0.1105) & (0.1057) \\
$^{(0)}\hat{\theta}_\beta$   & 0.1264 & 0.1122 & 0.1051 & 0.1029 & 0.1 \\
 & (0.1229) & (0.1118) & (0.105) & (0.1028) & (0.1) \\

 \hline
	$\gamma$

&    $0.2$

&    $ 0.4 $

&    $0.6 $

&    $ 0.8 $

& $1$ \\ \hline
$^{(1)}\hat{\theta}_\gamma$    & 0.1216 & 0.1245 & 0.1273 & 0.1303 & 0.133 \\
$^{(2)}\hat{\theta}_\gamma$    & 0.3227 & 0.4139 & 0.4916 & 0.5547 & 0.6059 \\
 & (0.1763) & (0.1854) & (0.1938) & (0.2015) & (0.2084) \\
$^{(0)}\hat{\theta}_\gamma$    & 0.2182 & 0.2851 & 0.3497 & 0.4055 & 0.4521 \\
  & (0.1444) & (0.1501) & (0.1555) & (0.1607) & (0.1655) \\
\hline
		\end{tabular}}
	 \label{tab:est2}
\end{table}
 \begin{table}

	\centering
	\caption{The

$p$-values for the two-sample drosophila data.
The outlier deleted

$p$-values are given in the second line of
each block

}
\scalebox{0.8}		{\begin{tabular}{c c c c c c c c c c c c} \hline
$\gamma$ \textdownarrow $\beta$ $\rightarrow$
& 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\

\hline

$-0.8$  & 0.262 & 0.341 & 0.426 & 0.509 & 0.582 & 0.642 & 0.695 & 0.738 & 0.775 & 0.806 & 0.831 \\
  & 0.264 & 0.343 & 0.428 & 0.511 & 0.583 & 0.643 & 0.696 & 0.739 & 0.776 & 0.807 & 0.832 \\
$-0.7$  & 0.235 & 0.325 & 0.417 & 0.506 & 0.582 & 0.644 & 0.698 & 0.74 & 0.776 & 0.807 & 0.831 \\
  & 0.239 & 0.327 & 0.42 & 0.508 & 0.584 & 0.646 & 0.7 & 0.741 & 0.777 & 0.807 & 0.832 \\
$-0.6$  & 0.218 & 0.314 & 0.412 & 0.501 & 0.579 & 0.645 & 0.699 & 0.742 & 0.779 & 0.807 & 0.831 \\
  & 0.221 & 0.317 & 0.414 & 0.503 & 0.581 & 0.647 & 0.7 & 0.744 & 0.78 & 0.808 & 0.832 \\
$-0.5$  & 0.207 & 0.306 & 0.406 & 0.499 & 0.578 & 0.646 & 0.7 & 0.745 & 0.779 & 0.807 & 0.831 \\
  & 0.211 & 0.309 & 0.408 & 0.5 & 0.58 & 0.647 & 0.701 & 0.746 & 0.78 & 0.808 & 0.832 \\
$-0.4$  & 0.199 & 0.301 & 0.402 & 0.496 & 0.579 & 0.647 & 0.704 & 0.746 & 0.782 & 0.809 & 0.831 \\
  & 0.208 & 0.305 & 0.404 & 0.497 & 0.581 & 0.649 & 0.705 & 0.747 & 0.783 & 0.81 & 0.832 \\
$-0.3$  & 0.187 & 0.294 & 0.398 & 0.494 & 0.578 & 0.648 & 0.702 & 0.747 & 0.782 & 0.81 & 0.831 \\
  & 0.211 & 0.303 & 0.403 & 0.498 & 0.58 & 0.649 & 0.704 & 0.749 & 0.783 & 0.811 & 0.832 \\
$-0.2$  & 0.149 & 0.284 & 0.395 & 0.493 & 0.578 & 0.647 & 0.705 & 0.749 & 0.785 & 0.81 & 0.831 \\
  & 0.22 & 0.307 & 0.402 & 0.496 & 0.58 & 0.648 & 0.706 & 0.75 & 0.786 & 0.811 & 0.832 \\
$-0.1$  & 0.058 & 0.251 & 0.388 & 0.491 & 0.576 & 0.648 & 0.705 & 0.749 & 0.785 & 0.811 & 0.831 \\
  & 0.235 & 0.315 & 0.404 & 0.497 & 0.579 & 0.651 & 0.706 & 0.75 & 0.786 & 0.812 & 0.832 \\
0 & 0.003 & 0.156 & 0.367 & 0.488 & 0.575 & 0.649 & 0.707 & 0.753 & 0.786 & 0.811 & 0.831 \\
   & 0.141 & 0.328 & 0.411 & 0.496 & 0.578 & 0.65 & 0.708 & 0.754 & 0.787 & 0.812 & 0.832 \\
0.1 & 0 & 0.03 & 0.306 & 0.479 & 0.575 & 0.648 & 0.707 & 0.753 & 0.788 & 0.814 & 0.831 \\
  & 0.293 & 0.345 & 0.419 & 0.5 & 0.579 & 0.65 & 0.708 & 0.754 & 0.789 & 0.814 & 0.832 \\
0.2 & 0 & 0.001 & 0.172 & 0.454 & 0.572 & 0.648 & 0.707 & 0.754 & 0.788 & 0.814 & 0.831 \\
  & 0.338 & 0.369 & 0.429 & 0.504 & 0.58 & 0.651 & 0.708 & 0.755 & 0.789 & 0.815 & 0.832 \\
0.3 & 0 & 0 & 0.037 & 0.391 & 0.566 & 0.648 & 0.708 & 0.754 & 0.79 & 0.815 & 0.831 \\
  & 0.392 & 0.398 & 0.444 & 0.51 & 0.583 & 0.65 & 0.709 & 0.755 & 0.791 & 0.815 & 0.832 \\
0.4 & 0 & 0 & 0.002 & 0.261 & 0.55 & 0.647 & 0.709 & 0.755 & 0.79 & 0.815 & 0.831 \\
  & 0.46 & 0.435 & 0.462 & 0.517 & 0.585 & 0.652 & 0.71 & 0.756 & 0.791 & 0.816 & 0.832 \\
0.5 & 0 & 0 & 0 & 0.101 & 0.513 & 0.644 & 0.709 & 0.757 & 0.792 & 0.816 & 0.831 \\
  & 0.54 & 0.478 & 0.483 & 0.526 & 0.588 & 0.652 & 0.71 & 0.758 & 0.793 & 0.817 & 0.832 \\
0.6 & 0 & 0 & 0 & 0.019 & 0.436 & 0.639 & 0.709 & 0.758 & 0.793 & 0.816 & 0.831 \\
  & 0.629 & 0.528 & 0.508 & 0.539 & 0.592 & 0.653 & 0.711 & 0.759 & 0.794 & 0.817 & 0.832 \\
0.7 & 0 & 0 & 0 & 0.002 & 0.3 & 0.629 & 0.709 & 0.757 & 0.793 & 0.817 & 0.831 \\
  & 0.727 & 0.584 & 0.538 & 0.552 & 0.597 & 0.655 & 0.712 & 0.758 & 0.794 & 0.818 & 0.832 \\
0.8 & 0.003 & 0 & 0 & 0 & 0.147 & 0.605 & 0.709 & 0.758 & 0.794 & 0.817 & 0.831 \\
  & 0.827 & 0.646 & 0.571 & 0.568 & 0.603 & 0.657 & 0.712 & 0.759 & 0.795 & 0.818 & 0.832 \\

\hline
		\end{tabular}}
\label{signed_test}
\end{table}

\section{Conclusion}
\label{Sec_Conclusion}

Logarithmic super divergence family acts as a super family of both LPD and LDPD family.
The theoretical properties of this new family of divergences have been established
for discrete models and similar results under continuous set-up can be done
in subsequent works.

config: arxiv


TITLE: What would a 300% decrease in something actually mean?
QUESTION [3 upvotes]: I was looking at a site search product and one of the testimonials was
300% drop in "no results" searches
What would this manifest as ?
All I can come up with is -2 "no results searches" for every one that did exist but that doesn't make a whole lot of sense.
Or is this likely just a misuse of percentages to make the product look good ?

REPLY [1 votes]: It's just a misuse of percentages. It could be unknowingly, or deliberately, as you just said.

A quantity can not decrease by more than 100% of itself.

By the way, you could have Googled it before asking on SE. Here's the first Google Search result that came up for me: https://answers.yahoo.com/question/index?qid=20110122151445AApNB9I


TITLE: Finding generators of an ideal / showing an ideal is prime
QUESTION [7 upvotes]: Suppose I have an ideal like $\mathrm{rad}(\langle y - x^2, z - x^3 \rangle)$. How do I go about finding generators for the ideal?

If I could show that $\langle y - x^2, z - x^3 \rangle$ is a prime ideal then obviously $y-x^2$ and $z-x^3$ would generate $\mathrm{rad}(\langle y - x^2, z - x^3 \rangle)$, but I don't see how to do this.
Thanks.

REPLY [3 votes]: You could start by showing the ideal $I=(y-x^2,z-x^3)$ is in fact radical so that finding generators for its radical becomes easy!
So let $f\in k[x,y,z]$ be such that there is an $n\geq0$ such that $f^n\in I$.
It is easy to see that there exist $g$, $h\in k[x,y,z]$ and $r\in k[x]$ such that
$$(\clubsuit)\qquad\qquad f(x,y,z)=g(x,y,z)\cdot(y-x^2)+h(x,y,z)\cdot(z-x^3)+r(x),$$
and using Newton's formula for powers of trinomials— we see immediately from this that $$f(x,t,z)^n\equiv r(x)^n\mod I,$$ so that in fact $r(x)^n\in I$. Suppose for a second that we manage to prove that

$(\star)\qquad\qquad$ we have $k[x]\cap I=0.$

Then, since $r(x)^n$ is in that intersection, we have $r(x)^n=0$ and therefore $r(x)=0$. Looking back at $(\clubsuit)$ we see now that $f\in I$, and we are happy.
Can you prove $(\star)$?


text:
\begin{document}

\title{On Lin's condition for products of random variables with joint singular distribution}\author{Alexander Il'inskii$^{1}$ and  Sofiya  Ostrovska$^{2}$}
\date{}

\maketitle

\begin{center}
 {\it $^1$Department of Mathematics and Informatics, Karazin National University, Kharkov, Ukraine\\
 E-mail: iljinskii@univer.kharkov.ua }
\end{center}
\begin{center}
 {\it $^2$Department of  Mathematics, Atilim University, Ankara, TURKEY\\
E-mail: sofia.ostrovska@atilim.edu.tr}\\
\end{center}

\begin{abstract} Lin's condition is used to establish the moment determinacy/indeterminacy of absolutely continuous probability distributions. Recently, a number of papers related to Lin's condition for functions of random variables have emerged. In this work, Lin's condition is studied for the product of random variables with given densities in the case when their joint distribution is singular.\end{abstract}

\noindent {\small \textbf{Keywords:} random variable, singular  distribution, Lin's condition
\vspace{0.2cm}}

\noindent {\small \textbf{Mathematics Subject Classifications:}
60E05}\vspace{0.2cm}

\section{Introduction}

Lin's condition plays a significant role in the investigation of the moment determinacy of absolutely continuous probability distributions. Generally speaking, it is used along with Krein's logarithmic integral to establish so-called ``converse criteria". See, for example, Theorems 5--10 in \cite[Section 5]{recent}.
This condition was introduced and applied by G. D. Lin \cite{lin}, while the name `Lin's condition' was put forth in
\cite{bernoulli}. Let us recall the pertinent notions.

\begin{definition} Let $f$ be a probability density continuously differentiable on $(0,\infty)$. The function
\begin{equation} \label{linfun}
L_f(x):=-\frac{xf^\prime(x)}{f(x)}
\end{equation}
is called \textit{Lin's function} of $f$.
\end{definition}

Formula \eqref{linfun} implies that Lin's function of $f$ is defined only at the points where $f$ does not vanish. In this article, we consider   probability densities of positive random variables whose Lin's functions are defined for all $x>0.$

\begin{definition}\label{lincond} Let $f\in C^1(0,\infty)$ be a probability density of a positive random variable. It is said that $f$ satisfies \textit{ Lin's condition} on $(x_0,\infty)$ if $L_f(x)$ is monotone increasing on $(x_0,\infty)$ and $\displaystyle \lim_{x\rightarrow +\infty}L_f(x)=+\infty$.
\end{definition}

 Due to the importance of this condition for establishing the moment (in)determinacy of absolutely continuous probability distributions (see, \cite{recent, stirzaker} and references therein), it has  to be examined how operations on random variables impact Lin's condition. Recently, Kopanov and Stoyanov in \cite{kopanov} proved that if a density $f$ of a positive random variable $X$ satisfies Lin's condition, then the densities of $X^r, r>0$ and $\ln X$  satisfy this condition, too. In addition, if $L_f(x)/x\rightarrow +\infty$ as $x\rightarrow +\infty,$ then the density of $e^X$ also satisfies Lin's condition as well.

The moment problem for products of random variables leads naturally to the
question concerning Lin's condition for the densities of products. This question was placed in
\cite{kopanov}, where it was inquired whether Lin's condition is inherited by the products of the random variables whose densities satisfy Lin's condition. It was also conjectured that the answer is affirmative   in the case of an absolutely continuous joint distribution. Although this is valid for independent random variables, the results of \cite{arxiv} show that, in general, this assertion may not be true. For more information  on the moment problem for products we refer to \cite{linst1, linst2, counter} and \cite[Section 6]{recent}.

In the present paper, this problem is investigated for the case when joint distribution of factors is singular rather than absolutely continuous. The main result of this work is the next statement.

\begin{theorem}\label{th1} Let  $f_1$ and $f_2$  be  densities of  positive random variables, both satisfying Lin's condition on $(0,+\infty)$.
Then, there exists a random vector $(\xi_1,\xi_2)$ possessing a singular distribution $P$ and satisfying the following conditions:
\begin{enumerate} \item $\xi_1$ and $\xi_2$ have densities $f_1$ and $f_2$ respectively;

\item the density $g$ of the product $\xi_1\cdot\xi_2$ is continuously differentiable on $(0,+\infty);$

\item the following equalities hold: \begin{equation}\label{limsupinf}
\limsup_{x\rightarrow +\infty} L_g(x)=+\infty\,,\quad\liminf_{x\rightarrow +\infty} L_g(x)=-\infty\,.
\end{equation}
\end{enumerate}
\end{theorem}

Obviously, equalities \eqref{limsupinf} imply that $g$ does not satisfy Lin's condition on any interval $(x_0,\infty)$.

\begin{remark} Assuming that  the densities of random variables of $\xi_1$ and $\xi_2$ satisfy Lin's condition on $(0,+\infty),$ we deduce that, in general, this property is not inherited by their product $\xi_1\xi_2$ in the case of a singular joint distribution of $\xi_1$ and $\xi_2.$\end{remark}

\section{Proof of the main theorem}

We start with the construction of a singular distribution with given marginal positive densities, which will be used in the sequel.

\begin{lemma}\label{lem1} Let $f_1$ and $f_2$ be continuous probability densities positive on $(0,+\infty)$ and vanishing elsewhere. Then, there exists a singular distribution $\tilde{P}$ concentrated on a curve $x_2=\varphi(x_1)$, where $\varphi$  is a continuously differentiable  strictly increasing function, and so that the projections of $\tilde{P}$ on the coordinate axes have the given densities $f_1(x_1)$ and $f_2(x_2)$.\end{lemma}

\begin{proof} To begin with, consider the set of points $\{u_{ij}:i\in \mathbb{N}_0, 0\leqslant j\leqslant 2^i\}$ defined by the conditions:
\begin{equation}\label{upoints}
u_{i0}=0,\quad\int_{u_{ij}}^{u_{i,j+1}}f_1(x_1)dx_1=\frac{1}{2^i}\;\;(j=0,1,\dots ,2^i-1),\quad u_{i,2^i}=+\infty\,.
\end{equation}
Clearly \eqref{upoints} defines all of the points $u_{ij}$ uniquely. Besides, for all $i\in \mathbb{N}_0,$ one has:
\begin{equation}\label{ineq} 0=u_{i0}<u_{i1}<\dots <u_{i, 2^i}=+\infty\;\;\mathrm{and}\;\; u_{ij}=u_{i+1, 2j}, \;0\leq j\leq 2^i.\end{equation}
Likewise, let  $\{v_{ij}:i\in \mathbb{N}_0, 0\leqslant j\leqslant 2^i\}$ be the set of points specified by  the conditions:
\begin{equation*}\label{vpoints}
v_{i0}=0,\quad\int_{v_{ij}}^{v_{i,j+1}}f_2(x_2)dx_2=\frac{1}{2^i}\;\;(j=0,1,\dots ,2^i-1),\quad v_{i,2^i}=+\infty\,.
\end{equation*}
Conditions similar to \eqref{ineq} are satisfied for these points, too.

For each $m\in \mathbb{N},$ consider the set
\begin{equation*}\label{km}
K_m:=\bigcup_{j=0}^{2^m-1}(u_{mj},u_{m,j+1})\times
(v_{mj},v_{m,j+1})\subset \mathbb{R}^2
\end{equation*}
and the two-dimensional probability distribution $P_m$ whose density is given by:
\begin{equation*}\label{denfm}f_m(x_1,x_2)=\left\{\begin{array}{cl}2^m f_1(x_1)f_2(x_2)&\text{when}\;\;(x_1,x_2)\in K_m,\\
0&\text{elsewhere}.\end{array}\right.\end{equation*}
The projections of $P_m$ on the $x_1$- and $x_2$-axes have densities $f_1(x_1)$ and $f_2(x_2)$, respectively. Indeed, for any given $m\in \mathbb{N},$ opt for $x_1^*\in (u_{mj}, u_{m,j+1})$ and find $$\int_{-\infty}^{+\infty} f_m(x_1^*,x_2)dx_2=
2^m f_1(x_1^*)\int_{v_{mj}}^{v_{m,j+1}}f_2(x_2)dx_2=f_1(x_1^*).$$ In the same manner, it can be established that the projection of $P_m$ on the $x_2$-axis has density $f_2.$
The sequence $P_m$ of probability distributions is weakly convergent to a two-di\-men\-sional distribution $\tilde{P}$ whose support $\displaystyle K=\cap_{m=1}^\infty K_m $ has zero Lebesgue's measure in $\mathbb{R}^2.$ What is more, $K$ is a continuous curve passing through all points $(u_{ij}, v_{ij}).$

Next, let us derive an equation of $x_2=\varphi(x_1)$ of the curve $K.$ Select a point $(a,b)\in K$ and consider the sequence of rectangles $I_m=J_m\times H_m,$ where $J_m=(u_{mj},u_{m,j+1}), H_m=(v_{mj}, v_{m,j+1})\;\text{with}\;j\; \text{such\; that}\; (a,b)\in I_m.$ Directly from  the construction of sets $K_m,$ one has:
$$
\int_{J_m}f_1(x_1)dx_1=\int_{H_m}f_2(x_2)dx_2,
$$
whence, by the Mean Value Theorem:
$$
f_1(x_1^*)|J_m|=f_2(x_2^*)|H_m|\;\text{for\;some}\;x_1^*\in J_m,\;x_2^*\in H_m.
$$
Consequently,
$$
\frac{|H_m|}{|J_m|}=\frac{f_1(x_1^*)}{f_2(x_2^*)}.
$$
Since all rectangles $I_m$ contain $(a,b),$ taking $m\to\infty,$ implies that $x_1^*\to a, x_2^*\to b$ and the ratio $\displaystyle\frac{|H_m|}{|J_m|}$ approaches the slope $k$ of the tangent line to $K$ at $(a,b).$  Thence, after passing to limit as $m\to\infty,$ one obtains:
$$k=\frac{f_1(a)}{f_2(b)}$$ or, equivalently, $$\varphi^\prime(a)=\frac{f_1(a)}{f_2(\varphi(a))}. $$ Due to the arbitrary selection of $a$, it follows that $x_2=\varphi(x_1)$ satisfies the initial value problem below:
\begin{equation}
\label{ivp}(x_2)^\prime=\frac{f_1(x_1)}{f_2(x_2)},\quad x_2(0)=0.
\end{equation}
Since $\varphi^\prime(x_1)>0$ for all $x_1\in (0,\infty),$ it follows that $\varphi$ is strictly increasing.
\end{proof}

\begin{example}
If $f_1=f_2$, that is random variables $\xi_1$ and $\xi_2$ are identically distributed, then $K$ is the straight line $x_2=x_1.$\end{example}

\begin{lemma}\label{lem2}Let $(\xi_1,\xi_2)$ be a random vector with distribution $\tilde{P}$ constructed  above. Then, the density
$g(z,\tilde{P})$ of the product $\xi_1\cdot \xi_2$ is given by:
\begin{equation}\label{pz}
g(z,\tilde{P})=f_1(x_1(z))\frac{x_1(z)}{z+\varphi^\prime(x_1(z))x_1^2(z)}.
\end{equation}\end{lemma}

\begin{proof} First, consider the distribution function of the product $\xi_1\cdot \xi_2$:
$$
F(z)=\textbf{P}\{\xi_1\cdot\xi_2<z\},\;z>0
$$
and the set of hyperbolae in the first quadrant
$$
\Gamma_z:=\{(x_1,x_2):x_1>0,x_2>0,x_1x_2=z\}.
$$
Given $z>0,$ denote $K\cap\Gamma_z=(x_1(z),x_2(z)).$ Select a small $h>0$ and obtain:
\begin{equation*}
\begin{split}
F(z+h)-F(z)&=\textbf{P}\big((\xi_1,\xi_2)\in \{(x_1,x_2):
x_2=\varphi(x_1),\;x_1(z)\leqslant x_1\leqslant x_1(z+h)\}\big)\\
&=\int_{x_1(z)}^{x_1(z+h)}f_1(x_1)dx_1=\int_{x_2(z)}^{x_2(z+h)}f_2(x_2)dx_2.
\end{split}
\end{equation*}
Applying the Mean Value Theorem yields:
$$
f_1(x_1^*)\left[x_1(z+h)-x_1(z)\right]=f_2(x_2^*)\left[x_2(z+h)-x_2(z)\right]
$$
for some $x_1^*$ and $x_2^*$
within the intervals of integration. As $h\to 0,$ one has: $x_1^*\to x_1(z)$ and, correspondingly,  $f_1(x_1^*)\to f_1(x_1(z)).$
Now, one can write that
$$
x_2(z+h)-x_2(z)=\left[\varphi^\prime(x_1(z))+o(1)\right]\left(x_1(z+h)-x_1(z)\right),
$$
while, on the other hand,
$$
x_2(z+h)-x_2(z)=\frac{z+h}{x_1(z+h)}-\frac{z}{x_1(z)}.
$$
Equating the expressions in the right-hand sides of the last two formulae leads to:
\begin{align*}
x_1(z+h)-x_1(z)=\frac{hx_1(z)}{z+\left[\varphi^\prime(x_1(z))+o(1)\right]x_1(z+h)x_1(z)}\,.
\end{align*}
Passing to limit as $h\rightarrow 0,$ one arrives at:
\begin{equation*}
\begin{split}
g(z,P)&=\lim_{h\to 0}\frac{F(z+h)-F(z)}{h}=\lim_{h\to 0} f_1(x_1^*)\frac{x_1(z)}{z+\left[\varphi^\prime(x_1(z))+o(1)\right] x_1(z+h)x_1(z)}\\
&=f_1(x_1(z))\frac{x_1(z)}{z+\varphi^\prime(x_1(z))x_1^2(z)}\,.
\end{split}
\end{equation*}
As a result, we obtain the density $g(z,P)$ of the product $\xi_1\cdot \xi_2$ in the form \eqref{pz}.
Similarly,  the expression for $g(z, P)$ in terms of $f_2$ can be derived.\end{proof}

\begin{example}\label{ex2}If $f_1=f_2=f,$ then $K=\{(x_1, x_2):x_1=x_2\},\;x_1(z)=\sqrt{z}$, and \eqref{pz} implies that
$$
g(z,\tilde P)=\frac{f(\sqrt{z})}{2\sqrt{z}},
$$
whence
$$
L_g(z)=\frac{1}{2}+\frac{1}{2}L_{f}(\sqrt{z}),
$$
demonstrating that $g$ satisfies Lin's condition whenever $f_1$ does.\end{example}

Finally, we present the proof of Theorem \ref{th1}.

\begin{proof}
To construct a singular probability distribution $P$ in $\mathbb{R}^2$ satisfying the conditions of Theorem \ref{th1}, first consider its construction inside a single rectangle $\Pi=[a,b]\times [\varphi (a), \varphi(b)].$
Denote by $Q$ the restriction of  distribution $\tilde P$ constructed in Lemma \ref{lem1} on rectangle $\Pi$.
Select a small $\delta >0$. Conditions on the value of $\delta$ will be specified later.

For each $x>0$ and $z>0,$ define the function $x=\rho(z)$ by the condition:
 \begin{equation}\label{rhoz}z=x\varphi(x).\end{equation}
  Obviously, $\rho$ is a continuous strictly increasing function on $(0,\infty).$

Then $a=\rho (s), b=\rho (t)$
for some $s<t$, and the curve $K$ has the following parametric representation in terms of $z$:
$$
\overrightarrow{\bf{x}}(z)= \left(x_1(z), x_2(z)\right)=\left(\rho(z),\varphi(\rho(z))\right).
$$
Choose $b^{\,\prime}=\rho(t-3\delta)<b$ close to $b$ and consider a non-negative function $ \tau\in C^\infty(0,+\infty)$ satisfying the conditions:
\begin{align}\label{tau}
\tau(z)=\left\{\begin{array}{ccl}0&\text{when}&z\notin [t-3\delta, t],\\
\varepsilon \sin^2(\nu z)&\text{when}&z\in [t-2\delta, t-\delta].\end{array}\right.
\end{align}
Here $\varepsilon$ and $\nu$ are positive real numbers satisfying certain restrictions which will be explained in the sequel.
Consider the arc
\begin{align*}\label{arcl}
l_\delta^{(1)}=\{(x_1,x_2)\in K: b^{\,\prime}\leqslant x_1\leqslant b\}
=\{\overrightarrow{\bf{x}}(z):t-3\delta\leqslant z\leqslant t\}
\end{align*}
and determine on the arc measure $\mu_1$ whose density with respect to $z$ equals
$\tau(z).$
Select $\varepsilon$ in \eqref{tau} so that $Q-\mu_1$ is a measure in $\Pi.$
Next, shift the
mass on $l_\delta^{(1)}$
with density $\tau(z)$
by $d=\varphi (b^{\,\prime}) - \varphi (a)$ units down so that the shifted arc $l_\delta^{(2)}=l_\delta^{(1)}-(0,d)$ has starting point at $(b^{\,\prime}, \varphi(a))$ and terminal point at $(b,\varphi (b)-d).$
Denote this measure $\mu_2$:
$$
\mu_2(B)=\mu_1(B+(0,d))
$$
where
$B$ is a Borel set in the plane and
$B+(0,d)= \{(x_1,x_2+d):(x_1,x_2)\in B\}.$
Arc $l_\delta^{(1)}$ has to be small enough
to ensure that
$\varphi(b)-\varphi(b^{\,\prime})<\varphi(b^{\,\prime})-\varphi(a)$ (this follows from the smallness of $\delta$).
The performed shifting defines a singular
measure $q_1$ in  rectangle $\Pi$ by:
\begin{equation*}
q_1(B)=(Q-\mu_1)(B)+\mu_2(B),
\end{equation*}
The construction of $q_1$ implies that the projections of $Q$ and $q_1$ on the $x_1$-axis are the same.
However, to guarantee the equal projections on both coordinate axes, the procedure has to be continued. Consider measure $\mu_2$ concentrated on $l_\delta^{(2)}$ and find $a^{\,\prime}>a$
from the condition $\varphi(a^{\,\prime})=\varphi(a)+(\varphi(b)-\varphi(b^{\,\prime})).$ When $\delta$ is small enough, it can be stated that $a^{\,\prime}<b^{\,\prime}.$ Let $\mu_3$ be the measure on arc
\begin{equation*}
l_\delta^{(3)}=\{(x_1,x_2):x_2=\varphi(x_1), a\leqslant x_1\leqslant a^{\,\prime}\}
\end{equation*}
possessing the same projection on the $x_2$-axis as $\mu_2.$
Measure $\mu_3$ is well-defined since function $\varphi$ is strictly increasing. It has to be noticed that, when $\varepsilon >0$ is small enough, the difference $Q-\mu_3$ is a measure on
$l_\delta^{(3)}.$ Finally, shift the mass defined by $\mu_3$ on $l_\delta^{(3)}$ as a whole by $d$ units up, that is, define measure $\mu_4(B)=\mu_3(B-(0,d))$ concentrated on $l_\delta^{(4)}=l_\delta^{(3)}+(0,d)$. As a result, we obtain a singular
measure on rectangle $\Pi=[a,b]\times [\varphi (a), \varphi(b)]$ given by:
\begin{equation*}\label{pnew}
\tilde{Q}=Q-\mu_1+\mu_2-\mu_3+\mu_4\,.
\end{equation*}
Here, $\delta$ and $\varepsilon$ are assumed to be sufficiently small to stipulate all the conditions mentioned above and the positivity of $\tilde{Q}.$ In addition, the projections of $\tilde{Q}$ on the coordinate axes coincide with the distributions of $\xi_1$ and $\xi_2$
on  $[a,b]$ and  $[\varphi(a),\varphi(b)]$ respectively,
that is, possess the given densities $f_1$ and $f_2$
on  $[a,b]$ and  $[\varphi(a),\varphi(b)]$. As before, denote by $g(z,\tilde Q)$ the density of $\xi_1\cdot\xi_2$ in the case when a joint distribution of $\xi_1$ and $\xi_2$ coincides with $\tilde Q$ in $\Pi.$
If $\delta >0$ is  small enough to ensure that arcs $l_\delta^{(2)}$ and $l_\delta^{(4)}$ have no intercepts with $\Gamma_z,$ then $g(z,\tilde Q)=g(z,\tilde P)$,  where $g(z,\tilde P)$ is given by \eqref{pz}.  To prove that  when $\nu$ is large enough, the derivative of $g(z,\tilde Q)$ with respect to $z$ can take  arbitrary large positive and large negative values, first notice that,
  for $z\in [t-2\delta, t-\delta]$ the following equality holds:
\begin{equation*}
g(z,\tilde Q)=(f_1(x_1(z))-\varepsilon \sin^2(\nu z))\frac{x_1(z)}{z+\varphi^\prime(x_1(z))x_1^2(z)}.
\end{equation*} Applying \eqref{rhoz}, this can be restated as follows:
\begin{equation}\label{gzgz}
g(z,\tilde Q)=(f_1(\rho(z))-\varepsilon \sin^2(\nu z))\frac{\rho(z)}{z+\varphi^\prime(\rho(z))\rho^2(z)}.
\end{equation}
It has to be pointed out that $x_1=\rho(z)$ is a continuously differentiable function of $z$. To check this, recall that $\rho(z)\cdot \varphi(\rho(z))=z,$ whence
\begin{align*} \rho(z+h)-\rho(z)=\frac{z+h}{\varphi(\rho(z+h))}-\frac{z}{\varphi(\rho(z))}\\
=\frac{-z\left(\varphi(\rho(z+h))-\varphi(\rho(z))\right)+h\varphi(\rho(z))}{
\varphi(\rho(z+h))\varphi(\rho(z))}\\
=\frac{-z\varphi(\rho(z^*))\left[(\rho(z+h))-(\rho(z))\right]+h\varphi(\rho(z))}{
\varphi(\rho(z+h))\varphi(\rho(z))}.\end{align*}
The last equality is justified by The Mean Value Theorem since $\varphi$ is a differentiable function. By plain calculations one can obtain that:
\begin{equation*}\frac{\rho(z+h)-\rho(z)}{h}=\frac{\varphi(\rho(z))}{z\varphi(\rho(z^*))+\varphi(\rho(z+h))
\varphi(\rho(z))}.\end{equation*}
As $h\rightarrow 0,$ this leads to
\begin{equation*} \frac{d}{dz}\rho(z)=\frac{\varphi(\rho(z))}{z\varphi^\prime (\rho(z))+\varphi^2(\rho(z))}.\end{equation*}
Notice that the denominator in the right-hand side is strictly positive. Furthermore, with the help of \eqref{ivp} it can be concluded that $\varphi^{\prime\prime}$ exists, too.

Now, consider Lin's function for density \eqref{gzgz}:
\begin{equation*}L_g(z)=-z\frac{g^\prime(z;\tilde Q)}{g(z;\tilde Q)}.\end{equation*}
For $z\in [t-2\delta, t-\delta]$, the ratio $z/g(z;\tilde Q)$ is bounded, while the behaviour of
$g^\prime(z;\tilde Q)$ is governed by $\left(\varepsilon \sin^2(\nu z)\right)^\prime=\nu \varepsilon \sin (2\nu z),$ which oscillates rapidly taking arbitrary large in magnitude positive and negative values on $[t-2\delta, t-\delta]$ when $\nu$ is sufficiently large.

\bigskip Finally, in order to find a joint distribution of $\xi_1$ and $\xi_2,$ consider an infinite sequence of a disjoint
rectangles $\{\Pi_n\},\;\Pi_n=[a_n,b_n]\times [\varphi(a_n),\varphi(b_n)], a_n\to \infty.$
Denote by  $Q_n, n\in \mathbb{N},$ the restriction of measure $P$ on rectangle $\Pi_n$.
For each rectangle $\Pi_n,$ construct $\tilde{Q}_n$ as described above, making, at every step the derivatives $p^\prime(z_n^*)$ and $p^\prime(z_n^{**})$ as large in magnitude as necessary and of opposite signs. To complete the picture, set
\begin{equation*}
 P= \tilde P-\sum_{n=1}^{\infty}Q_n+ \sum_{n=1}^\infty\tilde Q_n.
\end{equation*}
It is clear that measure $ P$ satisfies all conditions stated in Theorem~\ref{th1}. \end{proof}


text:
\section{Related Work}
\subsection{IoT Privacy}

Due to the importance of privacy in smart cities, connected vehicles, smart homes, and connected healthcare devices, there has been many works on inventing new methods or improving the existing ones~\cite{wernke2014classification, wang2015privacy, niu2015enhancing, cai2015cloaking,kido2005anonymous, lu2008pad, shokri2011quantifying, shokri2011quantifying2, shokri2012protecting, wang2015toward, 8ukil2015privacy, 12sivaraman2015network, 10harris2016security, 9dalipi2016security, 11al2015security, 14sadeghi2015security}. The PPM mechanisms used in these papers can be broken down into two  main subcategories: identity perturbation PPMs (anonymization) ~\cite{1corser2016evaluating,hoh2005protecting,freudiger2007mix, ma2009location, shokri2011quantifying, Naini2016} and data perturbation LPMs (obfuscation)~\cite{shokri2012protecting, gruteser2003anonymous,  bordenabe2014optimal}. ~\cite{hoh2005protecting, beresford2003location, freudiger2009optimal, beresford2003location, palanisamy2011mobimix} used identity perturbation PPMs and changed users' psedudonyms within mix-zones area. ~\cite{18shokri2014hiding,8zurbaran2015near,hoh2007preserving, gruteser2003anonymous, gedik2005location, gedik2008protecting, bordenabe2014optimal, duckham2005formal, duckham2006spatiotemporal, xue2009location, wernke2014classification, cai2015cloaking, chow2011spatial, cheng2006preserving, mokbel2006new, kalnis2006preserving, khoshgozaran2011location, bamba2008supporting, um2010advanced, zhangwei2010distributed} used cloaking technique which is a location perturbation PPMs to achieve  privacy or ~\cite{kido2005protection, krumm2009survey, shankar2009privately, chow2009faking, kido2005anonymous, lu2008pad} added dummy data to perturb users' data sets.

Also, some approaches are propose for PPM by using differential privacy. In particular, differential privacy aims at protecting queries on aggregated data.
Several identity perturbation~\cite{lee2012differential, bordenabe2014optimal, chatzikokolakis2015geo, nguyen2013differential, machanavajjhala2008privacy} and location perturbation ~\cite{chatzikokolakis2013broadening,shokri2014optimal, chatzikokolakis2015location,andres2013geo,bordenabe2014optimal} PPMs employed differential privacy techniques in order to achieve privacy.

~\cite{shokri2011quantifying, shokri2011quantifying2, ma2009location, shokri2012protecting, 6li2016privacy, 4olteanu2016quantifying, 14zhang2014privacy, diff2017} pursued quantitative approaches toward privacy, yet all these works lack a solid theoretical framework for privacy. In ~\cite{salamatian2013hide, csiszar1996almost, calmon2015fundamental, sankar2013utility, DBLP:journals/corr/LiaoSCT17,sankar, yamamoto1983source, hyposankar} mutual information is considered as privacy metric, but in different framework and problem formulation in comparison to our work. Here, we employ both anonymization and obfuscation techniques as privacy-preserving mechanism for a large set of time-series data that belong to different users with different statistical patterns.

mentioned studies illustrate the growing importance of privacy, but none of them consider the effect of presence of correlation in achieving privacy.

\subsection{Privacy in the Presence of Correlation}

In \cite{corgeo}, the obfuscation technique is employed to achieve privacy, however, in the case of continuous LBS queries, there are some correlation between locations. This paper shows the effect of exploiting the correlation of the user's obfuscated locations on the location privacy. Adaptive noise level is employed to achieve more privacy while still has a an acceptable utility level. In \cite{cordummy}, spatiotemporal correlation between neighboring location sets ruin the privacy that achieved using dummy-based LPPM, so to solve this problem, a spatiotemporal correlation-aware privacy protection is proposed to perturb the spatiotemporal correlation between neighboring locations. In \cite{corPLP}, Protecting Location Privacy (PLP)  against correlation-analysis attack in crowd sensing is employed in order to model the potential correlation between user's locations and filter the ones that disclose the user's private locations. In \cite{cordiff}, locations of only one user are temporally correlated and $\delta$-location set based differential privacy is proposed to achieve location privacy at every timestamps. All of the mentioned studies don't consider the correlation between multi-users, and only consider data correlation about one user.

However, ~\cite{cordiff8, cordata1, cordiff2, cordiff3, cordiff4} consider multi-user location correlations which is used to obtain social correlations between multiple users.  The main purpose of these papers is protecting queries on aggregated data. \cite{cordiff8} shows ignoring the correlation between data can impact privacy of released data. In \cite{cordiff2}, location correlation coefficient is used to quantifying the correlation between two users and then a privacy trajectory mechanism based on differential privacy is proposed to protect location correlation between two users. One of the main drawback of this paper is that the correlation between more than two users aren't considered. In addition, the authors assumed the adversary don't have access to association graph between users.