# A unified study for estimation of order restricted location/scale parameters
under the generalized Pitman nearness criterion

Naresh Garg and Neeraj Misra
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Kanpur-208016, Uttar Pradesh, India

## Abstract

We consider component-wise estimation of order restricted location/scale
parameters of a general bivariate location/scale distribution under the
generalized Pitman nearness criterion (GPN). We develop some general results
that, in many situations, are useful in finding improvements over
location/scale equivariant estimators. In particular, under certain
conditions, these general results provide improvements over the unrestricted
Pitman nearest location/scale equivariant estimators and restricted maximum
likelihood estimators. The usefulness of the obtained results is illustrated
through their applications to specific probability models. A simulation study
has been considered to compare how well different estimators perform under the
GPN criterion with a specific loss function.

Keywords: Generalised Pitman Nearness (GPN) Criterion; Location equivariant
estimator; Pitman Nearness (PN) Criterion; Restricted Parameter Space; Scale
equivariant estimator; Unrestricted Parameter Space.

##  1\. Introduction

The problem of estimating real-valued parameters of a set of distributions,
when it is known apriori that these parameters follow certain order
restrictions, is of great relevance. For example, in a clinical trial, where
estimating average blood pressures of two groups of Hypertension patients, one
treated with a standard drug and the other with a placebo, is of interest, it
can be assumed that the average blood pressure of Hypertension patients
treated with the standard drug is lower than the average blood pressure of
Hypertension patients treated with the placebo. Such estimation problems have
been extensively studied in the literature. For a detailed account of work
carried out in this area, one may refer to Barlow et al. (1972), Robertson et
al. (1988) and van Eeden (2006).

Early work on estimation of order restricted parameters deals with obtaining
isotonic regression estimators or restricted maximum likelihood estimators
(MLE) (see Brunk (1955), van Eeden (, , 1957, 1958) and Robertson et al.
(1988)). Subsequently, a lot of work was carried out using the decision
theoretic approach under different loss functions. Some of the key
contributions in this direction are due to Katz (1963), Cohen and Sackrowitz
(1970), Brewster-Zidek (1974), Lee (1981), Kumar and Sharma (1988, 1989,1992),
Kelly (1989), Kushary and Cohen (1989), Kaur and Singh (1991), Gupta and Singh
(1992), Vijayasree and Singh (1993), Hwang and Peddada (1994), Kubokawa and
Saleh (1994), Vijayasree et al. (1995), Misra and Dhariyal (1995), Garren
(2000), Misra et al. (2002, 2004), Peddada et al. (2005), Chang and Shinozaki
(2015) and Patra (2017).

A popular alternative criterion to compare different estimators is the Pitman
nearness criterion, due to Pitman (1937). This criterion of comparing two
estimators is based on the probability that one estimator is closer to the
estimand than the other estimator under the $L_{1}$ distance (i.e., absolute
error loss function). Rao (1981) pointed out advantages of the Pitman nearness
(PN) criterion over the mean squared error criterion. Keating (1985) further
advocated Rao’s findings through certain examples. Keating and Mason (1985)
provided some practical examples where the PN criterion is more relevant than
minimizing the risk function. Also, Peddada (1985) and Rao et al. (1986)
extended the notion of PN criterion by defining the generalised Pitman
Criterion (GPN) based on a general loss function (in place of the absolute
error loss function). For a detailed study of the PN criterion, one may check
out the monograph by Keating et al. (1993).

The PN criterion has been extensively used in the literature for different
estimation problems (see Nayak (1990) and Keating (1993)). However, there are
only a limited number of studies on use of the PN criterion in estimating
order restricted parameters. For component-wise estimation of order restricted
means of two independent normal distributions, having a common unknown
variance, Gupta and Singh (1992) showed that restricted MLEs are nearer to the
respective population means than unrestricted MLEs under the PN criterion.
Analogous result was also proved for the estimation of the common variance,
which is also considered in Misra et al. (2004). Chang et al. (2017, 2020)
considered estimation of order restricted means of a bivariate normal
distribution, having a known covariance matrix, and established that, under a
modified PN criterion, the restricted MLEs are nearer to respective population
means than some of the estimators proposed by Hwang and Peddada (1994) and Tan
and Peddada (2000). Ma and Liu (2014) considered the problem of estimating
order restricted scale parameters of two independent gamma distributions.
Under the PN criterion, they have compared restricted MLEs of scale parameters
with the best unbiased estimators. Some other studies in this direction are
due to Misra and van der Meulen (1997), Misra et al. (2004), and Chang and
Shinozaki (2015). Most of these studies are centered around specific
probability distributions (mostly, normal and gamma) and the absolute error
loss in the PN criterion. In this paper, we aim to unify these studies by
considering the problem of estimating order restricted location/scale
parameters of a general bivariate location/scale model under the GPN criterion
with a general loss function. We will develop some general results that, in
certain situations, are useful in finding improvements over location/scale
equivariant estimators. As a consequence of these general results, we will
obtain estimators improving upon the unrestricted Pitman nearest
location/scale equivariant estimators (PNLEE/PNSEE). We also consider some
applications of these general results to specific probability models and
obtain improvements over the PNLEE/PNSEE and the restricted MLEs.

The rest of the paper is organized as follows. In Section 2, we introduce some
useful notations, definitions, and results that are used later in the paper.
Sections 3.1 and 3.2 (4.1 and 4.2), respectively, deal with estimating the
smaller and larger location (scale) parameters under the criterion of GPN. In
Section 5, we present a simulation study to compare performances of various
competing estimators.

##  2\. Some Useful Notations, Definitions and Results

The following notion of the Pitman nearness criterion was first introduced by
Pitman (1937).

Definition 2.1 Let $\mathbb{X}$ be a random vector having a distribution
involving an unknown parameter $\boldsymbol{\theta}\in\Theta$
($\boldsymbol{\theta}$ may be vector valued). Let $\delta_{1}$ and
$\delta_{2}$ be two estimators of a real-valued estimand
$\tau(\boldsymbol{\theta})$. Then, the Pitman nearness (PN) of $\delta_{1}$
relative to $\delta_{2}$ is defined by

$PN(\delta_{1},\delta_{2};\boldsymbol{\theta})=P_{\boldsymbol{\theta}}[|\delta_{1}-\tau(\boldsymbol{\theta})|<|\delta_{2}-\tau(\boldsymbol{\theta})|],\;\;\boldsymbol{\theta}\in\Theta,$

and the estimator $\delta_{1}$ is said to be nearer to
$\tau(\boldsymbol{\theta})$ than $\delta_{2}$ if
$PN(\delta_{1},\delta_{2};\boldsymbol{\theta})\geq\frac{1}{2},\;\forall\;\boldsymbol{\theta}\in\Theta$,
with strict inequality for some $\boldsymbol{\theta}\in\Theta$.

Two drawbacks of the above criterion are that it does not take into account
that estimators $\delta_{1}$ and $\delta_{2}$ may coincide over a subset of
the sample space and that it is only based on the $L_{1}$ distance (absolute
error loss). To take care of these deficiencies, Nayak (1990) and Kubokawa
(1991) modified the Pitman (1937) nearness criterion and defined the
generalized Pitman nearness (GPN) criterion based on general loss function
$L(\boldsymbol{\theta},\delta).$

Definition 2.2 Let $\mathbb{X}$ be a random vector having a distribution
involving an unknown parameter $\boldsymbol{\theta}\in\Theta$ and let
$\tau(\boldsymbol{\theta})$ be a real-valued estimand. Let $\delta_{1}$ and
$\delta_{2}$ be two estimators of the estimand $\tau(\boldsymbol{\theta})$.
Also, let $L(\boldsymbol{\theta},a)$ be a specified loss function for
estimating $\tau(\boldsymbol{\theta})$. Then, the generalized Pitman nearness
(GPN) of $\delta_{1}$ relative to $\delta_{2}$ is defined by

$GPN(\delta_{1},\delta_{2};\boldsymbol{\theta})=P_{\boldsymbol{\theta}}[L(\boldsymbol{\theta},\delta_{1})<L(\boldsymbol{\theta},\delta_{2})]+\frac{1}{2}P_{\boldsymbol{\theta}}[L(\boldsymbol{\theta},\delta_{1})=L(\boldsymbol{\theta},\delta_{2})],\;\;\boldsymbol{\theta}\in\Theta.$

The estimator $\delta_{1}$ is said to be nearer to $\tau(\boldsymbol{\theta})$
than $\delta_{2}$, under the GPN criterion, if
$GPN(\delta_{1},\delta_{2};\boldsymbol{\theta})\geq\frac{1}{2},\;\forall\;\boldsymbol{\theta}\in\Theta$,
with strict inequality for some $\boldsymbol{\theta}\in\Theta$.

Definition 2.3 Let $\mathcal{D}$ be a class of estimators of a real-valued
estimand $\tau(\boldsymbol{\theta})$. Let $L(\boldsymbol{\theta},a)$ be a
given loss function. Then, an estimator $\delta^{*}$ is said to be the Pitman
nearest within the class $\mathcal{D}$, if

$GPN(\delta^{*},\delta;\boldsymbol{\theta})\geq\frac{1}{2},\;\forall\;\delta\in\mathcal{D},\;\boldsymbol{\theta}\in\Theta,$

with strict inequality for some $\boldsymbol{\theta}\in\Theta$.

The following result, famously known as Chebyshev’s inequality, will be used
in our study (see Marshall and Olkin (2007)).

Proposition 2.1 Let $S$ be random variable (r.v.) and let $k_{1}(\cdot)$ and
$k_{2}(\cdot)$ be real-valued monotonic functions defined on the
distributional support of the r.v. $S$. If $k_{1}(\cdot)$ and $k_{2}(\cdot)$
are monotonic functions of the same (opposite) type, then

$E[k_{1}(S)k_{2}(S)]\geq(\leq)E[k_{1}(S)]E[k_{2}(S)],$

provided the above expectations exist.

##  3\. Improved Estimators for Restricted Location Parameters

Let $\mathbb{X}=(X_{1},X_{2})$ be a random vector with a joint probability
density function (p.d.f.)

(3.1)
$f_{\boldsymbol{\theta}}(x_{1},x_{2})=f(x_{1}-\theta_{1},x_{2}-\theta_{2}),\;\;\;(x_{1},x_{2})\in\Re^{2},$

where $f(\cdot,\cdot)$ is a specified Lebesgue p.d.f. on $\Re^{2}$ and
$\boldsymbol{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0}=\\{(t_{1},t_{2})\in\Re^{2}:t_{1}\leq
t_{2}\\}$ is the vector of unknown restricted location parameters; here $\Re$
denotes the real line and $\Re^{2}=\Re\times\Re$. Generally,
$\mathbb{X}=(X_{1},X_{2})$ would be a minimal-sufficient statistic based on a
bivariate random sample or two independent random samples, as the case may be.

Consider estimation of the location parameter $\theta_{i}$ under the GPN
criterion with a general loss function
$L_{i}(\boldsymbol{\theta},a)=W(a-\theta_{i}),\;\boldsymbol{\theta}\in\Theta_{0},\;a\in\mathcal{A},\;i=1,2,$
where $\mathcal{A}=\Re$ and $W:\Re\rightarrow[0,\infty)$ is a specified non-
negative function such that $W(0)=0$, $W(t)$ is strictly decreasing on
$(-\infty,0)$ and strictly increasing on $(0,\infty)$. Throughout this
section, whenever term ”general loss function” is used, it refers to a loss
function having the above properties. Also, in this section, the GPN criterion
is considered with a general loss function described above.

The problem of estimating restricted location parameter $\theta_{i}\,(i=1,2),$
under the GPN criterion, is invariant under the group of transformations
$\mathcal{G}=\\{g_{c}:\,c\in\Re\\},$ where $g_{c}(x_{1},x_{2})$
$=(x_{1}+c,x_{2}+c),\;(x_{1},x_{2})\in\Re^{2},\;c\in\Re.$ Under the group of
transformations $\mathcal{G}$, any location equivariant estimator of
$\theta_{i}$ has the form

(3.2) $\delta_{\psi}(\mathbb{X})=X_{i}-\psi(D),$

for some function $\psi:\,\Re\rightarrow\Re\,,\;i=1,2,$ where $D=X_{2}-X_{1}$.
Let $f_{D}(t|\lambda)$ be the p.d.f. of r.v. $D=X_{2}-X_{1}$, where
$\lambda=\theta_{2}-\theta_{1}\in[0,\infty)$. Note that the distribution of
$D$ depends on $\boldsymbol{\theta}\in\Theta_{0}$ only through
$\lambda=\theta_{2}-\theta_{1}\in[0,\infty)$. Exploiting the prior information
of order restriction on parameters $\theta_{1}$ and $\theta_{2}$
($\theta_{1}\leq\theta_{2}$), we aim to obtain estimators that are Pitman
nearer to $\theta_{i},\,i=1,2$.

The following lemma will be useful in proving the main results of this section
(also see Nayak (1990) and Zhou and Nayak (2012)).

Lemma 3.1 Let $Y$ be a r.v. having the Lebesgue p.d.f. and let $m_{Y}$ be the
median of $Y$. Let $W:\Re\rightarrow[0,\infty)$ be a function such that
$W(0)=0$, $W(t)$ is strictly decreasing on $(-\infty,0)$ and strictly
increasing on $(0,\infty)$. Then, for $-\infty<c_{1}<c_{2}\leq m_{Y}$ or
$-\infty<m_{Y}\leq c_{2}<c_{1}$,
$GPN=P[W(Y-c_{2})<W(Y-c_{1})]+\frac{1}{2}P[W(Y-c_{2})=W(Y-c_{1})]>\frac{1}{2}$.

###### Proof.

. We have the following two cases:

Case 1: $-\infty<c_{1}<c_{2}\leq m_{Y}<\infty$

In this case $Y-m_{Y}\leq Y-c_{2}<Y-c_{1}$ and, thus, $Y-m_{Y}\geq 0$ implies
that $W(Y-c_{2})<W(Y-c_{1})$. Consequently,

$GPN\geq P[W(Y-c_{2})<W(Y-c_{1})]>P[Y\geq m_{Y}]=\frac{1}{2}.$

Case 2: $-\infty<m_{Y}\leq c_{2}<c_{1}<\infty$

In this case $Y-m_{Y}\geq Y-c_{2}>Y-c_{1}$. Thus, $Y-m_{Y}\leq 0$ implies that
$W(Y-c_{2})<W(Y-c_{1})$. Hence

$GPN\geq P[W(Y-c_{2})<W(Y-c_{1})]>P[Y\leq m_{Y}]=\frac{1}{2}.$

∎

Note that, in the unrestricted case (parameter space $\Theta=\Re^{2}$), the
problem of estimating $\theta_{i},\;i=1,2,$ under the GPN criterion is
invariant under the group of transformations
$\mathcal{G}_{0}=\\{g_{c_{1},c_{2}}:\,(c_{1},c_{2})\in\Re^{2}\\},$ where
$g_{c_{1},c_{2}}(x_{1},x_{2})$
$=(x_{1}+c_{1},x_{2}+c_{2}),\;(x_{1},x_{2})\in\Re^{2},\;(c_{1},c_{2})\in\Re^{2}.$
Any location equivariant estimator is of the form
$\delta_{i,c}(\mathbb{X})=X_{i}-c,\;c\in\Re,\;i=1,2$. An immediate consequence
of Lemma 3.1 is that, under the unrestricted parameter space $\Theta=\Re^{2}$,
the Pitman nearest location equivariant estimator (PNLEE) of $\theta_{i}$,
under the GPN criterion, is $\delta_{i,PNLEE}(\mathbb{X})=X_{i}-m_{0,i}$,
where $m_{0,i}$ is the median of the r.v. $Z_{i}=X_{i}-\theta_{i},\;i=1,2$.

###  3.1. Estimation of the Smaller Location Parameter $\theta_{1}$

Let $Z_{1}=X_{1}-\theta_{1}$, $\lambda=\theta_{2}-\theta_{1}$ and
$f_{D}(t|\lambda)$ be the p.d.f. of r.v. $D=X_{2}-X_{1}$. Let
$\delta_{\xi}(\mathbb{X})=X_{1}-\xi(D)$ and
$\delta_{\psi}(\mathbb{X})=X_{1}-\psi(D)$ be two location equivariant
estimators of $\theta_{1}$, where $\xi$ and $\psi$ are real-valued functions
defined on $\Re$. Then, the GPN of $\delta_{\xi}(\mathbb{X})$ relative to
$\delta_{\psi}(\mathbb{X})$ is given by

$\displaystyle GPN(\delta_{\xi},\delta_{\psi};\boldsymbol{\theta})$
$\displaystyle=P_{\boldsymbol{\theta}}[W(Z_{1}-\xi(D))<W(Z_{1}-\psi(D))]$
$\displaystyle\quad+\frac{1}{2}P_{\boldsymbol{\theta}}[W(Z_{1}-\xi(D))=W(Z_{1}-\psi(D))],\;\;\boldsymbol{\theta}\in\Theta_{0},$
$\displaystyle=\int_{-\infty}^{\infty}g_{1,\lambda}(\xi(t),\psi(t),t)f_{D}(t|\lambda)dt,\;\;\lambda\geq
0,$

where, for $\lambda\geq 0$ and fixed $t$ in the support of the distribution of
r.v. $D$,

$\displaystyle g_{1,\lambda}(\xi(t),\psi(t),t)$
$\displaystyle=P_{\boldsymbol{\theta}}[W(Z_{1}-\xi(t))<W(Z_{1}-\psi(t))|D=t]$
(3.3)
$\displaystyle\qquad+\frac{1}{2}P_{\boldsymbol{\theta}}[W(Z_{1}-\xi(t))\\!=\\!W(Z_{1}-\psi(t))|D=t].$

For any fixed $\lambda\geq 0$ and $t$, let $m_{\lambda}^{(1)}(t)$ denote the
median of the conditional distribution of $Z_{1}$ given $D=t$. For any fixed
$t$, the conditional p.d.f. of $Z_{1}$ given $D=t$ is
$f_{\lambda}(s|t)=\frac{f(s,s+t-\lambda)}{f_{D}(t|\lambda)}$ and
$f_{D}(t|\lambda)=\int_{-\infty}^{\infty}f(y,y+t-\lambda)dy$, $\lambda\geq 0$.
Thus,
$\int_{-\infty}^{m_{\lambda}^{(1)}(t)}\,f(s,s+t-\lambda)ds=\frac{1}{2}\int_{-\infty}^{\infty}f(s,s+t-\lambda)ds$.
For any fixed $t$, using Lemma 3.1, we have
$g_{1,\lambda}(\xi(t),\psi(t),t)>\frac{1}{2},\;\forall\;\lambda\geq 0$, if
$\psi(t)<\xi(t)\leq m_{\lambda}^{(1)}(t),\;\forall\;\lambda\geq 0$, or if
$m_{\lambda}^{(1)}(t)\leq\xi(t)<\psi(t),\;\forall\;\lambda\geq 0$. Also, note
that, for any fixed $t$,
$g_{1,\lambda}(\psi(t),\psi(t),t)=\frac{1}{2},\;\forall\;\lambda\geq 0$. These
observations, along with Lemma 3.1, yield the following result.

Theorem 3.1.1. Let $\delta_{\psi}(\mathbb{X})=X_{1}-\psi(D)$ be a location
equivariant estimator of $\theta_{1}$, where $\psi:\Re\rightarrow\Re$. Let
$l^{(1)}(t)$ and $u^{(1)}(t)$ be functions such that $l^{(1)}(t)\leq
m_{\lambda}^{(1)}(t)\leq u^{(1)}(t),\;\forall\;\lambda\geq 0$ and any $t$. For
any fixed $t$, define
$\psi^{*}(t)\\!=\\!\max\\{l^{(1)}(t),\min\\{\psi(t),u^{(1)}(t)\\}\\}$. Then,
under the GPN criterion with a general loss function, the estimator
$\delta_{\psi^{*}}(\mathbb{X})\\!=\\!X_{1}-\psi^{*}(D)$ is Pitman nearer to
$\theta_{1}$ than the estimator $\delta_{\psi}(\mathbb{X})=X_{1}-\psi(D)$, for
all $\boldsymbol{\theta}\in\Theta_{0}$, provided
$P_{\boldsymbol{\theta}}[l^{(1)}(D)\leq\psi(D)\leq
u^{(1)}(D)]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$.

###### Proof.

. The GPN of the estimator
$\delta_{\psi^{*}}(\mathbb{X})=X_{1}-\psi^{*}(D)$ relative to
$\delta_{\psi}(\mathbb{X})=X_{1}-\psi(D)$ can be written as

$\displaystyle
GPN(\delta_{\psi^{*}},\delta_{\psi};\boldsymbol{\theta})=\int_{-\infty}^{\infty}g_{1,\lambda}(\psi^{*}(t),\psi(t),t)f_{D}(t|\lambda)dt,\;\;\lambda\geq
0,$

where $g_{1,\lambda}(\cdot,\cdot,\cdot)$ is defined by (3.3).

Let $A=\\{t:\psi(t)<l^{(1)}(t)\\}$, $B=\\{t:l^{(1)}(t)\leq\psi(t)\leq
u^{(1)}(t)\\}$ and $C=\\{t:\psi(t)>u^{(1)}(t)\\}$. Clearly

$\psi^{*}(t)=\begin{cases}l^{(1)}(t),&t\in A\\\ \psi(t),&t\in B\\\
u^{(1)}(t),&t\in C\end{cases}.$

Since $l^{(1)}(t)\leq m_{\lambda}^{(1)}(t)\leq
u^{(1)}(t),\;\forall\;\lambda\geq 0$ and $t$, using Lemma 3.1, we have
$g_{1,\lambda}(\psi^{*}(t),\psi(t),t)>\frac{1}{2},\;\forall\;\lambda\geq 0$,
provided $t\in A\cup C$. Also, for $t\in B$,
$g_{1,\lambda}(\psi^{*}(t),\psi(t),t)=\frac{1}{2},\;\forall\;\lambda\geq 0$.
Since $P_{\underline{\theta}}(A\cup
C)>0,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$, we have

$GPN(\delta_{\psi^{*}},\delta_{\psi};\boldsymbol{\theta})$

$\displaystyle\\!=\\!\int_{A}\\!g_{1,\lambda}(\psi^{*}(t),\psi(t),t)f_{D}(t|\lambda)dt\\!+\\!\int_{B}\\!g_{1,\lambda}(\psi^{*}(t),\psi(t),t)f_{D}(t|\lambda)dt\\!$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\\!\int_{C}\\!g_{1,\lambda}(\psi^{*}(t),\psi(t),t)f_{D}(t|\lambda)dt$
$\displaystyle>\frac{1}{2},\;\;\boldsymbol{\theta}\in\Theta_{0}.$

∎

Using arguments similar to those used in the proof of Theorem 3.1.1, one can,
in fact, obtain a class of estimators improving over an arbitrary location
equivariant estimator, under certain conditions. The proof of the following
corollary is contained in the proof of Theorem 3.1.1, and hence skipped.

Corollary 3.1.1. Let $\delta_{\psi}(\mathbb{X})=X_{1}-\psi(D)$ be a location
equivariant estimator of $\theta_{1}$ such that
$P_{\boldsymbol{\theta}}[l^{(1)}(D)\leq\psi(D)\leq
u^{(1)}(D)]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$, where
$l^{(1)}(\cdot)$ and $u^{(1)}(\cdot)$ are as defined in Theorem 3.1.1. Let
$\psi_{1,0}:\Re\rightarrow\Re$ be such that $\psi(t)<\psi_{1,0}(t)\leq
l^{(1)}(t)$, whenever $\psi(t)<l^{(1)}(t)$, and
$u^{(1)}(t)\leq\psi_{1,0}(t)<\psi(t)$, whenever $u^{(1)}(t)<\psi(t)$. Also let
$\psi_{1,0}(t)=\psi(t)$, whenever $l^{(1)}(t)\leq\psi(t)\leq u^{(1)}(t)$. Let
$\delta_{\psi_{1,0}}(\mathbb{X})=X_{1}-\psi_{1,0}(D)$. Then,
$GPN(\delta_{\psi_{1,0}},\delta_{\psi};\boldsymbol{\theta})>\frac{1}{2},\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$.

Ironically, the result stated in Corollary 3.1.1 is general than the one
stated in Theorem 3.1.1. However, among all the improved estimators provided
through Corollary 3.1.1, the maximum improvement is provided by the one
covered under Theorem 3.1.1.

The following corollary provides improvements over the unrestricted PNLEE
$\delta_{1,PNLEE}(\mathbb{X})=X_{1}-m_{0,1}$, under the restricted parameter
space $\Theta_{0}$.

Corollary 3.1.2. Let $l^{(1)}(t)$ and $u^{(1)}(t)$ be as defined in Theorem
3.1.1. Suppose that $P_{\boldsymbol{\theta}}[l^{(1)}(D)\leq m_{0,1}\leq
u^{(1)}(D)]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$. Define, for any
fixed $t$,
$\xi^{*}(t)\\!=\\!\max\\{l^{(1)}(t),\min\\{m_{0,1},u^{(1)}(t)\\}\\}$. Then,
for every $\boldsymbol{\theta}\in\Theta_{0}$, the estimator
$\delta_{\xi^{*}}(\mathbb{X})\\!=\\!X_{1}-\xi^{*}(D)$ is Pitman nearer to
$\theta_{1}$ than the PNLEE $\delta_{1,PNLEE}(\mathbb{X})=X_{1}-m_{0,1}$,
under the GPN criterion.

In order to identify $l^{(1)}(t)$ and $u^{(1)}(t)$, satisfying $l^{(1)}(t)\leq
m_{\lambda}^{(1)}(t)\leq u^{(1)}(t),\;\forall\;\lambda\geq 0\text{ and }t$,
the following lemma will be useful.

Lemma 3.1.1. If, for every fixed $\lambda\geq 0$ and $t$,
$f(s,s+t-\lambda)/f(s,s+t)$ is increasing (decreasing) in $s$ (wherever the
ratio is not of the form $0/0$), then, for every fixed $t$,
$m_{\lambda}^{(1)}(t)$ is an increasing (decreasing) function of
$\lambda\in[0,\infty)$.

###### Proof.

. Let us fix $t$, $\lambda_{1}$ and $\lambda_{2}$, such that
$0\leq\lambda_{1}<\lambda_{2}<\infty$. Then, the hypothesis of the lemma
implies that $f_{\lambda_{2}}(s|t)/f_{\lambda_{1}}(s|t)$ is increasing
(decreasing) in $s$. Take $k_{1}(s)=I_{(-\infty,m_{\lambda_{2}}^{(1)}(t))}(s)$
and $k_{2}(s)=f_{\lambda_{2}}(s|t)/f_{\lambda_{1}}(s|t)$, where $I_{A}(\cdot)$
denotes the indicator function of set $A\subseteq\Re$. Here $k_{1}(s)$ is
decreasing in $s$ and $k_{2}(s)$ is increasing (decreasing) in $s$. Using
Proposition 2.1, we get

$\displaystyle\frac{1}{2}=$
$\displaystyle\int_{-\infty}^{\infty}k_{1}(s)k_{2}(s)f_{\lambda_{1}}(s|t)ds\leq(\geq)\left(\int_{-\infty}^{\infty}k_{1}(s)f_{\lambda_{1}}(s|t)ds\right)\left(\int_{-\infty}^{\infty}k_{2}(s)f_{\lambda_{1}}(s|t)ds\right)$
$\displaystyle\implies$
$\displaystyle\int_{-\infty}^{m_{\lambda_{2}}^{(1)}(t)}f_{\lambda_{2}}(s|t)ds=\int_{-\infty}^{m_{\lambda_{1}}^{(1)}(t)}f_{\lambda_{1}}(s|t)ds=\frac{1}{2}\leq(\geq)\int_{-\infty}^{m_{\lambda_{2}}^{(1)}(t)}f_{\lambda_{1}}(s|t)ds$
$\displaystyle\implies$ $\displaystyle
m_{\lambda_{1}}^{(1)}(t)\leq(\geq)\,m_{\lambda_{2}}^{(1)}(t),$

establishing the assertion. ∎

Under the assumptions of Lemma 3.1.1, for any fixed $t$, one may take

(3.4) $\displaystyle l^{(1)}(t)$ $\displaystyle=\inf_{\lambda\geq
0}m_{\lambda}^{(1)}(t)=m_{0}^{(1)}(t)\;(=\lim_{\lambda\to\infty}m_{\lambda}^{(1)}(t))$
(3.5) $\displaystyle\;\text{and}\;\;u^{(1)}(t)$
$\displaystyle=\sup_{\lambda\geq
0}m_{\lambda}^{(1)}(t)=\lim_{\lambda\to\infty}m_{\lambda}^{(1)}(t)\;(=m_{0}^{(1)}(t)),$

while applying Theorem 3.1.1 and Corollary 3.1.1.

While applying Theorem 3.1.1 and Corollaries 3.1.1-3.12, for any fixed $t$,
the commonly used choice for $(l^{(1)}(t),u^{(1)}(t))$ is given by
$l^{(1)}(t)=\inf_{\lambda\geq 0}m_{\lambda}^{(1)}(t)$ and
$u^{(1)}(t)=\sup_{\lambda\geq 0}m_{\lambda}^{(1)}(t)$. Now we will provide
some applications of Theorem 3.1.1 and Corollaries 3.1.1-3.1.2.

Example 3.1.1. Let $\mathbb{X}=(X_{1},X_{2})$ follow a bivariate normal
distribution with joint p.d.f. (3.1), where, for known positive real numbers
$\sigma_{1}$ and $\sigma_{2}$ and known $\rho\in(-1,1),$ the joint p.d.f. of
$\mathbb{Z}=(Z_{1},Z_{2})=(X_{1}-\theta_{1},X_{2}-\theta_{2})$ is

$f(z_{1},z_{2})=\frac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1-\rho^{2}}}e^{-\frac{1}{2(1-\rho^{2})}\left[\frac{z_{1}^{2}}{\sigma_{1}^{2}}-2\rho\,\frac{z_{1}z_{2}}{\sigma_{1}\sigma_{2}}+\frac{z_{2}^{2}}{\sigma_{2}^{2}}\right]},\;\;\;\mathbb{z}=(z_{1},z_{2})\in\Re^{2}.$

Consider estimation of $\theta_{1}$ under the GPN criterion with a general
loss function (i.e.,
$L_{1}(\boldsymbol{\theta},a)=W(a-\theta_{1}),\;\boldsymbol{\theta}\in\Theta_{0},\;a\in\Re$,
where $W(0)=0$, $W(t)$ is strictly decreasing on $(-\infty,0)$ and strictly
increasing on $(0,\infty)$). In this case, the PNLEE is
$\delta_{1,PNLEE}(\mathbb{X})=X_{1}$. Also, for any fixed $t\in\Re$ and
$\lambda\geq 0$, the conditional distribution of $Z_{1}$ given $D=t$ is
$N\left((1-\alpha)(\lambda-t),\frac{(1-\rho^{2})\sigma_{1}^{2}\sigma_{2}^{2}}{\tau^{2}}\right),\text{
where }\tau^{2}=\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}$ and
$\alpha=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\tau^{2}}$.

Here the restricted MLE is
$\delta_{1,RMLE}(\mathbb{X})=X_{1}-(1-\alpha)\max\\{0,-D\\}$. Hwang and
Peddada (1994) and Tan and Peddada (2000) proposed alternative estimators for
$\theta_{1}$ as $\delta_{1,HP}(\mathbb{X})=\min\\{X_{1},\alpha
X_{1}+(1-\alpha)X_{2}\\}=X_{1}-\max\\{0,(\alpha-1)D\\}$ and
$\delta_{1,PDT}(\mathbb{X})=\min\\{X_{1},\beta(\alpha)X_{1}+(1-\beta(\alpha))X_{2}\\}=X_{1}-(1-\beta(\alpha))\max\big{\\{}0,-D\big{\\}}$,
receptively, where
$\beta(\alpha)=\min\\{1,\max\\{0,\alpha\\}\\},\;\alpha\in\Re$.

The median of the conditional conditional distribution of random variable
$Z_{1}$, given $D=t$, is
$m_{\lambda}^{(1)}(t)=(1-\alpha)(\lambda-t),\;t\in\Re,\;\lambda\geq 0.$
Clearly, $m_{\lambda}^{(1)}(t)$ is increasing in $\lambda\in[0,\infty)$, if
$\alpha<1$ and decreasing in $\lambda\in[0,\infty)$, if $\alpha>1$. Thus, for
$t\in\Re$, as in (3.4) and (3.5), we may take

$\displaystyle l^{(1)}(t)$ $\displaystyle=\inf_{\lambda\geq
0}m_{\lambda}^{(1)}(t)=\begin{cases}-(1-\alpha)t,\;\;&\alpha\leq 1\\\
-\infty,\;\;&\alpha>1\end{cases}$ $\displaystyle\text{and}\quad u^{(1)}(t)$
$\displaystyle=\sup_{\lambda\geq
0}m_{\lambda}^{(1)}(t)=\begin{cases}\infty,\;\;&\alpha<1\\\
-(1-\alpha)t,\;\;&\alpha\geq 1\end{cases}.$

Consider the following cases:

Case-I: $0\leq\alpha<1$

In this case the Hwang and Peddada (1994) estimator
$\delta_{1,HP}(\mathbb{X})$, Tan and Peddada (2000) estimator
$\delta_{1,PDT}(\mathbb{X})$ and the restricted MLE
$\delta_{1,RMLE}(\mathbb{X})$ are the same and no improvement is possible over
these estimators using our results.

We have $l^{(1)}(t)=-(1-\alpha)t\;\text{and}\;u^{(1)}(t)=\infty,\;t\in\Re.$
Use of Theorem 3.1.1 and Corollary 3.1.1 leads us to following conclusions:

(i) The estimator $\delta_{1,RMLE}(\mathbb{X})=X_{1}-(1-\alpha)\max\\{0,-D\\}$
($=\delta_{1,HP}(\mathbb{X})=\delta_{1,PDT}(\mathbb{X})$) is nearer to
$\theta_{1}$ than the estimator $\delta_{1,PNLEE}(\mathbb{X})=X_{1}$.

(ii) Let $\psi_{1,0}(\cdot)$ be such that
$0<\psi_{1,0}(t)\leq-(1-\alpha)t,\;\forall\;t<0$, and
$\psi_{1,0}(t)=0,\;\forall\;t\geq 0$. Then the estimator
$\delta_{\psi_{1,0}}(\mathbb{X})=X_{1}-\psi_{1,0}(D)$ is nearer to
$\theta_{1}$ than $\delta_{1,PNLEE}(\mathbb{X})$. In particular, for the
choice,

$\psi_{1,\nu}(t)=\begin{cases}-(1-\nu)t,\;\;&t\leq 0\\\
0,\;\;&t>0\end{cases},\quad\alpha\leq\nu<1,$

the estimator $\delta_{\psi_{1,\nu}}(\mathbb{X})=X_{1}-\psi_{1,\nu}(D)$ is
nearer to $\theta_{1}$ than $\delta_{1,PNLEE}(\mathbb{X})$.

Case-II: $\alpha_{1}=1$

In this case
$\delta_{1,RMLE}(\mathbb{X})=\delta_{1,HP}(\mathbb{X})=\delta_{1,PDT}(\mathbb{X})=\delta_{1,PNLEE}(\mathbb{X})=X_{1}$.
Also $l^{(1)}(t)=u^{(1)}(t)=0,\;\forall\;t\in\Re$. Each of the above
estimators are nearer to $\theta_{1}$ than any other location equivariant
estimator.

Case-III $\alpha>1$

In this case, $\delta_{1,RMLE}(\mathbb{X})=X_{1}-(1-\alpha)\max\\{0,-D\\}$,
$\delta_{1,HP}(\mathbb{X})=X_{1}-(\alpha-1)\max\\{0,D\\}$ and
$\delta_{1,PDT}(\mathbb{X})=\delta_{1,PNLEE}(\mathbb{X})=X_{1}$. Also,
$l^{(1)}(t)=-\infty\;\text{and}\;u^{(1)}(t)=-(1-\alpha)t,\;\forall\;t\in\Re.$
We have the following consequences of Theorem 3.1.1 and Corollary 3.1.1:

(i) Estimators $\delta_{1,RMLE}(\mathbb{X})$ and $\delta_{1,HP}(\mathbb{X})$
are nearer to $\theta_{1}$ than the estimator
$\delta_{1,PDT}(\mathbb{X})\;(=\delta_{1,PNLEE}(\mathbb{X}))$.

(ii) The Estimator $\delta_{1,HP}^{*}(\mathbb{X})=\alpha
X_{1}+(1-\alpha)X_{2}$ is nearer to $\theta_{1}$ than the estimator
$\delta_{1,HP}(\mathbb{X})$.

(iii) Let $\psi_{1,0}(\cdot)$ be such that
$-(1-\alpha)t\leq\psi_{1,0}(t)<0,\;\forall\;t<0$, and
$\psi_{1,0}(t)=0,\;\forall\;t\geq 0$. Then the estimator
$\delta_{\psi_{1,0}}(\mathbb{X})=X_{1}-\psi_{1,0}(D)$ is nearer to
$\theta_{1}$ than $\delta_{1,PNLEE}(\mathbb{X})$. In particular, for the
choice,

$\psi_{1,\nu}(t)=\begin{cases}-(1-\nu)t,\;\;&t\leq 0\\\ 0,\;\;&t\geq
0\end{cases},\quad 1<\nu\leq\alpha,$

the estimator $\delta_{\psi_{1,\nu}}(\mathbb{X})=X_{1}-\psi_{1,\nu}(D)$ is
nearer to $\theta_{1}$ than $\delta_{1,PNLEE}(\mathbb{X})$.

(iv) Let $\psi_{1,0}(\cdot)$ be such that $-(1-\alpha)t\leq\psi_{1,0}(t)\leq
0,\;\forall\;t<0$, and $\psi_{1,0}(t)=-(1-\alpha)t,\;\forall\;t\geq 0$. Then
the estimator $\delta_{\psi_{1,0}}(\mathbb{X})=X_{1}-\psi_{1,0}(D)$ is nearer
to $\theta_{1}$ than $\delta_{1,HP}(\mathbb{X})$. In particular, for the
choice,

$\psi_{1,\nu}(t)=\begin{cases}-(1-\nu)t,\;\;&t\leq 0\\\
-(1-\alpha)t,\;\;&t>0\end{cases},\quad 1<\nu\leq\alpha,$

the estimator

$\delta_{\psi_{1,\nu}}(\mathbb{X})=\begin{cases}\nu
X_{1}+(1-\nu)X_{2},\;\;&X_{1}\geq X_{2}\\\ \alpha
X_{1}+(1-\alpha)X_{2},\;\;&X_{1}<X_{2}\end{cases}$

is nearer to $\theta_{1}$ than $\delta_{1,HP}(\mathbb{X})$.

Case-IV $\alpha<0$

Here
$\delta_{1,RMLE}(\mathbb{X})=\delta_{1,HP}(\mathbb{X})=X_{1}-(1-\alpha)\max\\{0,-D\\}$
and $\delta_{1,PDT}(\mathbb{X})=X_{1}-\max\\{0,-D\\}$. Also, we have
$l^{(1)}(t)=-(1-\alpha)t\;\text{and}\;u^{(1)}(t)=\infty,\;\forall\;t\in\Re.$
The following observations are evident from Theorem 3.1.1 and Corollary 3.1.1:

(i) Estimators $\delta_{1,RMLE}(\mathbb{X})\;(=\delta_{1,HP}(\mathbb{X}))$ and
$\delta_{1,PDT}(\mathbb{X})$ are nearer to $\theta_{1}$ than the estimator
$\delta_{1,PNLEE}(\mathbb{X})$.

(ii) The estimator $\delta_{1,RMLE}(\mathbb{X})\;(=\delta_{1,HP}(\mathbb{X}))$
is nearer to $\theta_{1}$ than $\delta_{1,PDT}(\mathbb{X})$.

(iii) Let $\psi_{1,0}(\cdot)$ be such that
$0<\psi_{1,0}(t)\leq-(1-\alpha)t,\;\forall\;t<0$, and
$\psi_{1,0}(t)=0,\;\forall\;t\geq 0$. Then the estimator
$\delta_{\psi_{1,0}}(\mathbb{X})=X_{1}-\psi_{1,0}(D)$ is nearer to
$\theta_{1}$ than $\delta_{1,PNLEE}(\mathbb{X})=X_{1}$. In particular, for the
choice,

$\psi_{1,\nu}(t)=\begin{cases}-(1-\nu)t,\;\;&t\leq 0\\\ 0,\;\;&t\geq
0\end{cases},\quad\alpha\leq\nu<0,$

the estimator

$\delta_{\psi_{1,\nu}}(\mathbb{X})=\begin{cases}\nu
X_{1}+(1-\nu)X_{2},\;\;&X_{1}>X_{2}\\\ X_{1},\;\;&X_{1}\leq X_{2}\end{cases}$

is nearer to $\theta_{1}$ than $\delta_{1,PNLEE}(\mathbb{X})=X_{1}$.

(iv) Let $\psi_{1,0}(\cdot)$ be such that
$-t\leq\psi_{1,0}(t)\leq-(1-\alpha)t,\;\forall\;t<0$, and
$\psi_{1,0}(t)=0,\;\forall\;t\geq 0$. Then the estimator
$\delta_{\psi_{1,0}}(\mathbb{X})=X_{1}-\psi_{1,0}(D)$ is nearer to
$\theta_{1}$ than $\delta_{1,PDT}(\mathbb{X})$. In particular, for the choice,

$\psi_{1,\nu}(t)=\begin{cases}-(1-\nu)t,\;\;&t\leq 0\\\ 0,\;\;&t\geq
0\end{cases},\quad\alpha\leq\nu<0,$

the estimator

$\delta_{\psi_{1,\nu}}(\mathbb{X})=\begin{cases}\nu
X_{1}+(1-\nu)X_{2},\;\;&X_{1}>X_{2}\\\ X_{1},\;\;&X_{1}\leq X_{2}\end{cases}$

is nearer to $\theta_{1}$ than $\delta_{1,PDT}(\mathbb{X})$.

For the bivariate normal model, some of the above results have also been
reported in Chang et al. (2017, 2020) under specific $W(t)=|t|,\;t\in\Re$. The
findings reported in the above example hold for any general $W(\cdot)$ such
that $W(0)=0$, $W(t)$ is increasing in $(0,\infty)$ and decreasing in
$(-\infty,0)$.

Example 3.1.2. Let $X_{1}$ and $X_{2}$ be independent random variables with a
joint p.d.f.
$f(x_{1}-\theta_{1},x_{2}-\theta_{2})=\frac{1}{\sigma_{1}\sigma_{2}}\,e^{-\frac{x_{1}-\theta_{1}}{\sigma_{1}}}e^{-\frac{x_{2}-\theta_{2}}{\sigma_{2}}},\;x_{1}>\theta_{1},\;x_{2}>\theta_{2},\;(\theta_{1},\theta_{2})\in\Theta_{0}$,
where $\sigma_{1}$ and $\sigma_{2}$ are known positive constants. Here the
PNLEE is $\delta_{1,PNLEE}(\mathbb{X})=X_{1}-\sigma_{1}\ln(2)$ and the
restricted MLE is
$\delta_{1,RMLE}(\mathbb{X})=\min\\{X_{1},X_{2}\\}=X_{1}-\max\\{0,-D\\}$.

For any fixed $t\in\Re$, the conditional p.d.f. of $Z_{1}$ given $D=t$ is

$\displaystyle f_{\lambda}(s|t)$
$\displaystyle=\frac{\sigma_{1}+\sigma_{2}}{\sigma_{1}\sigma_{2}}\,e^{-\left(\frac{1}{\sigma_{1}}+\frac{1}{\sigma_{2}}\right)(s-\max\\{-t+\lambda,0\\})},\;s\geq\max\\{-t+\lambda,0\\}$
$\displaystyle\text{and}\;\;\;\;m_{\lambda}^{(1)}(t)$
$\displaystyle=\max\\{-t+\lambda,0\\}+\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2),\;\;t\in\Re.$

Clearly, for every fixed $t\in\Re$, the median $m_{\lambda}^{(1)}(t)$ is an
increasing function of $\lambda\in[0,\infty)$ (this also follows from Lemma
3.1.1 as, for every fixed $\lambda\geq 0$ and $t\in\Re$,
$f(s,s+t-\lambda)/f(s,s+t)$ is increasing in $s\in(0,\infty)$). Thus, we may
take $l^{(1)}(t)=\\!\inf_{\lambda\geq
0}m_{\lambda}^{(1)}(t)=\max\\{0,-t\\}+\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2)\text{
and }u^{(1)}(t)\\!=\sup_{\lambda\geq 0}m_{\lambda}^{(1)}(t)=\infty,\;t\in\Re.$

The following conclusions immediately follow from Theorem 3.1.1 and Corollary
3.1.1:

(i) The estimator
$\delta_{1,PNLEE}^{*}(\mathbb{X})=\min\big{\\{}X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2),X_{1}-\sigma_{1}\ln(2)\big{\\}}$
is nearer to $\theta_{1}$ than the PNLEE $\delta_{1,PNLEE}(\mathbb{X})$.

(ii) The estimator
$\delta_{1,RMLE}^{*}(\mathbb{X})=\min\\{X_{1},X_{2}\\}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2)$
is nearer to $\theta_{1}$ than the restricted MLE $\delta_{1,RMLE}$.

(iii) Let $\psi_{1,0}(t)$ be such that
$\sigma_{1}\ln(2)<\psi_{1,0}(t)\leq-t+\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2),\;\forall\;t\leq\frac{-\sigma_{1}^{2}}{\sigma_{1}+\sigma_{2}}\ln(2)$,
and
$\psi_{1,0}(t)=\sigma_{1}\ln(2),\;\forall\;t>\frac{-\sigma_{1}^{2}}{\sigma_{1}+\sigma_{2}}\ln(2)$.
Then the estimator $\delta_{\psi_{1,0}}(\mathbb{X})=X_{1}-\psi_{1,0}(D)$ is
nearer to $\theta_{1}$ than the PNLEE
$\delta_{1,PNLEE}(\mathbb{X})=X_{1}-\sigma_{1}\ln(2)$.

(iv) Let $\psi_{1,0}(t)$ be such that
$-t\leq\psi_{1,0}(t)\leq-t+\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2),\;\forall\;t\leq
0$, and
$0\leq\psi_{1,0}(t)\leq\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2),\;\forall\;t>0$.
Then the estimator $\delta_{\psi_{1,0}}(\mathbb{X})=X_{1}-\psi_{1,0}(D)$ is
nearer to $\theta_{1}$ than the restricted MLE
$\delta_{1,RMLE}(\mathbb{X})=\min\\{X_{1},X_{2}\\}$.

It is worth mentioning that findings of Theorem 3.1.1 and Corollaries
3.1.1-3.1.2, and hence those of Examples 3.1.1 and 3.1.2, hold under any
general loss function
$L_{1}(\boldsymbol{\theta},a)=W(a-\theta_{1}),\;\boldsymbol{\theta}\in\Theta_{0},\;a\in\mathcal{A}=\Re,$
where $W:\Re\rightarrow[0,\infty)$ is such that $W(0)=0$, $W(t)$ is strictly
decreasing on $(-\infty,0)$ and strictly increasing on $(0,\infty)$.

###  3.2. Estimation of the Larger Location Parameter $\theta_{2}$

Consider estimation of the larger location parameter $\theta_{2}$ under the
GPN criterion with the general loss function
$L_{2}(\boldsymbol{\theta},a)=W(a-\theta_{2}),\;\boldsymbol{\theta}\in\Theta_{0},\;a\in\mathcal{A}=\Re$,
when it is known apriori that $\boldsymbol{\theta}\in\Theta_{0}$. The form of
any location equivariant estimator of $\theta_{2}$ is
$\delta_{\psi}(\mathbb{X})=X_{2}-\psi(D),$ for some function
$\psi:\,\Re\rightarrow\Re$, where $D=X_{2}-X_{1}$.

Let $Z_{2}=X_{2}-\theta_{2}$, $\lambda=\theta_{2}-\theta_{1}$ and
$f_{D}(t|\lambda)$ be the p.d.f. of r.v. $D=X_{2}-X_{1}$. Let
$\delta_{\xi}(\mathbb{X})=X_{2}-\xi(D)$ and
$\delta_{\psi}(\mathbb{X})=X_{2}-\psi(D)$ be two location equivariant
estimators of $\theta_{2}$. Then, the GPN of $\delta_{\xi}(\mathbb{X})$
relative to $\delta_{\psi}(\mathbb{X})$ is given by

$\displaystyle
GPN(\delta_{\xi},\delta_{\psi};\boldsymbol{\theta})=\int_{-\infty}^{\infty}g_{2,\lambda}(\xi(t),\psi(t),t)f_{D}(t|\lambda)dt,\;\;\lambda\geq
0,$

where, for $\lambda\geq 0$,

$\displaystyle g_{2,\lambda}(\xi(t),\psi(t),t)$
$\displaystyle=P_{\boldsymbol{\theta}}[W(Z_{2}-\xi(t))<W(Z_{2}-\psi(t))|D=t]$
$\displaystyle\qquad+\frac{1}{2}P_{\boldsymbol{\theta}}[W(Z_{2}-\xi(t))\\!=\\!W(Z_{2}-\psi(t))|D=t].$

Let $m_{\lambda}^{(2)}(t)$ denote the median of the conditional distribution
of $Z_{2}$ given $D=t$, where $\lambda\geq 0$ and $t$ belongs to the support
of r.v. $D$. For any fixed $t\in\Re$, the conditional p.d.f. of $Z_{2}$ given
$D=t$ is $f_{\lambda}(s|t)=\frac{f(s-t+\lambda,s)}{f_{D}(t|\lambda)}$ and
$f_{D}(t|\lambda)=\int_{-\infty}^{\infty}f(y-t+\lambda,y)dy$, $\lambda\geq 0$.
Thus
$\int_{-\infty}^{m_{\lambda}^{(2)}(t)}\,f(s-t+\lambda,s)ds=\frac{1}{2}\int_{-\infty}^{\infty}f(s-t+\lambda,s)ds$.
For any fixed $t$ and $\lambda\geq 0$, using Lemma 3.1, we have
$g_{2,\lambda}(\xi(t),\psi(t),t)>\frac{1}{2}$, provided $\psi(t)<\xi(t)\leq
m_{\lambda}^{(2)}(t)$ or if $m_{\lambda}^{(2)}(t)\leq\xi(t)<\psi(t)$. Also,
for any fixed $t$,
$g_{2,\lambda}(\psi(t),\psi(t),t)=\frac{1}{2},\;\forall\;\lambda\geq 0$. Now
using arguments similar to the ones used in proving Theorem 3.1.1, we get the
following results.

Theorem 3.2.1. Let $\delta_{\psi}(\mathbb{X})=X_{2}-\psi(D)$ be a location
equivariant estimator of $\theta_{2}$. Let $l^{(2)}(t)$ and $u^{(2)}(t)$ be
functions such that $l^{(2)}(t)\leq m_{\lambda}^{(2)}(t)\leq
u^{(2)}(t),\;\forall\;\lambda\geq 0$ and any $t$. For any fixed $t$, define
$\psi^{*}(t)\\!=\\!\max\\{l^{(2)}(t),\min\\{\psi(t),u^{(2)}(t)\\}\\}$. Let
$\delta_{\psi^{*}}(\mathbb{X})\\!=\\!X_{2}-\psi^{*}(D)$. Then,
$GPN(\delta_{\psi^{*}},\delta_{\psi};\boldsymbol{\theta})>\frac{1}{2},\;\forall\;\boldsymbol{\theta}\in\Theta_{0},$
provided $P_{\boldsymbol{\theta}}[l^{(2)}(D)\leq\psi(D)\leq
u^{(2)}(D)]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$.

Corollary 3.2.1. Let $\delta_{\psi}(\mathbb{X})=X_{2}-\psi(D)$ be a location
equivariant estimator of $\theta_{2}$ such that
$P_{\boldsymbol{\theta}}[l^{(2)}(D)\leq\psi(D)\leq
u^{(2)}(D)]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$, where
$l^{(2)}(\cdot)$ and $u^{(2)}(\cdot)$ are as defined in Theorem 3.2.1. Let
$\psi_{2,0}:\Re\rightarrow\Re$ be such that $\psi(t)<\psi_{2,0}(t)\leq
l^{(2)}(t)$, whenever $\psi(t)<l^{(2)}(t)$, or
$u^{(2)}(t)\leq\psi_{2,0}(t)<\psi(t)$, whenever $u^{(2)}(t)<\psi(t)$. Also let
$\psi_{2,0}(t)=\psi(t)$, whenever $l^{(2)}(t)\leq\psi(t)\leq u^{(2)}(t)$. Let
$\delta_{\psi_{2,0}}(\mathbb{X})=X_{2}-\psi_{2,0}(D)$. Then,
$GPN(\delta_{\psi_{2,0}},\delta_{\psi};\boldsymbol{\theta})>\frac{1}{2},\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$.

The following corollary provides improvements over the PNLEE
$\delta_{2,PNLEE}(\mathbb{X})=X_{2}-m_{0,2}$, under the restricted parameter
space $\Theta_{0}$

Corollary 3.2.2. Let
$\xi^{*}(t)\\!=\\!\max\\{l^{(2)}(t),\min\\{m_{0,2},u^{(2)}(t)\\}\\},\;t\in\Re$,
where $l^{(2)}(\cdot)$ and $u^{(2)}(\cdot)$ are as defined in Theorem 3.2.1.
Let $\delta_{\xi^{*}}(\mathbb{X})\\!=\\!X_{2}-\xi^{*}(D)$. Then,
$GPN(\delta_{\xi^{*}},\delta_{2,PNLEE};\boldsymbol{\theta})>\frac{1}{2},\;\forall\;\boldsymbol{\theta}\in\Theta_{0},$
provided $P_{\boldsymbol{\theta}}[l^{(2)}(D)\leq m_{0,2}\leq
u^{(2)}(D)]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$.

The following lemma describes the behaviour of $m_{\lambda}^{(2)}(t)$, for any
fixed $t$. The proof of the lemma, being similar to the proof of Lemma 3.1.1,
is skipped.

Lemma 3.2.1. If, for every fixed $\lambda\geq 0$ and $t$,
$f(s-t+\lambda,s)/f(s-t,s)$ is increasing (decreasing) in $s$ (wherever the
ratio is not of the form $0/0$), then, for every fixed $t$,
$m_{\lambda}^{(2)}(t)$ is an increasing (decreasing) function of
$\lambda\in[0,\infty)$.

Under the assumptions of Lemma 3.2.1, one may take, for any fixed $t$,

(3.6) $\displaystyle l^{(2)}(t)$ $\displaystyle=\inf_{\lambda\geq
0}m_{\lambda}^{(2)}(t)=m_{0}^{(2)}(t)\;(=\lim_{\lambda\to\infty}m_{\lambda}^{(2)}(t))$
(3.7) $\displaystyle\;\text{and}\;\;u^{(2)}(t)$
$\displaystyle=\sup_{\lambda\geq
0}m_{\lambda}^{(2)}(t)=\lim_{\lambda\to\infty}m_{\lambda}^{(2)}(t)\;(=m_{0}^{(2)}(t)),$

while applying Theorem 3.2.1 and Corollary 3.2.1.

As in Section 3.1, we will now apply Theorem 3.2.1 and Corollaries 3.2.1-3.2.2
to estimation of the larger location parameter $\theta_{2}$ in probability
models considered in Examples 3.1.1-3.1.2.

Example 3.2.1. Let $\mathbb{X}=(X_{1},X_{2})$ have a bivariate normal
distribution as described in Example 3.1.1. Consider estimation of
$\theta_{2}$ under the GPN criterion with a general loss function
$L_{2}(\boldsymbol{\theta},a)=W(a-\theta_{2}),\;\boldsymbol{\theta}\in\Theta_{0},\;a\in\Re$,
where $W(0)=0$, $W(t)$ is strictly decreasing on $(-\infty,0)$ and strictly
increasing on $(0,\infty)$. Here, for any fixed $t\in\Re$, $Z_{2}|D=t\sim
N\left(\alpha(t-\lambda),\frac{(1-\rho^{2})\sigma_{1}^{2}\sigma_{2}^{2}}{\tau^{2}}\right),\text{
where }\tau^{2}=\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}$ and
$\alpha=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\tau^{2}}$. Thus, for
$\lambda\geq 0$, $m_{\lambda}^{(2)}(t)=\alpha(t-\lambda),\;t\in\Re,$ and as in
(3.6) and (3.7), we may take

$\displaystyle l^{(2)}(t)=\begin{cases}\alpha t,\;\;&\alpha\leq 0\\\
-\infty,\;\;&\alpha>0\end{cases}\;\;\text{and}\quad
u^{(2)}(t)=\begin{cases}\infty,\;\;&\alpha\leq 0\\\ \alpha
t,\;\;&\alpha>0\end{cases}.$

The unrestricted PNLEE of $\theta_{2}$ is $\delta_{2,PNLEE}(\mathbb{X})=X_{2}$
(as $m_{0,2}=0$) and the restricted MLE of $\theta_{2}$ is
$\delta_{2,RMLE}(\mathbb{X})=X_{2}+\alpha\max\\{0,-D\\}$. Hwang and Peddada
(1994) and Tan and Peddada (2000) proposed alternative estimators for
$\theta_{2}$ as $\delta_{2,HP}(\mathbb{X})=X_{2}+\max\\{0,-\alpha D\\}$ and
$\delta_{2,PDT}(\mathbb{X})=X_{2}+\beta(\alpha)\max\big{\\{}0,-D\big{\\}}$,
respectively, where
$\beta(\alpha)=\min\\{1,\max\\{0,\alpha\\}\\},\;\alpha\in\Re$.

Using Corollary 3.2.2, we conclude that, under the GPN criterion, the
estimator

$\delta_{2,PNLEE}^{*}(\mathbb{X})=\delta_{2,RMLE}^{*}(\mathbb{X})=\begin{cases}\min\Big{\\{}X_{2},\alpha
X_{1}+(1-\alpha)X_{2}\Big{\\}},&\alpha\leq 0\\\ \leavevmode\nobreak\ \\\
\max\Big{\\{}X_{2},\alpha X_{1}+(1-\alpha)X_{2}\Big{\\}},&\alpha>0\end{cases}$

is nearer to $\theta_{2}$ than $\delta_{2,PNLEE}(\mathbb{X})=X_{2}.$ In the
same line as in Example 3.1.1, estimators dominating over
$\delta_{2,HP}(\mathbb{X})$ and $\delta_{2,PDT}(\mathbb{X})$ can be obtained
in certain situations.

Example 3.2.2. Let $X_{1}$ and $X_{2}$ be independent exponential random
variables as considered in Example 3.1.2. Consider estimation of $\theta_{2}$
under the GPN criterion. Here the PNLEE of $\theta_{2}$ is
$\delta_{2,PNLEE}(\mathbb{X})=X_{2}-\sigma_{2}\ln(2)$ and the restricted MLE
of $\theta_{2}$ is $\delta_{2,RMLE}(\mathbb{X})=X_{2}$. Also, for any fixed
$\lambda\geq 0$ and $t\in\Re$, the conditional p.d.f. of $Z_{2}$ given $D=t$
is

$f_{\lambda}(s|t)=\frac{\sigma_{1}+\sigma_{2}}{\sigma_{1}\sigma_{2}}\,e^{-\left(\frac{1}{\sigma_{1}}+\frac{1}{\sigma_{2}}\right)(s-\max\\{t-\lambda,0\\})},\text{
if}\;s\geq\max\\{t-\lambda,0\\}.$

Consequently,

$m_{\lambda}^{(2)}(t)=\max\\{0,t-\lambda\\}+\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2),\;t\in\Re,\;\;\lambda\geq
0,$

and, as in (3.6) and (3.7), we may take
$l^{(2)}(t)=\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2),\;t\in\Re,$
and
$u^{(2)}(t)=\max\\{t,0\\}+\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2),\;t\in\Re.$

The following conclusions are evident from Theorem 3.2.1 and Corollary 3.2.1:

(i) The estimator
$\delta_{2,RMLE}^{*}(\mathbb{X})=X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2)$
is nearer to $\theta_{2}$ than the restricted MLE
$\delta_{2,RMLE}(\mathbb{X})=X_{2}$.

(ii) The estimator

$\displaystyle\delta_{2,PNLEE}^{*}(\mathbb{X})=\begin{cases}X_{2}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2),&\text{if
}X_{2}<X_{1}\\\
X_{1}-\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2),&\text{if
}X_{1}\leq X_{2}<X_{1}+\frac{\sigma_{2}^{2}}{\sigma_{1}+\sigma_{2}}\ln(2)\\\
X_{2}-\sigma_{2}\ln(2),&\text{if }X_{2}\geq
X_{1}+\frac{\sigma_{2}^{2}}{\sigma_{1}+\sigma_{2}}\ln(2)\end{cases}$

is nearer to $\theta_{2}$ than $\delta_{2,PNLEE}(\mathbb{X})$.

(iii) Let $\psi_{2,0}(t)$ be such that
$\max\\{0,t\\}+\frac{\sigma_{1}\sigma_{2}}{\sigma_{1}+\sigma_{2}}\ln(2)\leq\psi_{2,0}(t)<\sigma_{2}\ln(2),\;\forall\;t\leq\frac{\sigma_{2}^{2}}{\sigma_{1}+\sigma_{2}}\ln(2)$,
and
$\psi_{2,0}(t)=\sigma_{2}\ln(2),\;\forall\;t>\frac{\sigma_{2}^{2}}{\sigma_{1}+\sigma_{2}}\ln(2)$.
Then the estimator $\delta_{\psi_{2,0}}(\mathbb{X})=X_{2}-\psi_{2,0}(D)$ is
nearer to $\theta_{2}$ than the PNLEE
$\delta_{2,PNLEE}(\mathbb{X})=X_{2}-\sigma_{2}\ln(2)$.

##  4\. Improved Estimators for Restricted Scale Parameters

Let $\mathbb{X}=(X_{1},X_{2})$ be a random vector having a joint p.d.f.

(4.1)
$f_{\boldsymbol{\theta}}(x_{1},x_{2})=\frac{1}{\theta_{1}\theta_{2}}f\left(\frac{x_{1}}{\theta_{1}},\frac{x_{2}}{\theta_{2}}\right),\;\;\;(x_{1},x_{2})\in\Re^{2},$

where $f(\cdot,\cdot)$ is a specified Lebesgue p.d.f. and, for
$\Re_{++}=(0,\infty)\times(0,\infty)$,
$\boldsymbol{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0}=\\{(t_{1},t_{2})\in\Re_{++}^{2}:t_{1}\leq
t_{2}\\}$ is the vector of unknown restricted scale parameters. For the sake
of simplicity, throughout this section, we assume that the distributional
support of $\mathbb{X}=(X_{1},X_{2})$ is a subset of $\Re_{++}^{2}$.
Generally, $\mathbb{X}=(X_{1},X_{2})$ would be a minimal-sufficient statistic
based on a bivariate random sample or two independent random samples.

For estimation of $\theta_{i}$, under the restricted parameter space
$\Theta_{0}$, we consider the GPN criterion with the loss function
$L_{i}(\boldsymbol{\theta},a)=W(\\!\frac{a}{\;\theta_{i}}\\!),\;\boldsymbol{\theta}\in\Theta_{0},\;\;a\in\mathcal{A}=\Re_{++}=(0,\infty),\;i=1,2,$
where $W:\Re_{++}\rightarrow[0,\infty)$ is a function such that $W(1)=0$,
$W(t)$ is strictly decreasing on $(0,1)$ and strictly increasing on
$(1,\infty)$. Every time the word ”general loss function” is used in this
section, it refers to the loss function as defined above.

The problem of estimating $\theta_{i}$, under the restricted parameter space
$\Theta_{0}$ and under the GPN criterion with a general loss function defined
above, is invariant under the group of transformations
$\mathcal{G}\\!=\\!\\{g_{b}:\,b\in(0,\infty)\\},$ where
$g_{b}(x_{1},x_{2})\\!=\\!(b\,x_{1},b\,x_{2}),\;(x_{1},x_{2})\in\Re^{2},\;b\in(0,\infty)$.
Any scale equivariant estimator of $\theta_{i}$ has the form

$\delta_{\psi}(\mathbb{X})\\!=\\!\psi(D)X_{i},$

for some function $\psi:\,\Re_{++}\rightarrow\Re_{++}\,,\;i=1,2,$ where
$D=\frac{X_{2}}{X_{1}}$. Let $f_{D}(t|\lambda)$ be the p.d.f. of r.v.
$D=\frac{X_{2}}{X_{1}}$, where
$\lambda=\frac{\theta_{2}}{\theta_{1}}\in[1,\infty)$. Note that the
distribution of $D$ depends on $\boldsymbol{\theta}\in\Theta_{0}$ only through
$\lambda=\frac{\theta_{2}}{\theta_{1}}\in[1,\infty)$. Exploiting the prior
information of order restriction on parameters $\theta_{1}$ and $\theta_{2}$
($\theta_{1}\leq\theta_{2}$), our aim is to obtain estimators that are nearer
to $\theta_{i},\,i=1,2$.

The following lemma, whose proof is similar to that of Lemma 3.1, will play an
important role in proving the main results of this section.

Lemma 4.1 Let $Y$ be a positive r.v. ($Pr(Y>0)=1$) having the Lebesgue p.d.f.
and let $m_{Y}>0$ be the median of $Y$. Let $W:\Re_{++}\rightarrow[0,\infty)$
be a function such that $W(1)=0$, $W(t)$ is strictly decreasing on $(0,1)$ and
strictly increasing on $(1,\infty)$. Then, for $0<c_{1}<c_{2}\leq m_{Y}$ or
$0<m_{Y}\leq c_{2}<c_{1}$,
$GPN=P\left[W\left(\frac{Y}{c_{2}}\right)<W\left(\frac{Y}{c_{1}}\right)\right]+\frac{1}{2}P\left[W\left(\frac{Y}{c_{2}}\right)=W\left(\frac{Y}{c_{1}}\right)\right]>\frac{1}{2}$.

Note that, in the unrestricted case (parameter space
$\Theta\\!=\\!\Re_{++}^{2}$), the problem of estimating the scale parameter
$\theta_{i},\,i=1,2$, under the GPN criterion with a general loss function, is
invariant under the multiplicative group of transformations
$\mathcal{G}_{0}=\\{g_{c_{1},c_{2}}:\,(c_{1},c_{2})\in\Re_{++}^{2}\\},$ where
$g_{c_{1},c_{2}}(x_{1},x_{2})=(c_{1}x_{1},c_{2}x_{2}),\;(x_{1},x_{2})\in\Re^{2},\;(c_{1},c_{2})\in\Re_{++}^{2}$.
Any scale equivariant estimator is of the form
$\delta_{i,c}(\mathbb{X})=cX_{i},\;c\in\Re_{++},\;i=1,2.$ Using Lemma 4.1, the
unrestricted PNSEE of $\theta_{i}$ is
$\delta_{i,PNSEE}(\mathbb{X})=\frac{X_{i}}{m_{0,i}}$, where $m_{0,i}>0$ is the
median of the r.v. $Z_{i}=\frac{X_{i}}{\theta_{i}},\,i=1,2$.

In the following subsections, we consider component-wise estimation of order
restricted scale parameters $\theta_{1}$ and $\theta_{2}$, under the GPN
criterion with a general loss function, and derive some general results.
Applications of main results are illustrated through various examples dealing
with specific probability models.

###  4.1. Estimation of The Smaller Scale Parameter $\theta_{1}$

Define $Z_{1}=\frac{X_{1}}{\theta_{1}}$ and
$\lambda=\frac{\theta_{2}}{\theta_{1}}$, so that $\lambda\geq 1$. Let
$f_{D}(t|\lambda)$ be the p.d.f. of r.v. $D=\frac{X_{2}}{X_{1}}$. Let
$\delta_{\psi}(\mathbb{X})=\psi(D)X_{1}$ and
$\delta_{\xi}(\mathbb{X})=\xi(D)X_{1}$ be two scale equivariant estimators of
$\theta_{1}$, where $\psi:\,\Re_{++}\rightarrow\Re_{++}$ and
$\xi:\,\Re_{++}\rightarrow\Re_{++}$ are specified functions. Then, the GPN of
$\delta_{\xi}(\mathbb{X})=\xi(D)X_{1}$ relative to
$\delta_{\psi}(\mathbb{X})=\psi(D)X_{1}$ is given by

$\displaystyle
GPN(\delta_{\xi},\delta_{\psi};\boldsymbol{\theta})=\int_{0}^{\infty}g_{1,\lambda}(\xi(t),\psi(t),t)f_{D}(t|\lambda)dt,\;\;\lambda\geq
1,$

where, for $\lambda\geq 1$ and fixed $t$ in support of r.v. $D$,

$\displaystyle g_{1,\lambda}(\xi(t),\psi(t),t)$
$\displaystyle=P_{\boldsymbol{\theta}}[W(\xi(t)Z_{1})<W(\psi(t)Z_{1})|D=t]$
(4.2)
$\displaystyle\qquad+\frac{1}{2}P_{\boldsymbol{\theta}}[W(\xi(t)Z_{1})\\!=\\!W(\psi(t)Z_{1})|D=t].$

For any fixed $\lambda\geq 1$ and $t$, let $m_{\lambda}^{(1)}(t)$ denote the
median of the conditional distributional of $Z_{1}$ given $D=t$. For any fixed
$t$, the conditional p.d.f. of $Z_{1}$ given $D=t$ is
$f_{\lambda}(s|t)=\frac{\frac{s}{\lambda}f(s,\frac{st}{\lambda})}{f_{D}(t|\lambda)}$
and
$f_{D}(t|\lambda)=\int_{0}^{\infty}\frac{y}{\lambda}f(y,\frac{yt}{\lambda})dy$,
$\lambda\geq 1$. Then
$\int_{0}^{m_{\lambda}^{(1)}(t)}\,sf(s,\frac{st}{\lambda})ds=\frac{1}{2}\int_{0}^{\infty}sf(s,\frac{st}{\lambda})ds$.
It follows from Lemma 4.1 that, for any fixed $t$ and $\lambda\geq 1$,
$g_{1,\lambda}(\xi(t),\psi(t),t)>\frac{1}{2}$, provided
$m_{\lambda}^{(1)}(t)\leq\frac{1}{\xi(t)}<\frac{1}{\psi(t)}$ or
$\frac{1}{\psi(t)}<\frac{1}{\xi(t)}\leq m_{\lambda}^{(1)}(t)$. Also, for any
fixed $t$,
$g_{1,\lambda}(\psi(t),\psi(t),t)=\frac{1}{2},\;\forall\;\lambda\geq 1$.

On similar lines as in Theorem 3.1.1, under certain conditions, the following
theorem provides shrinkage type improvements over an arbitrary scale
equivariant estimator under the GPN criterion with a general loss function.

Theorem 4.1.1. Let $\delta_{\psi}(\mathbb{X})=\psi(D)X_{1}$ be a scale
equivariant estimator of $\theta_{1}$. Let $l^{(1)}(t)$ and $u^{(1)}(t)$ be
functions such that $0<l^{(1)}(t)\leq m_{\lambda}^{(1)}(t)\leq
u^{(1)}(t),\;\forall\;\lambda\geq 1$ and any $t$. For any fixed $t$, define
$\psi^{*}(t)\\!=\\!\max\\{\frac{1}{u^{(1)}(t)},\min\\{\psi(t),\frac{1}{l^{(1)}(t)}\\}\\}$.
Then, under the GPN criterion, the estimator
$\delta_{\psi^{*}}(\mathbb{X})\\!=\\!\psi^{*}(D)X_{1}$ is nearer to
$\theta_{1}$ than the estimator $\delta_{\psi}(\mathbb{X})=\psi(D)X_{1}$,
provided
$P_{\boldsymbol{\theta}}\left[\frac{1}{u^{(1)}(D)}\leq\psi(D)\leq\frac{1}{l^{(1)}(D)}\right]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$.

###### Proof.

. The GPN of the estimator $\delta_{\psi^{*}}(\mathbb{X})=\psi^{*}(D)X_{1}$
relative to $\delta_{\psi}(\mathbb{X})=\psi(D)X_{1}$ can be written as

$\displaystyle
GPN(\delta_{\psi^{*}},\delta_{\psi};\boldsymbol{\theta})=\int_{0}^{\infty}g_{1,\lambda}(\psi^{*}(t),\psi(t),t)f_{D}(t|\lambda)dt,\;\;\lambda\geq
1,$

where, for $\lambda\geq 1$ and $t$ in support of r.v. $D$,
$g_{1,\lambda}(\cdot,\cdot,\cdot)$ is defined by (4.2).

Let $A=\\{t:\psi(t)<\frac{1}{u^{(1)}(t)}\\}$,
$B=\\{t:\frac{1}{u^{(1)}(t)}\leq\psi(t)\leq\frac{1}{l^{(1)}(t)}\\}$ and
$C=\\{t:\psi(t)>\frac{1}{l^{(1)}(t)}\\}$. Then

$\psi^{*}(t)=\begin{cases}\frac{1}{u^{(1)}(t)},&t\in A\\\ \psi(t),&t\in B\\\
\frac{1}{l^{(1)}(t)},&t\in C\end{cases}.$

Since $l^{(1)}(t)\leq m_{\lambda}^{(1)}(t)\leq u^{(1)}(t)\;(\text{or
}\frac{1}{u^{(1)}(t)}\leq\frac{1}{m_{\lambda}^{(1)}(t)}\leq\frac{1}{l^{(1)}(t)}),\;\forall\;\lambda\geq
1$ and $t$, using Lemma 4.1, we have
$g_{1,\lambda}(\psi^{*}(t),\psi(t),t)>\frac{1}{2},\;\forall\;\lambda\geq 1$,
whenever $t\in A\cup C$. Also, for $t\in B$,
$g_{1,\lambda}(\psi^{*}(t),\psi(t),t)=\frac{1}{2},\;\forall\;\lambda\geq 1$.
Since $P_{\boldsymbol{\theta}}(A\cup
C)>0,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$, we conclude that

$GPN(\delta_{\psi^{*}},\delta_{\psi};\boldsymbol{\theta})$

$\displaystyle\\!=\\!\int_{A}\\!g_{1,\lambda}(\psi^{*}(t),\psi(t),t)f_{D}(t|\lambda)dt\\!+\\!\int_{B}\\!g_{1,\lambda}(\psi^{*}(t),\psi(t),t)f_{D}(t|\lambda)dt\\!$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\\!\int_{C}\\!g_{1,\lambda}(\psi^{*}(t),\psi(t),t)f_{D}(t|\lambda)dt$
$\displaystyle>\frac{1}{2},\;\;\boldsymbol{\theta}\in\Theta_{0}.$

∎

The proof of the following corollary is contained in the proof of Theorem
4.1.1, and hence skipped.

Corollary 4.1.1. Let $\delta_{\psi}(\mathbb{X})=\psi(D)X_{1}$ be a scale
equivariant estimator of $\theta_{1}$. Let
$\psi_{1,0}:\Re_{++}\rightarrow\Re_{++}$ be such that
$\psi(t)<\psi_{1,0}(t)\leq\frac{1}{u^{(1)}(t)}$, whenever
$\psi(t)<\frac{1}{u^{(1)}(t)}$, and
$\frac{1}{l^{(1)}(t)}\leq\psi_{1,0}(t)<\psi(t)$, whenever
$\frac{1}{l^{(1)}(t)}<\psi(t)$, where $l^{(1)}(\cdot)$ and $u^{(1)}(\cdot)$
are as defined in Theorem 4.1.1. Also let $\psi_{1,0}(t)=\psi(t)$, whenever
$\frac{1}{u^{(1)}(t)}\leq\psi(t)\leq\frac{1}{l^{(1)}(t)}$. Then, the estimator
$\delta_{\psi_{1,0}}(\mathbb{X})=\psi_{1,0}(D)X_{1}$ is nearer to $\theta_{1}$
than $\delta_{\psi}(\mathbb{X})=\psi(D)X_{1}$, provided
$P_{\boldsymbol{\theta}}\left[\frac{1}{u^{(1)}(D)}\leq\psi(D)\leq\frac{1}{l^{(1)}(D)}\right]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$.

The following corollary provides improvements over the PNSEE
$\delta_{1,PNSEE}(\mathbb{X})=\frac{X_{1}}{m_{0,1}}$, under the restricted
parameter space $\Theta_{0}$.

Corollary 4.1.2. Let
$\xi^{*}(t)\\!=\\!\max\\{\frac{1}{u^{(1)}(t)},\min\\{\frac{1}{m_{0,1}},\frac{1}{l^{(1)}(t)}\\}\\}$,
where $l^{(1)}(\cdot)$ and $u^{(1)}(\cdot)$ are as defined in Theorem 4.1.1.
Then, under the GPN criterion, the estimator
$\delta_{\psi^{*}}(\mathbb{X})\\!=\\!\xi^{*}(D)X_{1}$ is nearer to
$\theta_{1}$ than the PNSEE
$\delta_{1,PNSEE}(\mathbb{X})=\frac{X_{1}}{m_{0,1}}$, provided
$P_{\boldsymbol{\theta}}\left[l^{(1)}(D)\leq m_{0,1}\leq
u^{(1)}(D)\right]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$.

In various applications of Theorem 4.1.1 and Corollaries 4.1.1-4.1.2, a common
choice for $(l^{(1)}(t),u^{(1)}(t))$ is given by $l^{(1)}(t)=\inf_{\lambda\geq
1}m_{\lambda}^{(1)}(t)$ and $u^{(1)}(t)=\sup_{\lambda\geq
1}m_{\lambda}^{(1)}(t)$.

In order to identify the behaviour of function $m_{\lambda}^{(1)}(t)$, for any
fixed $t$, the following lemma be useful in many situations. Since the proof
of the lemma is on the same lines as in Lemma 3.1.1, it is skipped.

Lemma 4.1.1. If, for every fixed $\lambda\geq 1$ and $t$,
$f(s,\frac{st}{\lambda})/f(s,st)$ is increasing (decreasing) in $s$ (wherever
the ratio is not of the form $0/0$), then, for every fixed $t$,
$m_{\lambda}^{(1)}(t)$ is an increasing (decreasing) function of
$\lambda\in[1,\infty)$.

Under the assumptions of Lemma 4.1.1, one may take, for any fixed $t$,

(4.3) $\displaystyle l^{(1)}(t)$ $\displaystyle=\inf_{\lambda\geq
1}m_{\lambda}^{(1)}(t)=m_{1}^{(1)}(t)\;(=\lim_{\lambda\to\infty}m_{\lambda}^{(1)}(t))$
(4.4) $\displaystyle\;\text{and}\;\;u^{(1)}(t)$
$\displaystyle=\sup_{\lambda\geq
1}m_{\lambda}^{(1)}(t)=\lim_{\lambda\to\infty}m_{\lambda}^{(1)}(t)\;(=m_{1}^{(1)}(t)),$

while applying Theorem 4.1.1 and Corollary 4.1.1.

Now we will consider some applications of Theorem 4.1.1 and Corollaries
4.1.1-4.1.2 to specific probability models.

Example 4.1.1. Let $X_{1}$ and $X_{2}$ be independent gamma random variables
with joint p.d.f. (4.1), where,
$f(z_{1},z_{2})=\frac{z_{1}^{\alpha_{1}-1}z_{2}^{\alpha_{2}-1}e^{-z_{1}}e^{-z_{2}}}{\Gamma(\alpha_{1})\Gamma(\alpha_{2})},\;(z_{1},z_{2})\in\Re_{++}^{2}$,
for known positive constants $\alpha_{1}$ and $\alpha_{2}.$

Consider estimation of the smaller scale parameter $\theta_{1}$, under the GPN
criterion with a general loss function, when it is known apriori that
$\boldsymbol{\theta}\in\Theta_{0}$. Here the restricted MLE of $\theta_{1}$ is
$\delta_{1,RMLE}(\mathbb{X})\\!=\\!\min\big{\\{}\\!\frac{X_{1}}{\alpha_{1}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}}\\!\big{\\}}=X_{1}\psi_{1,RMLE}(D)$,
where
$\psi_{1,RMLE}(D)=\min\big{\\{}\frac{1}{\alpha_{1}},\frac{1+D}{\alpha_{1}+\alpha_{2}}\\}$
and the unrestricted PNSEE of $\theta_{1}\text{ is
}\delta_{1,PNSEE}(\mathbb{X})=\frac{X_{1}}{m_{0,1}}$, where $m_{0,1}$ is such
that
$\frac{1}{\Gamma(\alpha_{1})}\int_{0}^{m_{0,1}}t^{\alpha_{1}-1}e^{-t}\,dt=\frac{1}{2}$.

For $t\in\Re_{++}$ and $\lambda\geq 1$, the conditional p.d.f. of $Z_{1}$
given $D=t$ is

$\displaystyle
f_{\lambda}(s|t)=\begin{cases}\frac{(1+\frac{t}{\lambda})^{\alpha_{1}+\alpha_{2}}\,s^{\alpha_{1}+\alpha_{2}-1}\,e^{-(1+\frac{t}{\lambda})s}}{\Gamma{(\alpha_{1}+\alpha_{2})}},&\text{
if}\;\;0<s<\infty\\\ 0,&\text{ otherwise}\end{cases}.$

For $t\in(0,\infty)$ and $\lambda\geq 1$, let $m_{\lambda}^{(1)}(t)$ be the
median of the p.d.f. $f_{\lambda}(s|t),\;t\in(0,\infty)$ and $\lambda\geq 1.$
For $\alpha>0$, let $\nu(\alpha)$ denote the median of Gamma($\alpha$,1)
distribution, i.e.
$\frac{1}{\Gamma(\alpha)}\int_{0}^{\nu(\alpha)}t^{\alpha-1}e^{-t}\,dt=\frac{1}{2}$.
Then, $m_{0,1}=\nu(\alpha_{1})$ and, for any $t>0$ and $\lambda\geq 1$,
$\left(\frac{\lambda+t}{\lambda}\right)m_{\lambda}^{(1)}(t)=\nu(\alpha_{1}+\alpha_{2})$,
$m_{1}^{(1)}(t)=\frac{\nu(\alpha_{1}+\alpha_{2})}{1+t}$ and
$\lim_{\lambda\to\infty}m_{\lambda}(t)=\nu(\alpha_{1}+\alpha_{2})$. From Chen
and Rubin (1986), we have
$\alpha_{1}+\alpha_{2}-\frac{1}{3}<\nu(\alpha_{1}+\alpha_{2})<\alpha_{1}+\alpha_{2}$.

Thus, as in (4.3) and (4.4), we may take
$l^{(1)}(t)=m_{1}^{(1)}(t)=\frac{\nu(\alpha_{1}+\alpha_{2})}{1+t}\text{ and
}u^{(1)}(t)\\!=\lim\limits_{\lambda\to\infty}m_{\lambda}^{(1)}(t)=\nu(\alpha_{1}+\alpha_{2}).$

The following conclusions are immediate from Theorem 4.1.1 and Corollaries
4.1.1-4.1.2:

(i) The estimator
$\delta_{1,RMLE}^{*}(\mathbb{X})=\max\big{\\{}\frac{X_{1}}{\nu(\alpha_{1}+\alpha_{2})},\min\big{\\{}\frac{X_{1}}{\alpha_{1}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}}\big{\\}}\big{\\}}$
is nearer to $\theta_{1}$ than the restricted MLE
$\delta_{1,RMLE}(\mathbb{X})=\min\\{\frac{X_{1}}{\alpha_{1}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}}\\}$.
Clearly, for $\nu(\alpha_{1}+\alpha_{2})\geq\alpha_{1}$,

$\delta_{1,RMLE}^{*}(\mathbb{X})=\begin{cases}\frac{X_{1}}{\nu(\alpha_{1}+\alpha_{2})},&\text{
if
}0<\frac{X_{2}}{X_{1}}\leq\frac{\alpha_{1}+\alpha_{2}}{\nu(\alpha_{1}+\alpha_{2})}-1\\\
\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}},&\text{ if
}\frac{\alpha_{1}+\alpha_{2}}{\nu(\alpha_{1}+\alpha_{2})}-1<\frac{X_{2}}{X_{1}}\leq\frac{\alpha_{2}}{\alpha_{1}}\\\
\frac{X_{1}}{\alpha_{1}},&\text{ if
}\frac{X_{2}}{X_{1}}>\frac{\alpha_{2}}{\alpha_{1}}\end{cases}$

and, for $\nu(\alpha_{1}+\alpha_{1})<\alpha_{1}$,
$\delta_{1,RMLE}^{*}(\mathbb{X})=\frac{X_{1}}{\nu(\alpha_{1}+\alpha_{2})}$.
Since $\nu(\alpha_{1}+\alpha_{2})>\alpha_{1}+\alpha_{2}-\frac{1}{3}$, we have
$\nu(\alpha_{1}+\alpha_{2})>\alpha_{1}$, if $\alpha_{2}\geq\frac{1}{3}$.

(ii) The estimator
$\delta_{1,PNSEE}^{*}(\mathbb{X})=\max\big{\\{}\frac{X_{1}}{\nu(\alpha_{1}+\alpha_{2})},\min\big{\\{}\frac{X_{1}}{\nu(\alpha_{1})},\frac{X_{1}+X_{2}}{\nu(\alpha_{1}+\alpha_{2})}\big{\\}}\big{\\}}$
is nearer to $\theta_{1}$ than
$\delta_{1,PNSEE}(\mathbb{X})=\frac{X_{1}}{\nu(\alpha_{1})}$. Clearly

$\delta_{1,PNSEE}^{*}(\mathbb{X})=\begin{cases}\frac{X_{1}+X_{2}}{\nu(\alpha_{1}+\alpha_{2})},&\text{
if
}0<\frac{X_{2}}{X_{1}}\leq\frac{\nu(\alpha_{1}+\alpha_{2})}{\nu(\alpha_{1})}-1\\\
\frac{X_{1}}{\nu(\alpha_{1})},&\text{ if
}\frac{X_{2}}{X_{1}}>\frac{\nu(\alpha_{1}+\alpha_{2})}{\nu(\alpha_{1})}-1\end{cases}.$

(iii) The estimator
$\delta_{1,UE}^{*}(\mathbb{X})=\max\big{\\{}\frac{X_{1}}{\nu(\alpha_{1}+\alpha_{2})},\min\big{\\{}\frac{X_{1}}{\alpha_{1}},\frac{X_{1}+X_{2}}{\nu(\alpha_{1}+\alpha_{2})}\big{\\}}\big{\\}}$
is nearer to $\theta_{1}$ than the unbiased estimator
$\delta_{1,UE}(\mathbb{X})=\frac{X_{1}}{\alpha_{1}}$. Clearly, for
$\nu(\alpha_{1}+\alpha_{2})\geq\alpha_{1}$,

$\delta_{1,UE}^{*}(\mathbb{X})=\begin{cases}\frac{X_{1}+X_{2}}{\nu(\alpha_{1}+\alpha_{2})},&\text{
if
}0\leq\frac{X_{2}}{X_{1}}<\frac{\nu(\alpha_{1}+\alpha_{2})}{\alpha_{1}}-1\\\
\frac{X_{1}}{\alpha_{1}},&\text{ if
}\frac{X_{2}}{X_{1}}\geq\frac{\nu(\alpha_{1}+\alpha_{2})}{\alpha_{1}}-1\end{cases}$

and, for $\nu(\alpha_{1}+\alpha_{1})<\alpha_{1}$,
$\delta_{1,UE}^{*}(\mathbb{X})=\frac{X_{1}}{\nu(\alpha_{1}+\alpha_{2})}$.

(iv) For $\nu(\alpha_{1}+\alpha_{2})\leq\alpha_{1}$, the estimator
$\delta_{1,UE}(\mathbb{X})=\frac{X_{1}}{\alpha_{1}}$ is nearer to $\theta_{1}$
than the restricted MLE
$\delta_{1,RMLE}(\mathbb{X})=\min\\{\frac{X_{1}}{\alpha_{1}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}}\\}$.
Ma and Liu (2014) proved a similar result under the GPN criterion with a
specific loss function
$L_{1}(\boldsymbol{\theta},a)=|\frac{a}{\theta_{1}}-1|,\;\boldsymbol{\theta}\in\Theta_{0},\;a>0$.
In fact several results reported in Ma and Liu (2014) can be obtained as
particular cases of Corollary 4.1.1.

Example 4.1.2. Let $X_{1}$ and $X_{2}$ be independent random variables with
joint p.d.f. (4.1), where
$\boldsymbol{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0}$ and
$f\left(z_{1},z_{2}\right)=\alpha_{1}\alpha_{2}z_{1}^{\alpha_{1}-1}z_{2}^{\alpha_{2}-1},\text{
if }0<z_{1}<1,\,0<z_{2}<1;=0,\text{ otherwise}$, for known positive constants
$\alpha_{1}$ and $\alpha_{2}$.

Consider estimation of $\theta_{1}$ under the GPN criterion with general loss
function. Here the unrestricted PNSEE of $\theta_{1}\text{ is
}\delta_{1,PNSEE}(\mathbb{X})=2^{\frac{1}{\alpha_{1}}}X_{1}$. Also, for
$t\in\Re_{++}$, the conditional p.d.f. of $Z_{1}$ given $D=t$ is

$\displaystyle
f_{\lambda}(s|t)=\frac{(\alpha_{1}+\alpha_{2})s^{\alpha_{1}+\alpha_{2}-1}}{\left(\min\\{1,\frac{\lambda}{t}\\}\right)^{\alpha_{1}+\alpha_{2}}},\text{
if}\;0<s<\min\bigg{\\{}1,\frac{\lambda}{t}\bigg{\\}},\;\lambda\geq 1.$

The median of the above density is
$m_{\lambda}^{(1)}(t)=2^{\frac{-1}{\alpha_{1}+\alpha_{2}}}\min\big{\\{}1,\frac{\lambda}{t}\big{\\}},\;\forall\;t\in\Re_{++}$
and $\lambda\geq 1.$ Clearly, for $t\in\Re_{++}$, $m_{\lambda}^{(1)}(t)$ is
increasing in $\lambda\in[1,\infty)$. As in (4.3) and (4.4), we take
$l^{(1)}(t)=m_{1}^{(1)}(t)=2^{\frac{-1}{\alpha_{1}+\alpha_{2}}}\min\big{\\{}1,\frac{1}{t}\big{\\}}\text{
and
}u^{(1)}(t)\\!=\lim\limits_{\lambda\to\infty}m_{\lambda}^{(1)}(t)=2^{\frac{-1}{\alpha_{1}+\alpha_{2}}}.$

The following conclusions immediately follow from Theorem 4.1.1 and Corollary
4.1.1:

(i) Define
$\delta_{1,PNSEE}^{*}(\mathbb{X})=\max\\!\big{\\{}\\!2^{\frac{1}{\alpha_{1}+\alpha_{2}}},\min\\{2^{\frac{1}{\alpha_{1}+\alpha_{2}}}\max\\{1,t\\},2^{\frac{1}{\alpha_{1}}}\\}\big{\\}}X_{1}=$
$\min\\{2^{\frac{1}{\alpha_{1}+\alpha_{2}}}\max\\{1,t\\},2^{\frac{1}{\alpha_{1}}}\\}X_{1}$.
Then the estimator $\delta_{1,PNSEE}^{*}(\mathbb{X})$ is nearer to
$\theta_{1}$ than the PNSEE $\delta_{1,PNSEE}(\mathbb{X})$. It is easy to
verify that

$\delta_{1,PNSEE}^{*}(\mathbb{X})=\begin{cases}2^{\frac{1}{\alpha_{1}+\alpha_{2}}}X_{1},&\text{if
}0<\frac{X_{2}}{X_{1}}<1\\\ 2^{\frac{1}{\alpha_{1}+\alpha_{2}}}X_{2},&\text{if
}1\leq\frac{X_{2}}{X_{1}}<2^{\frac{\alpha_{2}}{\alpha_{1}(\alpha_{1}+\alpha_{2})}}\\\
2^{\frac{1}{\alpha_{1}}}X_{1},&\text{if }\frac{X_{2}}{X_{1}}\geq
2^{\frac{\alpha_{2}}{\alpha_{1}(\alpha_{1}+\alpha_{2})}}\end{cases}.$

(ii) Let $\psi_{1,0}(t)$ be such that
$2^{\frac{1}{\alpha_{1}+\alpha_{2}}}\max\\{1,t\\}\leq\psi_{1,0}(t)<2^{\frac{1}{\alpha_{1}}},\;\forall\;t\leq
2^{\frac{\alpha_{2}}{\alpha_{1}(\alpha_{1}+\alpha_{2})}}$, and
$\psi_{1,0}(t)=2^{\frac{1}{\alpha_{1}}},\;\forall\;t>2^{\frac{\alpha_{2}}{\alpha_{1}(\alpha_{1}+\alpha_{2})}}$.
Then the estimator $\delta_{\psi_{1,0}}(\mathbb{X})=\psi_{1,0}(D)X_{1}$ is
nearer to $\theta_{1}$ than the PNSEE $\delta_{1,PNSEE}(\mathbb{X})$.

###  4.2. Estimation of The Larger Scale Parameter $\theta_{2}$

In this section, we consider estimation of the larger scale parameter
$\theta_{2}$ under the GPN criterion with a general loss function
$L_{2}(\boldsymbol{\theta},a)=W(\frac{a}{\theta_{2}})$, when it is known that
$0<\theta_{1}\leq\theta_{2}<\infty$ (i.e.,
$\boldsymbol{\theta}\in\Theta_{0}$). Here $W:\Re_{++}\rightarrow\Re_{++}$ is
such that $W(1)=0$, $W(t)$ is strictly decreasing in $(0,1)$ and strictly
increasing in $(1,\infty)$. Here, any scale equivariant estimator of
$\theta_{2}$ is of the form $\delta_{\psi}(\mathbb{X})=\psi(D)X_{2},$ for some
function $\psi:\,\Re_{++}\rightarrow\Re_{++}$, where $D=\frac{X_{2}}{X_{1}}$.
Define $Z_{2}=\frac{X_{2}}{\theta_{2}}$,
$\lambda=\frac{\theta_{2}}{\theta_{1}}$ and $f_{D}(t|\lambda)$ the p.d.f. of
r.v. $D=\frac{X_{2}}{X_{1}}$. Let $\delta_{\xi}(\mathbb{X})=\xi(D)X_{2}$ and
$\delta_{\psi}(\mathbb{X})=\psi(D)X_{2}$ be two scale equivariant estimators
of $\theta_{2}$. Then, the GPN of $\delta_{\xi}(\mathbb{X})=\xi(D)X_{2}$
relative to $\delta_{\psi}(\mathbb{X})=\psi(D)X_{2}$ can be written as

$\displaystyle GPN(\delta_{\xi},\delta_{\psi};\boldsymbol{\theta})$
$\displaystyle=\int_{0}^{\infty}g_{2,\lambda}(\xi(t),\psi(t),t)f_{D}(t|\lambda)dt,\;\;\boldsymbol{\theta}\in\Theta_{0},$

where, for $\lambda\geq 1$,
$g_{2,\lambda}(\xi(t),\psi(t),t)=P_{\boldsymbol{\theta}}[W(\xi(t)Z_{2})<W(\psi(t)Z_{2})|D=t]+\frac{1}{2}P_{\boldsymbol{\theta}}[W(\xi(t)Z_{2})$
$=W(\psi(t)Z_{2})|D=t].$ For any fixed $\lambda\geq 1$ and $t$, let
$m_{\lambda}^{(2)}(t)$ denote the median of the conditional distributional of
$Z_{2}$ given $D=t$. For any fixed $t$, the conditional p.d.f. of $Z_{2}$
given $D=t$ is $f_{\lambda}(s|t)=\frac{\frac{\lambda s}{t^{2}}f(\frac{\lambda
s}{t},s)}{f_{D}(t|\lambda)}$ and
$f_{D}(t|\lambda)=\int_{0}^{\infty}\frac{\lambda y}{t^{2}}f(\frac{\lambda
y}{t},y)dy$, $\lambda\geq 1$. Thus
$\int_{0}^{m_{\lambda}^{(2)}(t)}sf(\frac{\lambda
s}{t},s)ds=\frac{1}{2}\int_{0}^{\infty}sf(\frac{\lambda s}{t},s)ds$. Using
Lemma 4.1, we have, for any fixed $t$ and $\lambda\geq 1$,
$g_{2,\lambda}(\xi(t),\psi(t),t)>\frac{1}{2}$, provided
$m_{\lambda}^{(2)}(t)\leq\frac{1}{\xi(t)}<\frac{1}{\psi(t)}$ or
$\frac{1}{\psi(t)}<\frac{1}{\xi(t)}\leq m_{\lambda}^{(2)}(t)$. Moreover, for
any fixed $t$,
$g_{2,\lambda}(\psi(t),\psi(t),t)=\frac{1}{2},\;\forall\;\lambda\geq 1$. These
arguments lead to the following results.

Theorem 4.2.1. Let $\delta_{\psi}(\mathbb{X})=\psi(D)X_{2}$ be a scale
equivariant estimator of $\theta_{2}$. Let $l^{(2)}(t)$ and $u^{(2)}(t)$ be
functions such that $0<l^{(2)}(t)\leq m_{\lambda}^{(2)}(t)\leq
u^{(2)}(t),\;\forall\;\lambda\geq 1$ and any $t$. For any fixed $t$, define
$\psi^{*}(t)\\!=\\!\max\\{\frac{1}{u^{(2)}(t)},\min\\{\psi(t),\frac{1}{l^{(2)}(t)}\\}\\}$.
Then, under the GPN criterion, the estimator
$\delta_{\psi^{*}}(\mathbb{X})\\!=\\!\psi^{*}(D)X_{2}$ is nearer to
$\theta_{2}$ than the estimator $\delta_{\psi}(\mathbb{X})=\psi(D)X_{2}$,
provided
$P_{\boldsymbol{\theta}}\left[\frac{1}{u^{(2)}(D)}\leq\psi(D)\leq\frac{1}{l^{(2)}(D)}\right]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$.

Corollary 4.2.1. Let $\delta_{\psi}(\mathbb{X})=\psi(D)X_{2}$ be a scale
equivariant estimator of $\theta_{2}$. Let
$\psi_{2,0}:\Re_{++}\rightarrow\Re_{++}$ be such that
$\psi(t)<\psi_{2,0}(t)\leq\frac{1}{u^{(2)}(t)}$, whenever
$\psi(t)<\frac{1}{u^{(2)}(t)}$, and
$\frac{1}{l^{(2)}(t)}\leq\psi_{2,0}(t)<\psi(t)$, whenever
$\frac{1}{l^{(2)}(t)}<\psi(t)$, where $l^{(2)}(\cdot)$ and $u^{(2)}(\cdot)$
are as defined in Theorem 4.2.1. Also let $\psi_{2,0}(t)=\psi(t)$, whenever
$\frac{1}{u^{(2)}(t)}\leq\psi(t)\leq\frac{1}{l^{(2)}(t)}$. Then,
$GPN(\delta_{\psi_{2,0}},\delta_{\psi};\boldsymbol{\theta})>\frac{1}{2},\;\forall\;\boldsymbol{\theta}\in\Theta_{0},$
provided
$P_{\boldsymbol{\theta}}\left[\frac{1}{u^{(2)}(D)}\leq\psi(D)\leq\frac{1}{l^{(2)}(D)}\right]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$,
where $\delta_{\psi_{2,0}}(\mathbb{X})=\psi_{2,0}(D)X_{2}$.

Note that, in the unrestricted case $\Theta\\!=\\!\Re_{++}^{2}$, the PNSEE of
$\theta_{2}$ is $\delta_{2,PNSEE}(\mathbb{X})=\frac{X_{2}}{m_{0,2}}$, where
$m_{0,2}>0$ is the median of the r.v. $Z_{2}=\frac{X_{2}}{\theta_{2}}$. The
following corollary provides improvements over the PNSEE under the restricted
parameter space.

Corollary 4.2.2. Let
$\xi^{*}(t)\\!=\\!\max\\{\frac{1}{u^{(2)}(t)},\min\\{\frac{1}{m_{0,2}},\frac{1}{l^{(2)}(t)}\\}\\}$,
where $l^{(2)}(\cdot)$ and $u^{(2)}(\cdot)$ are as defined in Theorem 4.2.1.
Then, under the GPN criterion, the estimator
$\delta_{\xi^{*}}(\mathbb{X})\\!=\\!\xi^{*}(D)X_{2}$ is nearer to $\theta_{2}$
than the PNSEE $\delta_{2,PNSEE}(\mathbb{X})=\frac{X_{2}}{m_{0,2}}$, provided
$P_{\boldsymbol{\theta}}\left[l^{(2)}(D)\leq m_{0,2}\leq
u^{(2)}(D)\right]<1,\;\forall\;\boldsymbol{\theta}\in\Theta_{0}$.

In order to identify the behaviour of $m_{\lambda}^{(2)}(t),$ for any fixed
$t$, we have the following lemma on the lines of Lemma 4.1.1.

Lemma 4.2.1. If, for every fixed $\lambda\geq 1$ and $t$, $f(\frac{\lambda
s}{t},s)/f(\frac{s}{t},s)$ is increasing (decreasing) in $s$ (wherever the
ratio is not of the form $0/0$), then, for every fixed $t$,
$m_{\lambda}^{(2)}(t)$ is an increasing (decreasing) function of
$\lambda\in[1,\infty)$.

Under the assumptions of Lemma 4.2.1, one may take, for any fixed $t$,

(4.5) $\displaystyle l^{(2)}(t)$ $\displaystyle=\inf_{\lambda\geq
1}m_{\lambda}^{(2)}(t)=m_{1}^{(2)}(t)\;(=\lim_{\lambda\to\infty}m_{\lambda}^{(2)}(t))$
(4.6) $\displaystyle\;\text{and}\;\;u^{(2)}(t)$
$\displaystyle=\sup_{\lambda\geq
1}m_{\lambda}^{(2)}(t)=\lim_{\lambda\to\infty}m_{\lambda}^{(2)}(t)\;(=m_{1}^{(2)}(t)),$

while applying Theorem 4.2.1 and Corollary 4.2.1.

As in Section 4.1, we will now apply Theorem 4.2.1 and Corollaries 4.2.1-4.2.2
to estimation of the larger scale parameter $\theta_{2}$ in scale probability
models considered in Examples 4.1.1-4.1.2.

Example 4.2.1. Let $X_{1}$ and $X_{2}$ be independent gamma random variables
as defined in Example 4.1.1. Consider estimation of $\theta_{2}$, under the
GPN criterion with a general loss function.

For $t\in\Re_{++}$ and $\lambda\geq 1$, the conditional p.d.f. of $Z_{2}$
given $D=t$ is

$\displaystyle
f_{\lambda}(s|t)=\begin{cases}\frac{(1+\frac{\lambda}{t})^{\alpha_{1}+\alpha_{2}}\,s^{\alpha_{1}+\alpha_{2}-1}\,e^{-(1+\frac{\lambda}{t})s}}{\Gamma{(\alpha_{1}+\alpha_{2})}},&\text{
if}\;\;0<s<\infty\\\ 0,&\text{ otherwise}\end{cases}.$

Let $\nu(\alpha)$ denote the median of Gamma($\alpha$,1) distribution. Then
$m_{\lambda}^{(2)}(t)=\left(\frac{t}{\lambda+t}\right)\nu(\alpha_{1}+\alpha_{2})$,
$\lambda\geq 1,\;t>0$, and, as in (4.5) and (4.6), we may take
$l^{(2)}(t)=0,\;t>0$ and
$u^{(2)}(t)=\frac{t}{1+t}\,\nu(\alpha_{1}+\alpha_{2}),\;t>0$. Also
$m_{0,2}=\nu(\alpha_{2})$ and the PNSEE of $\theta_{2}$ is
$\delta_{2,PNSEE}(\mathbb{X})=\frac{X_{2}}{\nu(\alpha_{2})}$. The restricted
MLE of $\theta_{2}$ is
$\delta_{2,RMLE}(\mathbb{X})=\max\\{\frac{X_{2}}{\alpha_{2}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}}\\}=\psi_{2,R}(D)X_{2}$,
where
$\psi_{2,R}(t)=\max\\{\frac{1}{\alpha_{2}},\frac{1+t}{t(\alpha_{1}+\alpha_{2})}\\},\;t>0$.
Using Theorem 4.2.2 and Corollary 4.2.1, the following conclusions are
evident:

(i) The estimator
$\delta_{2,RMLE}^{*}(\mathbb{X})=\max\big{\\{}\frac{X_{2}}{\alpha_{2}},\frac{X_{1}+X_{2}}{\nu(\alpha_{1}+\alpha_{2})}\big{\\}}$
is nearer to $\theta_{2}$ than the restricted MLE
$\delta_{2,RMLE}(\mathbb{X})=\max\\{\frac{X_{2}}{\alpha_{2}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}}\\}$.

(ii) The estimator
$\delta_{2,PNSEE}^{*}(\mathbb{X})=\max\big{\\{}\frac{X_{2}}{\nu(\alpha_{2})},\frac{X_{1}+X_{2}}{\nu(\alpha_{1}+\alpha_{2})}\big{\\}}$
is nearer to $\theta_{2}$ than the PNSEE
$\delta_{2,PNSEE}(\mathbb{X})=\frac{X_{2}}{\nu(\alpha_{2})}$.

(iii) The restricted MLE
$\delta_{2,RMLE}(\mathbb{X})=\max\\{\frac{X_{2}}{\alpha_{2}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}}\\}$
is nearer to $\theta_{2}$ than the unbiased estimator
$\delta_{2,UE}(\mathbb{X})=\frac{X_{2}}{\alpha_{2}}$. This result, under a
specific loss function
$L_{2}(\boldsymbol{\theta},a)=|\frac{a}{\theta_{2}}-1|,\;\boldsymbol{\theta}\in\Theta_{0},\;a>0$,
is proved in Ma and Liu (2014).

Example 4.2.2. Let $X_{1}$ and $X_{2}$ be independent random variables as
described in Example 4.1.2. Consider estimation of $\theta_{2}$ under the GPN
criterion with a general error loss function. Here the PNSEE of
$\theta_{2}\text{ is
}\delta_{2,PNSEE}(\mathbb{X})=2^{\frac{1}{\alpha_{2}}}X_{2}$. Also, for
$\lambda\geq 1$ and $t\in[0,\infty)$, the conditional p.d.f. of $Z_{2}$ given
$D=t$ is

$\displaystyle
f_{\lambda}(s|t)=\frac{(\alpha_{1}+\alpha_{2})s^{\alpha_{1}+\alpha_{2}-1}}{\left(\min\\{1,\frac{t}{\lambda}\\}\right)^{\alpha_{1}+\alpha_{2}}},\text{
if}\;0<s<\min\bigg{\\{}1,\frac{t}{\lambda}\bigg{\\}}.$

The median of the above density is
$m_{\lambda}^{(2)}(t)=2^{\frac{-1}{\alpha_{1}+\alpha_{2}}}\min\big{\\{}1,\frac{t}{\lambda}\big{\\}},\;\forall\;t\in\Re_{++}$
and $\lambda\geq 1.$ Clearly, for $t\in\Re_{++}$, $m_{\lambda}^{(2)}(t)$ is
decreasing in $\lambda\in[1,\infty)$. As in (4.5) and (4.6), we may take
$l^{(2)}(t)=\lim\limits_{\lambda\to\infty}m_{\lambda}^{(2)}(t)=0\text{ and
}u^{(2)}(t)\\!=m_{1}^{(2)}(t)=2^{\frac{-1}{\alpha_{1}+\alpha_{2}}}\min\\{1,t\\}.$

The following conclusions immediately follow from Theorem 4.2.1 and Corollary
4.2.1:

(i) The estimator
$\delta_{2,PNSEE}^{*}(\mathbb{X})=\max\\!\big{\\{}\\!2^{\frac{1}{\alpha_{1}+\alpha_{2}}}X_{1},2^{\frac{1}{\alpha_{2}}}X_{2}\big{\\}}$
is nearer to $\theta_{1}$ than the PNSEE
$\delta_{2,PNSEE}(\mathbb{X})=2^{\frac{1}{\alpha_{2}}}X_{2}$.

(ii) Let $\psi_{2,0}(t)$ be such that
$2^{\frac{1}{\alpha_{2}}}<\psi_{2,0}(t)\leq
2^{\frac{1}{\alpha_{1}+\alpha_{2}}}\max\big{\\{}1,\frac{1}{t}\big{\\}},\;\forall\;t\leq
2^{\frac{-\alpha_{1}}{\alpha_{2}(\alpha_{1}+\alpha_{2})}}$, and
$\psi_{2,0}(t)=2^{\frac{1}{\alpha_{2}}},\;\forall\;t>2^{\frac{-\alpha_{1}}{\alpha_{2}(\alpha_{1}+\alpha_{2})}}$.
Then the estimator $\delta_{\psi_{2,0}}(\mathbb{X})=\psi_{2,0}(D)X_{2}$ is
nearer to $\theta_{2}$ than the PNSEE
$\delta_{2,PNSEE}(\mathbb{X})=2^{\frac{1}{\alpha_{2}}}X_{2}$.

##  5\. Simulation Study

###  5.1. For Smaller Location Parameter $\theta_{1}$

In Example 3.1.1, we considered a bivariate normal distribution with unknown
means $\theta_{1}$ and $\theta_{2}$
($-\infty<\theta_{1}\leq\theta_{2}<\infty$), known variances
$\sigma_{1}^{2}>0$ and $\sigma_{2}^{2}>0$, and known correlation coefficient
$\rho$ ($-1<\rho<1$). For estimation of $\theta_{1}$ under GPN criterion, we
considered various estimators. To further evaluate the performances of these
estimators, in this section, we compare these estimators under the GPN
criterion with the absolute error loss (i.e., $W(t)=|t|,\;t\in\Re$),
numerically. For simulations, 10,000 random samples of size 1 were generated
from the relevant bivariate normal distribution. For various configurations of
$\alpha=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}}$,
using the Monte Carlo simulations, we obtained the GPN values of the
restricted MLE
($\delta_{1,RMLE}^{*}(\mathbb{X})=X_{1}-(1-\alpha)\max\\{0,-D\\}$) relative to
the PNLEE ($\delta_{1,PNLEE}(\mathbb{X})=X_{1}$), of the improved Hwang and
Peddada (HP) estimator ($\delta_{1,HP}^{*}(\mathbb{X})=\alpha
X_{1}+(1-\alpha)X_{2}$) relative to the Hwang and Peddada (HP) estimator
($\delta_{1,HP}(\mathbb{X})=X_{1}-\max\\{0,(\alpha-1)D\\}$) and of the
restricted MLE ($\delta_{1,RMLE}(\mathbb{X})=X_{1}-(1-\alpha)\max\\{0,-D\\}$)
relative to the Tan and Peddada (PDT) estimator
($\delta_{1,PDT}(\mathbb{X})=X_{1}-\max\\{0,-D\\}$). These values are
tabulated in Tables 1-3. The following observations are evident from Tables
1-3:

(i) All the GPN values are greater than $0.5$, which is in conformity with the
theoretical findings of Example 3.1.1.
(ii) From Table 1, we can observe that, when $\sigma_{1}$ is relatively larger
than $\sigma_{2}$, we get higher GPN values. Also, for negative $\rho$, the
GPN values are higher.
(iii) From Table 2, we can see that, as the value of
$\rho\sigma_{2}-\sigma_{1}$ ($>0$) increases, the GPN value also increases.
Also, from Table 3, we observe that as the value of
$\rho\sigma_{1}-\sigma_{2}$ ($>0$) increases, the GPN value also increases.

Table 1. The GPN values of the restricted MLE ($\delta_{1,RMLE}$) relative to the PNLEE ($\delta_{1,PNLEE}$) | ($\sigma_{1},\sigma_{2},\rho$)
---|---
$\theta_{2}-\theta_{1}$ | (3,0.5,-0.9) | (0.5,5,-0.5) | (1,1,0) | (15,2,0.2) | (1,30,0.5) | (30,1,0.9)
0.0 | 0.743 | 0.557 | 0.560 | 0.708 | 0.549 | 0.740
0.5 | 0.713 | 0.577 | 0.609 | 0.718 | 0.537 | 0.749
1.0 | 0.693 | 0.548 | 0.580 | 0.722 | 0.545 | 0.740
1.5 | 0.662 | 0.558 | 0.547 | 0.719 | 0.535 | 0.740
2.0 | 0.626 | 0.566 | 0.540 | 0.717 | 0.546 | 0.753
2.5 | 0.611 | 0.570 | 0.516 | 0.721 | 0.557 | 0.741
3.0 | 0.584 | 0.575 | 0.507 | 0.709 | 0.556 | 0.732
Table 2. The GPN values of the improved HP ($\delta_{1,HP}^{*}$) relative to the HP ($\delta_{1,HP}$) when $\alpha>1$ (i.e. $\sigma_{1}<\rho\sigma_{2}$) | ($\sigma_{1},\sigma_{2},\rho$)
---|---
$\theta_{2}-\theta_{1}$ | (0.1,5,0.2) | (1,25,0.2) | (0.5,2,0.5) | (5,15,0.5) | (0.5,5,0.9) | (2,15,0.9)
0 | 0.520 | 0.502 | 0.524 | 0.512 | 0.620 | 0.619
0.5 | 0.515 | 0.514 | 0.523 | 0.521 | 0.631 | 0.620
1 | 0.516 | 0.508 | 0.532 | 0.518 | 0.633 | 0.621
1.5 | 0.520 | 0.502 | 0.528 | 0.520 | 0.639 | 0.625
2 | 0.524 | 0.511 | 0.521 | 0.514 | 0.626 | 0.634
2.5 | 0.520 | 0.520 | 0.518 | 0.513 | 0.621 | 0.630
3 | 0.513 | 0.515 | 0.510 | 0.516 | 0.609 | 0.629
Table 3. The GPN values of the restricted MLE ($\delta_{1,RMLE}$) relative to the PDT ($\delta_{1,PDT}$) when $\alpha<0$ (i.e. $\sigma_{2}<\rho\sigma_{1}$) | ($\sigma_{1},\sigma_{2},\rho$)
---|---
$\theta_{2}-\theta_{1}$ | (5,0.1,0.2) | (25,1,0.2) | (2,0.5,0.5) | (15,5,0.5) | (5,0.5,0.9) | (15,2,0.9)
0 | 0.512 | 0.516 | 0.522 | 0.516 | 0.617 | 0.619
0.5 | 0.733 | 0.613 | 0.650 | 0.526 | 0.723 | 0.679
1 | 0.706 | 0.677 | 0.640 | 0.550 | 0.708 | 0.712
1.5 | 0.692 | 0.713 | 0.598 | 0.577 | 0.683 | 0.722
2 | 0.672 | 0.729 | 0.569 | 0.588 | 0.667 | 0.720
2.5 | 0.653 | 0.730 | 0.539 | 0.597 | 0.644 | 0.710
3 | 0.633 | 0.725 | 0.522 | 0.611 | 0.630 | 0.706

###  5.2. For Larger Scale Parameter $\theta_{2}$

In this section, for estimation of the larger scale parameter $\theta_{2}$,
under the GPN criterion with loss function
$L_{2}(\underline{\theta},a)=|\frac{a}{\theta_{2}}-1|,\;a\in\mathcal{A}=(0,\infty),\;\underline{\theta}\in\Theta_{0}$,
we numerically compare various estimators considered in Example 4.2.1. For
simulations, 10,000 random samples of size 1 were generated from relevant
gamma distributions. Using the Monte Carlo simulations, we obtained the GPN
values of the improved restricted MLE
($\delta_{2,RMLE}^{*}(\mathbb{X})=\max\big{\\{}\frac{X_{2}}{\alpha_{2}},\frac{X_{1}+X_{2}}{\nu(\alpha_{1}+\alpha_{2})}\big{\\}}$)
relative to the restricted MLE
($\delta_{2,RMLE}(\mathbb{X})=\max\big{\\{}\frac{X_{2}}{\alpha_{2}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}}\big{\\}}$),
of the improved PNSEE
($\delta_{2,PNSEE}^{*}(\mathbb{X})=\max\big{\\{}\frac{X_{2}}{\nu(\alpha_{2})},\frac{X_{1}+X_{2}}{\nu(\alpha_{1}+\alpha_{2})}\big{\\}}$)
relative to the PNSEE
($\delta_{2,PNSEE}(\mathbb{X})=\frac{X_{2}}{\nu(\alpha_{2})}$) and of the
restricted MLE
($\delta_{2,RMLE}(\mathbb{X})=\max\big{\\{}\frac{X_{2}}{\alpha_{2}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}}\big{\\}}$)
relative to the unbiased estimator
($\delta_{2,UE}(\mathbb{X})=\frac{X_{2}}{\alpha_{2}}$), as given in Table 4,
Table 5 and Table 6, receptively. The following observations are evident from
Tables 4-6:

(i) All the GPN values are greater than $0.5$, confirming our theoretical
findings of Example 4.2.1.
(ii) When $\alpha_{1}$ is relatively larger than $\alpha_{2}$, we get higher
GPN values.

Table 4. The GPN values of the modified restricted MLE ($\delta_{2,RMLE}^{*}$) relative to the restricted MLE ($\delta_{2,RMLE}$) | $(\alpha_{1},\alpha_{2})$
---|---
$\theta_{2}/\theta_{1}$ | (0.5,0.2) | (0.2,0.8) | (1,1) | (5,2) | (1,30) | (30,1)
1 | 0.552 | 0.538 | 0.518 | 0.515 | 0.508 | 0.503
1.5 | 0.613 | 0.567 | 0.603 | 0.640 | 0.519 | 0.736
2 | 0.664 | 0.580 | 0.627 | 0.635 | 0.522 | 0.699
2.5 | 0.692 | 0.592 | 0.634 | 0.605 | 0.522 | 0.666
3 | 0.715 | 0.600 | 0.635 | 0.586 | 0.522 | 0.643
3.5 | 0.731 | 0.595 | 0.627 | 0.568 | 0.519 | 0.622
4 | 0.740 | 0.606 | 0.623 | 0.554 | 0.515 | 0.610
Table 5. The GPN values of the modified PNSEE ($\delta_{2,PNSEE}^{*}$) relative to the PNSEE ($\delta_{2,PNSEE}$) | ($(\alpha_{1},\alpha_{2})$)
---|---
$\theta_{2}/\theta_{1}$ | (0.5,0.2) | (0.2,0.8) | (1,1) | (5,2) | (1,30) | (30,1)
1 | 0.573 | 0.522 | 0.568 | 0.600 | 0.514 | 0.695
1.5 | 0.612 | 0.543 | 0.595 | 0.631 | 0.522 | 0.686
2 | 0.632 | 0.545 | 0.601 | 0.599 | 0.519 | 0.648
2.5 | 0.645 | 0.548 | 0.596 | 0.575 | 0.514 | 0.620
3 | 0.650 | 0.551 | 0.585 | 0.558 | 0.510 | 0.602
3.5 | 0.659 | 0.549 | 0.581 | 0.544 | 0.508 | 0.591
4 | 0.656 | 0.549 | 0.571 | 0.537 | 0.506 | 0.580
Table 6. The GPN values of the restricted MLE ($\delta_{2,RMLE}$) relative to the unbiased estimator ($\delta_{2,UE}$) | $(\alpha_{1},\alpha_{2})$
---|---
$\theta_{2}/\theta_{1}$ | (0.5,0.2) | (0.2,0.8) | (1,1) | (5,2) | (1,30) | (30,1)
1 | 0.702 | 0.569 | 0.628 | 0.648 | 0.523 | 0.758
1.5 | 0.736 | 0.579 | 0.642 | 0.668 | 0.529 | 0.744
2 | 0.745 | 0.579 | 0.641 | 0.627 | 0.522 | 0.698
2.5 | 0.756 | 0.575 | 0.633 | 0.598 | 0.516 | 0.664
3 | 0.751 | 0.578 | 0.616 | 0.575 | 0.512 | 0.640
3.5 | 0.748 | 0.573 | 0.610 | 0.559 | 0.509 | 0.625
4 | 0.744 | 0.572 | 0.597 | 0.550 | 0.506 | 0.611

## Funding

This work was supported by the [Council of Scientific and Industrial Research
(CSIR)] under Grant [number 09/092(0986)/2018].

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