# Isotonic Regression Estimators For Simultaneous Estimation of Order
Restricted Location/Scale Parameters of a Bivariate Distribution: A Unified
Study

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

## Abstract

The problem of simultaneous estimation of location/scale parameters
$\theta_{1}$ and $\theta_{2}$ of a general bivariate location/scale model,
when the ordering between the parameters is known apriori (say,
$\theta_{1}\leq\theta_{2}$), has been considered. We consider isotonic
regression estimators based on the best location/scale equivariant estimators
(BLEEs/BSEEs) of $\theta_{1}$ and $\theta_{2}$ with general weight functions.
Let $\mathcal{D}$ denote the corresponding class of isotonic regression
estimators of $(\theta_{1},\theta_{2})$. Under the sum of the weighted squared
error loss function, we characterize admissible estimators within the class
$\mathcal{D}$, and identify estimators that dominate the BLEE/BSEE of
($\theta_{1}$,$\theta_{2}$). Our study unifies several studies reported in the
literature for specific probability distributions having independent
marginals. We also report a generalized version of the Katz (1963) result on
the inadmissibility of certain estimators under a loss function that is
weighted sum of general loss functions for component problems. A simulation
study is also carried out to validate the findings of the paper.

Keywords: Admissible estimators; BLEE; BSEE; Inadmissible estimators; Isotonic
regression; Location/Scale model; Mixed estimators.

##  1\. Introduction

Let $\underline{X}=(X_{1},X_{2})$ be a random vector having a joint
probability density function (p.d.f.)
$f_{\underline{\theta}}(x_{1},x_{2}),\;(x_{1},x_{2})\in\Re^{2},$ where
$\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta=\Re^{2}/\Re_{++}^{2}$ is
the vector of unknown location/scale parameters, $\Re_{++}=(0,\infty)$ and
$\Re^{2}$ denotes the two-dimensional Euclidean space. In many real life
situations, the ordering between the parameters $\theta_{1}$ and $\theta_{2}$
may be known apriori (say, $\theta_{1}\leq\theta_{2}$) and it may be of
interest to simultaneously estimate $\theta_{1}$ and $\theta_{2}$. For
example, in an engine efficiency measurement experiment where estimating the
average efficiency of an internal combustion engine (I.C. engine) and an
external combustion engine (E.C. engine) is of interest, it can be assumed
that the average efficiency of an I.C. engine is higher than the average
efficiency of an E.C. engine. For more real-life examples, one can see Barlow
et al. (1972) and Robertson et al. (1988).

Let $\Theta_{0}=\\{\underline{\theta}\in\Theta:\theta_{1}\leq\theta_{2}\\}$ be
the restricted parameter space. There is an extensive literature on estimation
of ($\theta_{1}$,$\theta_{2}$) (simultaneously as well as componentwise) when
it is known apriori that $\underline{\theta}\in\Theta_{0}$. Most of the early
studies were focussed around finding restricted maximum likelihood estimators
(under restriction that $\underline{\theta}\in\Theta_{0}$) and studying their
properties (see, for example, van Eeden (, , 1957, 1958) and Brunk (1955)).
Subsequently, a vast literature appeared on decision theoretic estimation of
order restricted estimators with special focus on risk properties of isotonic
regression and maximum likelihood estimators (see 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 and Kumar (2017)). For bivariate models,
isotonic regression estimators are also popularly called mixed estimators
(see, for example, Katz (1963), Kumar and Sharma (1988), and Vijayasree and
Singh (1991, 1993)). Most of the above mentioned studies are centred around
specific distributions with independent marginals and specific loss functions,
barring a few general studies (see, for example, Kelly (1989), Hwang and
Peddada (1994), and Kubokawa and Saleh (1994)). For a detailed account of
contributions in this area of research one may refer to the research monograph
by van Eeden (2006).

A natural goal in these problems is to use the prior information
$\underline{\theta}\in\Theta_{0}$ to find estimators that improve upon the
natural estimators for the unrestricted case (e.g., the best location/scale
estimators; unrestricted maximum likelihood estimators). This aspect of the
problem has been taken up by many researchers in the part for specific
probability models and/or specific loss functions (see, for example, Kumar and
Sharma (1989), Kushary and Cohen (1989), and Vijayasree and Singh (1991,
1993)).

Katz (1963) introduced mixed estimators for simultaneously estimating order
restricted parameters of two distributions. Under a general class of
reasonable loss functions (which includes squared error loss), he showed that
the natural estimators for the unrestricted case, that are not ordered in the
same way as parameters, are inadmissible and he proposed dominating
estimators. He further investigated a class of mixed estimators that retain
the ordering of parameters and obtained admissible and minimax estimators
under normality of the two distributions. Kumar and Sharma (1988) dealt with
simultaneous estimation of ordered means of two normal distributions having
known variances and characterized admissible estimators among mixed estimators
based on the BLEEs of $\theta_{1}$ and $\theta_{2}$. Jin and Pal (1991)
considered mixed estimators for simultaneous estimation of order restricted
location parameters of two independent exponential distributions and derived
estimators that are admissible within the class of mixed estimators. For
simultaneous estimation of order restricted scale parameters of two
independent exponential distributions, Vijayasree and Singh (1991) considered
the class of isotonic regression estimators based on unrestricted maximum
likelihood estimators and obtained estimators that are admissible within this
class of estimators. Patra and Kumar (2017) considered simultaneous estimation
of ordered means of a bivariate normal distribution under the sum of squared
error loss functions and discussed admissibility of isotonic regression
estimators based on BLEEs of $\theta_{1}$ and $\theta_{2}$.

In this paper we aim to unify some of the above studies for specific
probability models by considering a general bivariate location/scale model.
For simultaneous estimation of $\theta_{1}$ and $\theta_{2}$, under the
restricted parameter space and the sum of weighted squared error loss
functions, we consider the class of isotonic regression estimators based of
BLEEs/BSEEs. We will characterize admissible estimators within this class and
obtain subclass of estimators that dominate the BLEEs/BSEEs. The rest of the
paper is organized as follows. In Section 2, we introduce some useful
definitions and results that are used later in the paper. In Section 3, we
report a general result on the inadmissibility of certain estimators under a
loss function that is weighted sum of general loss functions. Section 4
(Section 5) deals with isotonic regression estimators of order restricted
location (scale) parameters.

##  2\. Some Useful Definitions and Results

We will discuss some properties of log-concave and log-convex functions (see
Pecaric et. al. (1992)), as they are relevant to our study. We begin with the
definitions of log-concave and log-convex functions.

Definition 2.2 Let $D$ be a convex subset of $\Re^{m}$, the $m$-dimensional
Euclidean space. A function $h:D\rightarrow[0,\infty)$ is said to be log-
concave (log-convex) if, for all $\underline{x},\,\underline{y}\in D$ and all
$\alpha\in[0,1]$,

$h(\alpha\underline{x}+(1-\alpha)\underline{y})\geq(\leq)(h(\underline{x}))^{\alpha}(h(\underline{y}))^{1-\alpha}.$

The following property is well known.

P1. Let $g:\Re\rightarrow[0,\infty)$ is a p.d.f. with the interior of support
as $(a,b)$ $(-\infty\leq a<b\leq\infty)$. Then $g$ is log-concave (log-convex)
on $(a,b)$ if, and only if,

(2.1) $g(x_{1})\,g(x_{2}-\delta)\geq(\leq)g(x_{1}-\delta)\,g(x_{2}),$

for all $0<\delta<b-a$ and for all $a+\delta<x_{1}<x_{2}<b$ (see Pecaric et.
al. (1992)).

Throughout the paper, whenever, we say that a function $k:A\rightarrow\Re$,
where $A\subseteq\Re$, is increasing (decreasing) it means that it is non-
decreasing (non-increasing). Moreover, $\Re$ will denote the real line,
$\Re_{++}^{2}=(0,\infty)\times(0,\infty)$ and, for any integer $m\;(\geq\,2)$,
$\Re^{m}$ will denote the $m$-dimensional Euclidean space.

The following result, taken from Misra and van der Meulen (2003), will be used
in proving the main results of the paper.

Proposition 2.1 Let $T_{1}$ and $T_{2}$ be random variables having
distributional supports $B_{1}$ and $B_{2}$, respectively. Let
$k_{i}:B_{1}\cup B_{2}\rightarrow\Re,\;i=1,2,$ be given functions and let
$\beta_{i}=E[k_{i}(T_{i})],\;i=1,2$. Suppose that $T_{1}\leq_{st}T_{2}$
($T_{2}\leq_{st}T_{1}$). Then
(i) $\beta_{1}\leq\beta_{2}$, provided $k_{1}(x)\leq k_{2}(x)$, $\forall\;x\in
B_{1}\cup B_{2}$, and $k_{1}(x)$ or $k_{2}(x)$ is an increasing (decreasing)
function of $x\in B_{1}\cup B_{2}$;
(ii) $\beta_{2}\leq\beta_{1}$, provided $k_{2}(x)\leq k_{1}(x)$,
$\forall\;x\in B_{1}\cup B_{2}$, and $k_{1}(x)$ or $k_{2}(x)$ is an decreasing
(increasing) function of $x\in B_{1}\cup B_{2}$.

In the following section we consider isotonic regression estimators of order
restricted location parameters $\theta_{1}$ and $\theta_{2}$
($\theta_{1}\leq\theta_{2}$) under a quite general loss function.

##  3\. A General Result for Inadmissibility of an Unrestricted Estimator

Let $\underline{X}=(X_{1},X_{2})$ be a random vector having the Lebesgue
probability density function (p.d.f.)
$f_{\underline{\theta}}(x_{1},x_{2}),\;(x_{1},x_{2})\in\Re^{2},\;\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0},$
where
$\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0}=\\{(x_{1},x_{2}):-\infty<x_{1}\leq
x_{2}<\infty\\}$ is an unknown parameter. Consider simultaneous estimation of
$\theta_{1}$ and $\theta_{2}$ under the loss function

(3.1)
$L(\underline{\theta},\underline{a})=p_{1}W(a_{1}-\theta_{1})+p_{2}W(a_{2}-\theta_{2}),\;\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0},\;\underline{a}=(a_{1},a_{2})\in\mathcal{A}=\Re^{2},$

where $p_{1}>0$ and $p_{2}>0$ are pre-specified constants and
$W:\Re\rightarrow[0,\infty)$ satisfies the following assumption:

Assumption D1. $W(\cdot)$ is a non-negative and strictly convex function,
defined on $\Re$, such that $W(0)=0$, $W(t)$ is strictly decreasing in
$t\in(-\infty,0)$ and strictly increasing in $t\in(0,\infty)$.

Under the assumption $D1$, $W(\cdot)$ is continuous everywhere and it is also
differentiable everywhere, except possibly at countable number of points.

For an estimator $\underline{\delta}=(\delta_{1},\delta_{2})$, its risk
function is defined by

$R(\underline{\theta},\underline{\delta})=E_{\underline{\theta}}[L(\underline{\theta},\underline{\delta})]=p_{1}E_{\underline{\theta}}[W_{1}(\delta_{1}(\underline{X})-\theta_{1})]+p_{2}E_{\underline{\theta}}[W_{2}(\delta_{2}(\underline{X})-\theta_{2})],\;\;\underline{\theta}\in\Theta_{0}.$

The following lemma generalizes the result stated in Theorem 1 of Katz (1963).

Lemma 3.1. Suppose that the assumption (D1) holds. Let
$\underline{\delta}=(\delta_{1},\delta_{2})\in\Re^{2}$ be such that
$\delta_{1}>\delta_{2}$. For $\alpha\in\Re$, define
$\delta_{1,\alpha}=\alpha\delta_{1}+(1-\alpha)\delta_{2}$,
$\delta_{2,\alpha}=\frac{p_{1}}{p_{2}}(1-\alpha)\delta_{1}+(1-\frac{p_{1}}{p_{2}}(1-\alpha))\delta_{2}$
and $\underline{\delta}_{\alpha}=(\delta_{1,\alpha},\delta_{2,\alpha})$. Then,

(a) for $\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\}<\alpha<1$,

$L(\underline{\theta},\underline{\delta}_{\alpha})<L(\underline{\theta},\underline{\delta}),\;\forall\;\underline{\theta}\in\Theta_{0};$

(b) for $\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\}\leq\alpha<1$,

$L(\underline{\theta},\underline{\delta}_{\alpha})<L(\underline{\theta},\underline{\delta}),\;\forall\;\underline{\theta}\in\Theta_{0},\text{
such that }\theta_{1}<\theta_{2};$

(c) for $\frac{p_{1}}{p_{1}+p_{2}}\leq\alpha_{1}<\alpha_{2}<\infty$,

$L(\underline{\theta},\underline{\delta}_{\alpha_{1}})<L(\underline{\theta},\underline{\delta}_{\alpha_{2}}),\;\forall\;\underline{\theta}\in\Theta_{0}.$

Proof. (a) and (b) Since $W(\cdot)$ is a strictly convex function $\Re$, we
have

(3.2)
$\frac{W(x_{2})-W(x_{1})}{x_{2}-x_{1}}<\frac{W(y_{2})-W(y_{1})}{y_{2}-y_{1}},$

provided $x_{1}\leq y_{1},\;x_{2}\leq y_{2}$, with at least one of the
inequalities being strict, $x_{1}\neq x_{2}$ and $y_{1}\neq y_{2}$. Fix
$\alpha\in[\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\},1)$, and
$\underline{\theta}\in\Theta_{0}$. Using (3.2), with
$x_{1}=\delta_{2}-\theta_{2}$,
$x_{2}=\frac{p_{1}}{p_{2}}(1-\alpha)\delta_{1}+(1-\frac{p_{1}}{p_{2}}(1-\alpha))\delta_{2}-\theta_{2}$,
$y_{1}=\alpha\delta_{1}+(1-\alpha)\delta_{2}-\theta_{1}$ and
$y_{2}=\delta_{1}-\theta_{1}$ (so that $x_{1}<y_{1}$, $x_{2}<y_{2}$,
$x_{1}\neq x_{2}$ and $y_{1}\neq y_{2}$), we get

$\displaystyle
L(\underline{\theta},\underline{\delta})-L(\underline{\theta},\underline{\delta}_{\alpha})=$
$\displaystyle
p_{1}[W(\delta_{1}-\theta_{1})-W(\alpha\delta_{1}+(1-\alpha)\delta_{2}-\theta_{1})]$
$\displaystyle+p_{2}\left[W(\delta_{2}-\theta_{2})-W\left(\frac{p_{1}}{p_{2}}(1-\alpha)\delta_{1}+(1-\frac{p_{1}}{p_{2}}(1-\alpha))\delta_{2}-\theta_{2}\right)\right]$
$\displaystyle=$ $\displaystyle
p_{1}(1-\alpha)(\delta_{1}-\delta_{2})\bigg{\\{}\frac{W(\delta_{1}-\theta_{1})-W(\alpha\delta_{1}+(1-\alpha)\delta_{2}-\theta_{1})}{(1-\alpha)(\delta_{1}-\delta_{2})}$
$\displaystyle\;-\frac{W(\frac{p_{1}}{p_{2}}(1-\alpha)\delta_{1}+(1-\frac{p_{1}}{p_{2}}(1-\alpha))\delta_{2}-\theta_{2})-W(\delta_{2}-\theta_{2})}{(1-\alpha)\frac{p_{1}}{p_{2}}(\delta_{1}-\delta_{2})}\bigg{\\}}$
$\displaystyle>$ $\displaystyle\;0,$

under the hypotheses of assertions (a) and (b).

(c) Fix $\underline{\theta}\in\Theta_{0}$ and consider

$\displaystyle Q(\alpha)=L(\underline{\theta},\underline{\delta}_{\alpha})=$
$\displaystyle p_{1}[W(\alpha\delta_{1}+(1-\alpha)\delta_{2}-\theta_{1})]$
$\displaystyle+p_{2}\left[W\left(\frac{p_{1}}{p_{2}}(1-\alpha)\delta_{1}+(1-\frac{p_{1}}{p_{2}}(1-\alpha))\delta_{2}-\theta_{2}\right)\right].$

For simplicity, we assume that $W(\cdot)$ is differentiable everywhere, so
that $W^{{}^{\prime}}(\cdot)$ is strictly increasing (as $W(\cdot)$ is
convex), Then

$\displaystyle Q^{{}^{\prime}}(\alpha)=p_{1}(\delta_{1}-\delta_{2})$
$\displaystyle\bigg{\\{}W^{{}^{\prime}}(\alpha\delta_{1}+(1-\alpha)\delta_{2}-\theta_{1})$
$\displaystyle\qquad\qquad-W^{{}^{\prime}}\left(\frac{p_{1}}{p_{2}}(1-\alpha)\delta_{1}+(1-\frac{p_{1}}{p_{2}}(1-\alpha))\delta_{2}-\theta_{2}\right)\bigg{\\}}>0,$

provided $\alpha\geq\frac{p_{1}}{p_{1}+p_{2}}$ and $\theta_{1}\leq\theta_{2}$,
with at least one of the inequalities being strict; here
$Q^{{}^{\prime}}(\cdot)$ denotes the derivative of $Q(\cdot)$.

The following theorem is an immediate consequence of the above lemma.

Theorem 3.1. Suppose that the assumption (D1) holds. Let
$\underline{\delta}(\underline{X})=(\delta_{1}(\underline{X}),\delta_{2}(\underline{X}))$
be an estimator of $\underline{\theta}=(\theta_{1},\theta_{2})$ such that
$P_{\underline{\theta}}[\delta_{1}(\underline{X})>\delta_{2}(\underline{X})]>0$,
for some $\underline{\theta}\in\Theta_{0}$, such that $\theta_{1}<\theta_{2}$.
For $\alpha\in\Re$, define
$\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M}),$
where

$\displaystyle\delta_{1,\alpha}^{M}(\underline{X})=\begin{cases}\delta_{1}(\underline{X}),&\text{
if}\quad\delta_{1}(\underline{X})\leq\delta_{2}(\underline{X})\\\
\alpha\delta_{1}(\underline{X})+(1-\alpha)\delta_{2}(\underline{X}),&\text{
if}\quad\delta_{1}(\underline{X})>\delta_{2}(\underline{X})\end{cases}$

and

$\displaystyle\delta_{2,\alpha}^{M}(\underline{X})=\begin{cases}\delta_{2}(\underline{X}),&\text{
if}\quad\delta_{1}(\underline{X})\leq\delta_{2}(\underline{X})\\\
\frac{p_{1}}{p_{2}}(1-\alpha)\delta_{1}(\underline{X})+(1-\frac{p_{1}}{p_{2}}(1-\alpha))\delta_{2}(\underline{X}),&\text{
if}\quad\delta_{1}(\underline{X})>\delta_{2}(\underline{X})\end{cases}.$

Then,

(a) for $\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\}\leq\alpha<1$,

$R(\underline{\theta},\underline{\delta}_{\alpha}^{M})\leq
R(\underline{\theta},\underline{\delta}),\;\forall\;\underline{\theta}\in\Theta_{0},$

with strict inequality for some $\underline{\theta}\in\Theta_{0}$;

(b) for $\frac{p_{1}}{p_{1}+p_{2}}\leq\alpha_{1}<\alpha_{2}<\infty$,

$R(\underline{\theta},\underline{\delta}_{\alpha_{1}}^{M})\leq
R(\underline{\theta},\underline{\delta}_{\alpha_{2}}^{M}),\;\forall\;\underline{\theta}\in\Theta_{0},$

with strict inequality for some $\underline{\theta}\in\Theta_{0}$.

Proof. Note that, for any $\alpha\in\Re$,
$L(\underline{\theta},\underline{\delta}_{\alpha}^{M}(\underline{X}))=L(\underline{\theta},\underline{\delta}(\underline{X})),\;\forall\;\underline{\theta}\in\Theta_{0},$
if $\delta_{1}(\underline{X})\leq\delta_{2}(\underline{X})$. Now the result
follows on using Lemma 3.1.

Let
$\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M}),\;\alpha\in\Re,$
be as defined in Theorem 3.1. We call
$\underline{\delta}_{\alpha}^{M},\;\alpha\in\Re,$ a mixed estimator (or
isotonic regression estimator) of $\underline{\theta}=(\theta_{1},\theta_{2})$
based on $\underline{\delta}=(\delta_{1},\delta_{2})$. The following points
are noteworthy.

Remark 3.1 (a) We have
$\underline{\delta}_{1}^{M}=(\delta_{1,1}^{M},\delta_{2,1}^{M})=(\delta_{1},\delta_{2}).$
Also, for $\alpha\leq 1$,
$\delta_{1,\alpha}^{M}(\underline{X})=\min\\{\delta_{1}(\underline{X}),\alpha\delta_{1}(\underline{X})+(1-\alpha)\delta_{2}(\underline{X})\\}$
and
$\delta_{2,\alpha}^{M}(\underline{X})=\max\\{\delta_{2}(\underline{X}),\frac{p_{1}}{p_{2}}(1-\alpha)\delta_{1}(\underline{X})+(1-\frac{p_{1}}{p_{2}}(1-\alpha))\delta_{2}(\underline{X})\\}$.

(b) For $\alpha>\frac{p_{1}}{p_{1}+p_{2}}=\alpha_{0}$, say,
$P_{\underline{\theta}}[\delta_{1,\alpha}^{M}(\underline{X})>\delta_{2,\alpha}^{M}(\underline{X})]=P_{\underline{\theta}}[\delta_{1}(\underline{X})>\delta_{2}(\underline{X})],\;\forall\;\underline{\theta}\in\Theta_{0}$.
Thus any estimator
$\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M})$,
with $\alpha>\frac{p_{1}}{p_{1}+p_{2}}$, is inadmissible for estimating
$\underline{\theta}\in\Theta_{0}$, provided
$P_{\underline{\theta}}[\delta_{1}(\underline{X})>\delta_{2}(\underline{X})]>0,$
for some $\underline{\theta}\in\Theta_{0}$. In this case the estimator
$\underline{\delta}_{\alpha_{0}}^{M}=(\delta_{1,\alpha_{0}}^{M},\delta_{2,\alpha_{0}}^{M})$
improves upon
$\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M})$,
for any $\alpha>\frac{p_{1}}{p_{1}+p_{2}}$.

(c) Suppose that the parameter space is
$\Theta_{0}=\\{(\theta_{1},\theta_{2})\in\Re_{++}^{2}:\theta_{1}\leq\theta_{2}\\}$.
For estimating $\underline{\theta}=(\theta_{1},\theta_{2})$, consider the loss
function

$L(\underline{\theta},\underline{a})=p_{1}W\left(\frac{a_{1}}{\theta_{1}}-1\right)+p_{2}W\left(\frac{a_{2}}{\theta_{2}}-1\right),\;\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0},\;\underline{a}=(a_{1},a_{2})\in\mathcal{A}=\Re_{++}^{2},$

where $p_{1}>0$ and $p_{2}>0$ are pre-specified constants and
$W:\Re\rightarrow[0,\infty)$ is a strictly convex function, such that
$W(1)=0$, $W(t)$ is strictly decreasing in $t\in(-\infty,1)$ and strictly
increasing in $t\in(1,\infty)$. Then Lemma 3.1 holds, provided
$\delta_{1}>\delta_{2}>0$. Also, Theorem 3.1 holds, provided
$P_{\underline{\theta}}[\delta_{i}(\underline{X})>0]=1,\;\forall\;\underline{\theta}\in\Theta_{0},\;i=1,2.$

##  4\. Admissibility of Isotonic regression estimators of Location Parameters
under weighted sum of squared errors loss

Let $\underline{X}=(X_{1},X_{2})$ be a random vector having the Lebesgue
probability density function (p.d.f.) belonging to the location family

(4.1)
$f_{\underline{\theta}}(x_{1},x_{2})=f(x_{1}-\theta_{1},x_{2}-\theta_{2}),\;(x_{1},x_{2})\in\Re^{2},\;\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0},$

where $f(\cdot,\cdot)$ is a specified bivariate Lebesgue p.d.f. on
$\Re^{2}=(-\infty,\infty)\times(-\infty,\infty)$ and
$\Theta_{0}=\\{(\theta_{1},\theta_{2}):-\infty<\theta_{1}\leq\theta_{2}<\infty\\}$.
Generally, $(X_{1},X_{2})$ is a minimal sufficient statistic based on a random
sample. For simultaneous estimation of $\theta_{1}$ and $\theta_{2}$
($\underline{\theta}\in\Theta_{0}$), consider the weighted sum of squared
errors loss function, given by

(4.2)
$L(\underline{\theta},\underline{a})=p_{1}(a_{1}-\theta_{1})^{2}+p_{2}(a_{2}-\theta_{2})^{2},\;\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0},\;\underline{a}=(a_{1},a_{2})\in\mathcal{A}=\Re^{2},$

where $p_{1}>0$ and $p_{2}>0$ are pre-specified constants.

Under the (unrestricted) parameter space $\Theta=\Re^{2}$, the problem of
simultaneously estimating $\theta_{1}$ and $\theta_{2}$ under the loss
function (4.2) is invariant under the additive group of transformations
$\mathcal{G}=\\{g_{c_{1},c_{2}}:(c_{1},c_{2})\in\Re^{2}\\}$, where
$g_{c_{1},c_{2}}(x_{1},x_{2})=(x_{1}+c_{1},x_{2}+c_{2})$,
$(x_{1},x_{2})\in\Re^{2},\;(c_{1},c_{2})\in\Re^{2}$. The group of
transformations $\mathcal{G}$ induces the group of transformations
$\overline{\mathcal{G}}=\\{\overline{g}_{c_{1},c_{2}}:(c_{1},c_{2})\in\Re^{2}\\}$
and
$\tilde{\mathcal{G}}=\\{\tilde{g}_{c_{1},c_{2}}:(c_{1},c_{2})\in\Re^{2}\\}$ on
the parameter space $\Theta$ and the action space $\mathcal{A}$, respectively,
where $\overline{g}_{c_{1},c_{2}}(x_{1},x_{2})=(x_{1}+c_{1},x_{2}+c_{2})$,
$\tilde{g}_{c_{1},c_{2}}(a_{1},a_{2})=(a_{1}+c_{1},a_{2}+c_{2})$,
$(x_{1},x_{2})\in\Theta=\Re^{2},\;(a_{1},a_{2})\in\mathcal{A}=\Re^{2},\;(c_{1},c_{2})\in\Re^{2}$.
Under the group of transformations $\mathcal{G}$, an estimator
$\underline{\delta}=(\delta_{1},\delta_{2})$ is invariant (location invariant)
for estimating $\underline{\theta}=(\theta_{1},\theta_{2})$ if, and only if,

(4.3)
$\underline{\delta}(\underline{X})=(\delta_{1}(\underline{X}),\delta_{2}(\underline{X}))=(X_{1}-c_{1},X_{2}-c_{2}),$

for some $\underline{c}=(c_{1},c_{2})\in\Re^{2}$. The best location
equivariant estimator (BLEE) of $(\theta_{1},\theta_{2})$ is

(4.4)
$\underline{\delta}_{0}(\underline{X})=(\delta_{1,0}(\underline{X}),\delta_{2,0}(\underline{X}))=(X_{1}-c_{0,1},X_{2}-c_{0,2}),$

where, for $Z_{i}=X_{i}-\theta_{i}$, $c_{0,i}=E[Z_{i}],\;i=1,2$.

Consider isotonic regression estimators
$\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M})\;(\alpha\in\Re),$
based on the BLEE $(X_{1}-c_{0,1},X_{2}-c_{0,2})$, where, for $\alpha\in\Re$,

(4.5)
$\displaystyle\delta_{1,\alpha}^{M}(\underline{X})=\begin{cases}X_{1}-c_{0,1},&\text{
if}\quad X_{1}-c_{0,1}\leq X_{2}-c_{0,2}\\\
\alpha(X_{1}-c_{0,1})+(1-\alpha)(X_{2}-c_{0,2}),&\text{ if}\quad
X_{1}-c_{0,1}>X_{2}-c_{0,2}\end{cases}$

and

(4.6)
$\displaystyle\delta_{2,\alpha}^{M}(\underline{X})=\begin{cases}X_{2}-c_{0,2},&\text{
if}\quad X_{1}-c_{0,1}\leq X_{2}-c_{0,2}\\\
\frac{p_{1}}{p_{2}}(1-\alpha)(X_{1}-c_{0,1})+(1-\frac{p_{1}}{p_{2}}(1-\alpha))(X_{2}-c_{0,2}),&\text{
if}\quad X_{1}-c_{0,1}>X_{2}-c_{0,2}\end{cases}.$

Let
$\mathcal{D}=\\{\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M}(\underline{X}),\delta_{2,\alpha}^{M}(\underline{X})):\alpha\in\Re\\}$
be the class of isotonic regression estimators (IREs), based on BLEEs. Define
$\lambda=\theta_{2}-\theta_{1}$ ($\lambda\geq 0$), $Z_{1}=X_{1}-\theta_{1}$,
$Z_{2}=X_{2}-\theta_{2}$, $Z=Z_{2}-Z_{1}$ and $\underline{Z}=(Z_{1},Z_{2})$.
The risk function of the estimator
$\underline{\delta}_{\alpha}^{M},\;\alpha\in\Re,$ is given by

$R(\underline{\theta},\underline{\delta}_{\alpha}^{M})$

$\displaystyle=p_{1}E_{\underline{\theta}}[(Z_{1}-c_{0,1})^{2}I_{(c_{0,2}-c_{0,1}-\lambda,\infty)}(Z)]$
$\displaystyle\quad+p_{1}E_{\underline{\theta}}[((Z_{2}+\lambda-
c_{0,2})-\alpha(Z+\lambda+c_{0,1}-c_{0,2}))^{2}I_{(-\infty,c_{0,2}-c_{0,1}-\lambda)}(Z)]$
$\displaystyle\quad+p_{2}E_{\underline{\theta}}[(Z_{2}-c_{0,2})^{2}I_{(c_{0,2}-c_{0,1}-\lambda,\infty)}(Z)]$
(4.7)
$\displaystyle\quad+p_{2}E_{\underline{\theta}}\left[\\!\left(\frac{p_{1}}{p_{2}}\alpha(Z+\lambda+c_{0,1}-c_{0,2})-\frac{p_{1}}{p_{2}}(Z+\lambda+c_{0,1}-c_{0,2})+Z_{2}-c_{0,2}\right)^{2}\\!I_{(-\infty,c_{0,2}-c_{0,1}-\lambda)}(Z)\\!\right],\,\underline{\theta}\in\Theta_{0},$

where, for any set $A$, $I_{A}(\cdot)$ denotes its indicator function. The
risk function $R(\underline{\theta},\underline{\delta}_{\alpha}^{M})$ depends
on $\underline{\theta}\in\Theta_{0}$ only through
$\lambda=\theta_{2}-\theta_{1}\;(\geq 0)$. Let $f_{Z}(\cdot)$ denote the
p.d.f. of $Z$, so that
$f_{Z}(z)=\int_{-\infty}^{\infty}f(s,s+z)\,ds,\;-\infty<z<\infty$. Clearly,
for any fixed $\underline{\theta}\in\Theta_{0}$ (or $\lambda\geq 0)$,
$R(\underline{\theta},\underline{\delta}_{\alpha}^{M})$ is minimized at
$\alpha=\alpha_{1}(\lambda)$, where, for $\lambda\geq 0$,

$\displaystyle\alpha_{1}(\lambda)$
$\displaystyle=\frac{p_{1}}{p_{1}+p_{2}}+\frac{p_{2}}{p_{1}+p_{2}}\frac{\lambda\,E_{\underline{\theta}}[(Z+\lambda+c_{0,1}-c_{0,2})I_{(-\infty,c_{0,2}-c_{0,1}-\lambda)}(Z)]}{E_{\underline{\theta}}[(Z+\lambda+c_{0,1}-c_{0,2})^{2}I_{(-\infty,c_{0,2}-c_{0,1}-\lambda)}(Z)]}$
(4.8)
$\displaystyle=\frac{p_{1}}{p_{1}+p_{2}}+\frac{p_{2}}{p_{1}+p_{2}}\alpha_{1}^{*}(\lambda);$
$\displaystyle\text{here}\;\;\;\alpha_{1}^{*}(\lambda)$
$\displaystyle=\frac{\lambda\,E_{\underline{\theta}}[(Z+\lambda+c_{0,1}-c_{0,2})I_{(-\infty,c_{0,2}-c_{0,1}-\lambda)}(Z)]}{E_{\underline{\theta}}[(Z+\lambda+c_{0,1}-c_{0,2})^{2}I_{(-\infty,c_{0,2}-c_{0,1}-\lambda)}(Z)]},\;\;\lambda\geq
0.\qquad\qquad$
$\displaystyle=\frac{\lambda\,\int_{-\infty}^{0}z\,f_{Z}(z+c_{0,2}-c_{0,1}-\lambda)dz}{\int_{-\infty}^{0}z^{2}\,f_{Z}(z+c_{0,2}-c_{0,1}-\lambda)dz},\;\;\lambda\geq
0,$ (4.9) $\displaystyle=E_{\lambda}[k(S_{\lambda},\lambda)],\;\;\lambda\geq
0,$

where $k(z,\lambda)=\frac{\lambda}{z},\;z<0,\;\lambda\geq 0,$ and
$S_{\lambda}$ is a r.v. having the p.d.f.

(4.10)
$h_{\lambda}(z)=\begin{cases}\frac{z^{2}f_{Z}(z+c_{0,2}-c_{0,1}-\lambda)}{\int_{-\infty}^{0}s^{2}f_{Z}(s+c_{0,2}-c_{0,1}-\lambda)\,ds},&\text{if}\;\;z<0\\\
0,&\text{otherwise}\end{cases},\;\;\lambda\geq 0.$

Using (2.1), it is easy to check that if $f_{Z}(\cdot)$ is log-concave on
$(-\infty,c_{0,2}-c_{0,1})$, then $S_{\lambda_{1}}\leq_{lr}S_{\lambda_{2}}$,
and consequently $S_{\lambda_{1}}\leq_{st}S_{\lambda_{2}}$, whenever
$0\leq\lambda_{1}<\lambda_{2}<\infty$.

The following lemma will be useful in proving the main result of this
subsection.

Lemma 4.1. Suppose that $f_{Z}(\cdot)$ is log-concave on
$(-\infty,c_{0,2}-c_{0,1})$. Then $\alpha_{1}^{*}(\lambda)$ (and hence
$\alpha_{1}(\lambda)$) is a decreasing function of $\lambda\in[0,\infty)$,
$\inf_{\lambda\geq
0}\alpha_{1}(\lambda)=\frac{p_{1}}{p_{1}+p_{2}}+\frac{p_{2}}{p_{1}+p_{2}}\lim_{\lambda\to\infty}\alpha_{1}^{*}(\lambda)=\alpha_{\infty},\text{
say, and }\sup_{\lambda\geq
0}\alpha_{1}(\lambda)=\frac{p_{1}}{p_{1}+p_{2}}+\frac{p_{2}}{p_{1}+p_{2}}\alpha_{1}^{*}(0)=\frac{p_{1}}{p_{1}+p_{2}}.$

###### Proof.

Let $0\leq\lambda_{1}<\lambda_{2}<\infty$, $T_{i}=S_{\lambda_{i}},\;i=1,2,$
and $m_{i}(z)=k(z,\lambda_{i})=\frac{\lambda_{i}}{z},\;z<0,\;i=1,2,$ where
$S_{\lambda}$ is a r.v. having p.d.f. given by (4.10). Then
$\alpha_{1}^{*}(\lambda_{i})=E_{\lambda_{i}}[m_{i}(T_{i})],\;i=1,2$. The
hypothesis that $f_{Z}(\cdot)$ is log-concave on $(-\infty,c_{0,2}-c_{0,1})$
implies that $T_{1}\leq_{lr}T_{2}$ and, consequently, $T_{1}\leq_{st}T_{2}$.
Also $m_{1}(t)\geq\,m_{2}(t),\;\forall\;t<0$, and $m_{i}(t)$ is a decreasing
function of $t\in(-\infty,0)$. Now, using Proposition 2.1 (a), it follows that

$\alpha_{1}^{*}(\lambda_{1})=E_{\lambda_{1}}[m_{1}(T_{1})]\geq
E_{\lambda_{2}}[m_{2}(T_{2})]=\alpha_{1}^{*}(\lambda_{2}).$

∎

In the following theorem we use the convention that
$[a_{\infty},\frac{p_{1}}{p_{1}+p_{2}}]=(a_{\infty},\frac{p_{1}}{p_{1}+p_{2}}]$,
if $a_{\infty}=-\infty$.

Theorem 4.1. Suppose that $f_{Z}(\cdot)$ is log-concave on
$(-\infty,c_{0,2}-c_{0,1})$. Then the estimators that are admissible within
the class
$\mathcal{D}=\\{\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M}):\alpha\in\Re\\}$
are
$\\{\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M}):\alpha\in[\alpha_{\infty},\frac{p_{1}}{p_{1}+p_{2}}]\\}$.
Moreover, for $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{\infty}<\infty$ or
$\frac{p_{1}}{p_{1}+p_{2}}\leq\alpha_{2}<\alpha_{1}<\infty$, the estimator
$\underline{\delta}_{\alpha_{2}}^{M}(\underline{X})$ dominates the estimator
$\underline{\delta}_{\alpha_{1}}^{M}(\underline{X})$, for any
$\underline{\theta}\in\Theta_{0}$.

###### Proof.

Let $\alpha_{1}(\lambda)$ be as defined by (4), so that, for any fixed
$\underline{\theta}\in\Theta_{0}$ (or fixed $\lambda\geq 0$), the risk
function $R(\underline{\theta},\underline{\delta}_{\alpha}^{M})$, given by
(4), is uniquely minimized at $\alpha=\alpha_{1}(\lambda)$. Since, for any
$\lambda\geq 0$, $\alpha_{1}(\lambda)$ is a continuous function of
$\lambda\in[0,\infty)$, it assumes all values in $\Re$ that are between
$\inf_{\lambda\geq 0}\alpha_{1}(\lambda)=\alpha_{\infty}$ and
$\sup_{\lambda\geq 0}\alpha_{1}(\lambda)=\frac{p_{1}}{p_{1}+p_{2}}$, as
$\lambda$ varies on $[0,\infty)$. It follows that each
$\alpha\in(\alpha_{\infty},\frac{p_{1}}{p_{1}+p_{2}}]$ uniquely minimizes the
risk function $R(\underline{\theta},\underline{\delta}_{\alpha}^{M})$ at some
$\underline{\theta}\in\Theta_{0}$ (or at some $\lambda\geq 0$). This proves
that the estimators
$\\{\underline{\delta}_{\alpha}^{M}:\alpha\in(\alpha_{\infty},\frac{p_{1}}{p_{1}+p_{2}}]\\}$
are admissible among the estimators in the class $\mathcal{D}$. When
$\alpha_{\infty}>-\infty$, the admissibility of the estimator
$\underline{\delta}_{\alpha_{\infty}}^{M}$, within the class $\mathcal{D}$ of
isotonic regression estimators, can be proved by contradiction. Also, note
that, for any fixed $\lambda\geq 0$,
$R(\underline{\theta},\underline{\delta}_{\alpha}^{M})$ is a strictly
decreasing function of $\alpha$ on $(-\infty,\alpha_{1}(\lambda))$ and a
strictly increasing function of $\alpha$ on $(\alpha_{1}(\lambda),\infty)$.
Since
$\alpha_{1}(\lambda)\in[\alpha_{\infty},\frac{p_{1}}{p_{1}+p_{2}}],\;\forall\;\lambda\geq
0$, the foregoing discussion implies that, for any
$\underline{\theta}\in\Theta_{0}$ (or for any $\lambda\geq 0$),
$R(\underline{\theta},\underline{\delta}_{\alpha}^{M})$ is a decreasing
function of $\alpha$ on $(-\infty,\alpha_{\infty}]$ (provided
$\alpha_{\infty}>-\infty$) and is an increasing function of $\alpha$ on
$[\frac{p_{1}}{p_{1}+p_{2}},\infty)$. This establishes the second assertion. ∎

Remark 4.1 Note that
$\underline{\delta}_{0}(\underline{X})=(\delta_{1,0}(\underline{X}),\delta_{2,0}(\underline{X}))=(X_{1}-c_{0,1},X_{2}-c_{0,2})$
is the BLEE of $\underline{\theta}=(\theta_{1},\theta_{2})$. If
$P_{\underline{\theta}}[\delta_{1,0}(\underline{X})>\delta_{2,0}(\underline{X})]=P_{\underline{\theta}}[X_{1}-c_{0,1}>X_{2}-c_{0,2}]>0,$
for some $\underline{\theta}\in\Theta_{0}$, then the mixed estimator
$\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M})$
improves upon $\underline{\delta}_{0}(\underline{X})$, provided
$\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\}\leq\alpha<1$.

###  4.1. Applications

Now we illustrate some applications of results of Sections 3 and 4.

Example 4.1.1. Let $\underline{X}=(X_{1},X_{2})\sim
N_{2}(\theta_{1},\theta_{2},\sigma_{1}^{2},\sigma_{2}^{2},\rho)$, where
$-\infty<\theta_{1}\leq\theta_{2}<\infty$, $\theta_{1}$ and $\theta_{2}$ are
unknown, and $\sigma_{i}>0,\;i=1,2,$ and $\rho\in(-1,1)$ are known. Consider
estimation of $\underline{\theta}=(\theta_{1},\theta_{2})$ under the sum of
squared error loss functions, given by (4.2). Here $(X_{1},X_{2})$ is the BLEE
of $(\theta_{1},\theta_{2})$, i.e., $c_{0,i}=0,\;i=1,2$. Also, we have
$Z_{i}\sim N(0,\sigma_{i}^{2}),\;i=1,2,$ and $Z\sim N(0,\tau^{2})$, where
$\tau^{2}=\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}$. Moreover,
$f_{Z}(z)$ is log-concave on $\Re$. For $c_{0,1}=c_{0,2}=0$, we have

$\sup_{\lambda\geq 0}\alpha_{1}(\lambda)=\frac{p_{1}}{p_{1}+p_{2}}.$

It can be verified that

$\alpha_{\infty}=\frac{p_{1}}{p_{1}+p_{2}}+\frac{p_{2}}{p_{1}+p_{2}}\lim_{\lambda\to\infty}(\alpha_{1}^{*}(\lambda))=\frac{p_{1}}{p_{1}+p_{2}}+\frac{p_{2}}{p_{1}+p_{2}}\lim_{\lambda\to\infty}\frac{\int_{-\infty}^{0}\lambda
z\frac{1}{\tau}\phi\left(\frac{z-\lambda}{\tau}\right)dz}{\int_{-\infty}^{0}z^{2}\frac{1}{\tau}\phi\left(\frac{z-\lambda}{\tau}\right)dz}=-\infty.$

Using Theorem 4.1, it follows that the estimators
$\\{\underline{\delta}_{\alpha}^{M}:\alpha\in(-\infty,\frac{p_{1}}{p_{1}+p_{2}}]\\}$
are admissible within the class
$\mathcal{D}=\\{\underline{\delta}_{\alpha}^{M}:-\infty<\alpha<\infty\\}$ (as
defined by (4.5) and (4.6)) of isotonic regression estimators of
$(\theta_{1},\theta_{2})$. Also, the estimators
$\\{\underline{\delta}_{\alpha}^{M}:\alpha>\frac{p_{1}}{p_{1}+p_{2}}\\}$ are
inadmissible and, for $\frac{p_{1}}{p_{1}+p_{2}}\leq\alpha_{2}<\alpha_{1}$,
the estimator $\underline{\delta}_{\alpha_{2}}^{M}$ dominates the estimator
$\underline{\delta}_{\alpha_{1}}^{M}$. Moreover, using Theorem 3.1, we
conclude that estimators
$\\{\underline{\delta}_{\alpha}^{M}:\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\}\leq\alpha<1\\}$
dominate the BLEE $(X_{1},X_{2})$ for simultaneous estimation of
$(\theta_{1},\theta_{2})$. For the special case $p_{1}=p_{2}=1$, the above
class of admissible estimators is also obtained in Patra and Kumar (2017).
Earlier Kumar and Sharma (1988) also obtained similar results for $\rho=0$ and
$p_{1}=p_{2}$. For
$\alpha_{0}=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}}$,
$p_{1}=\alpha_{0}$, $p_{2}=1-\alpha_{0}$ and
$\rho<\min\\{\frac{\sigma_{1}}{\sigma_{2}},\frac{\sigma_{2}}{\sigma_{1}}\\}$,
the restricted maximum likelihood estimator (MLE) of $\underline{\theta}$ is
$\underline{\delta}_{\alpha_{0}}^{M}$ (see Patra and Kumar (2017)) and it is
admissible within the class $\mathcal{D}$ of isotonic regression estimators.
In this case the restricted MLE is same as the Hwang and Peddada (1994)
estimator and Tan and Peddada (2000) estimator. Under
$\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}$ (say) and $p_{1}=p_{2}=1$, the
restricted MLE of $\underline{\theta}$ is $\underline{\delta}_{\nu_{0}}^{M}$,
where $\nu_{0}=\frac{1}{2}$, and it is admissible within the class
$\mathcal{D}$ of isotonic regression estimators of $\underline{\theta}$. For
$\rho=0$, this result is also discussed in Kumar and Sharma (1988). In general
(i.e., $p_{1}\neq p_{2}$) the restricted MLE of $(\theta_{1},\theta_{2})$ is

(4.11)
$\underline{\delta}_{R}(\underline{X})=\begin{cases}(X_{1},X_{2}),&\text{if
}X_{1}\leq X_{2}\\\
(\alpha_{0}X_{1}+(1-\alpha_{0})X_{2},\alpha_{0}X_{1}+(1-\alpha_{0})X_{2}),&\text{if
}X_{1}>X_{2}\end{cases},$

the Hwang and Peddada (1994) estimator is

(4.12)
$\underline{\delta}_{HP}(\underline{X})=(\min\\{X_{1},\alpha_{0}X_{1}+(1-\alpha_{0})X_{2}\\},\max\\{X_{1},\alpha_{0}X_{1}+(1-\alpha_{0})X_{2}\\})$

and the Tan and Peddada (2000) estimator is

(4.13)
$\underline{\delta}_{PDT}(\underline{X})=(\min\\{X_{1},\alpha_{0}^{+}X_{1}+(1-\alpha_{0}^{+})X_{2}\\},\max\\{X_{1},\alpha_{0}^{+}X_{1}+(1-\alpha_{0}^{+})X_{2}\\})$

where
$\alpha_{0}=\frac{\sigma_{2}(\sigma_{2}-\rho\sigma_{1})}{\sigma_{1}^{2}+\sigma_{2}^{2}-2\rho\sigma_{1}\sigma_{2}}$
and $\alpha_{0}^{+}=\max\\{0,\alpha_{0}\\}$. Clearly
$\underline{\delta}_{R}(\underline{X}),\;\underline{\delta}_{HP}(\underline{X}),\;\underline{\delta}_{PDT}(\underline{X})\notin\mathcal{D}$,
unless
$\rho<\min\\{\frac{\sigma_{1}}{\sigma_{2}},\frac{\sigma_{2}}{\sigma_{1}}\\}$,
$p_{1}=\alpha_{0}$ and $p_{2}=1-\alpha_{0}$.

Example 4.1.2. Let $X_{1}$ and $X_{2}$ be independently and identically
distributed such that $X_{i}$ follows exponential distribution with unknown
location parameter $\theta_{i}$ and known scale parameter
$\sigma_{i}>0,\;i=1,2,$ where it is known apriori that
$-\infty<\theta_{1}\leq\theta_{2}<\infty$. Then,
$f(z_{1},z_{2})=f_{1}(z_{1})f_{2}(z_{2}),\,\underline{z}=(z_{1},z_{2})\in\Re^{2},$
where, for known positive constants $\sigma_{1}>0$ and $\sigma_{2}>0$,
$f_{i}(z)=\frac{1}{\sigma_{i}}e^{-\frac{z}{\sigma_{i}}},\text{
if}\;z>0,i=1,2.$ Here $c_{0,i}=\sigma_{i},\;i=1,2$, and the BLEE of
$(\theta_{1},\theta_{2})$ is $(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$. Moreover,

$f_{Z}(z)=\begin{cases}\frac{1}{\sigma_{1}+\sigma_{2}}e^{\frac{z}{\sigma_{1}}},&\text{
if}\;\;z<0\\\ \frac{1}{\sigma_{1}+\sigma_{2}}e^{-\frac{z}{\sigma_{2}}},&\text{
if}\;\;z\geq 0\end{cases}$

is log-concave on $\Re$. Here, it is easy to verify that
$\alpha_{\infty}=\frac{p_{1}}{p_{1}+p_{2}}+\frac{p_{2}}{p_{1}+p_{2}}\lim_{\lambda\to\infty}\alpha_{1}^{*}(\lambda)=-\infty$.
Using Theorem 4.1, we conclude that the estimators
$\\{\underline{\delta}_{\alpha}^{M}:\alpha\in(-\infty,\frac{p_{1}}{p_{1}+p_{2}}]\\}$
are admissible within the class
$\\{\underline{\delta}_{\alpha}^{M}:-\infty<\alpha<\infty\\}$ of isotonic
regression estimators of $(\theta_{1},\theta_{2})$. Moreover, for
$\frac{p_{1}}{p_{1}+p_{2}}\leq\alpha_{2}<\alpha_{1}<\infty$, the isotonic
regression estimator $\underline{\delta}_{\alpha_{2}}^{M}$ dominates the
estimator $\underline{\delta}_{\alpha_{1}}^{M}$. In particular the BLEE
$\underline{\delta}_{0}(\underline{X})=(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$ is
inadmissible for estimating $(\theta_{1},\theta_{2})$ and is dominated by the
isotonic regression estimators
$\underline{\delta}_{\alpha}^{M}(\underline{X})=(\delta_{1,\alpha}^{M}(\underline{X}),\delta_{2,\alpha}^{M}(\underline{X})),\;\frac{p_{1}}{p_{1}+p_{2}}\leq\alpha<1$.
Also, using Theorem 3.1 the isotonic regression estimators
$\underline{\delta}_{\alpha}^{M}$, for
$\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\}\leq\alpha<1$, dominate the BLEE
$(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$.

###  4.2. Simulation Study For Estimation of Location Parameter
$(\theta_{1},\theta_{2})$

In Example 4.1.1, we considered estimation of the location parameters
$(\theta_{1},\theta_{2})$ of a bivariate normal distribution having order
restricted location parameters (i.e., $\theta_{1}\leq\theta_{2}$), known
variances and known correlation coefficient ($\sigma_{1}>0$, $\sigma_{2}>0$
and $\rho\in(-1,1)$). We have shown that the class of isotonic regression
estimators (IREs)
$\\{\underline{\delta}_{\alpha}^{M}:\alpha\in(-\infty,\frac{p_{1}}{p_{1}+p_{2}}]\\}$
is admissible within the class
$\mathcal{D}=\\{\underline{\delta}_{\alpha}^{M}:-\infty<\alpha<\infty\\}$ and
also, estimators in the class
$\\{\underline{\delta}_{\alpha}^{M}:\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\}\leq\alpha<1\\}$
dominate the BLEE $\underline{\delta}_{0}(\underline{X})=(X_{1},X_{2})$. To
further evaluate the performances of various estimators under the loss
function
$L(\underline{\theta},\underline{a})=\frac{1}{\sigma_{1}^{2}}(a_{1}-\theta_{1})^{2}+\frac{1}{\sigma_{2}^{2}}(a_{2}-\theta_{2})^{2},\;\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0},\;\underline{a}=(a_{1},a_{2})\in\Re^{2}$
(i.e., $p_{1}=\frac{1}{\sigma_{1}^{2}}$ and $p_{2}=\frac{1}{\sigma_{2}^{2}}$),
in this section, we compare the risk performances of the BLEE $(X_{1},X_{2})$,
the isotonic regression estimators (IREs)
$\underline{\delta}_{\alpha}^{M}(\underline{X})=(\delta_{1,\alpha}^{M}(\underline{X}),\delta_{2,\alpha}^{M}(\underline{X}))$
with
$\alpha=\frac{p_{1}}{p_{1}+p_{2}}=\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}$,
the restricted MLE $\underline{\delta}_{R}(\underline{X})$ (as defined by
(4.11)), the Hwang and Peddada (1994) estimator
$\underline{\delta}_{HP}(\underline{X})$ (as defined by (4.12)) and the Tan
and Peddada (2000) estimator $\underline{\delta}_{PDT}(\underline{X})$ (as
defined by (4.13)), numerically, through the Monte Carlo simulations. Note
that, for
$\rho<\min\\{\frac{\sigma_{1}}{\sigma_{2}},\frac{\sigma_{2}}{\sigma_{1}}\\}$,
the restricted MLE $\underline{\delta}_{R}(\underline{X})$, the Hwang and
Peddada (1994) estimator $\underline{\delta}_{HP}(\underline{X})$ and the Tan
and Peddada (2000) estimator $\underline{\delta}_{PDT}(\underline{X})$ are
same.

For simulations, we generated 50000 samples of size 1 each from relevant
bivariate distributions and computed the simulated risks of various
estimators.

(a) $\sigma_{1}=0.2$, $\sigma_{2}=0.5$ and $\rho=-0.9$.

(b) $\sigma_{1}=10$, $\sigma_{2}=1$ and $\rho=-0.5$.

(c) $\sigma_{1}=2$, $\sigma_{2}=10$ and $\rho=-0.2$.

(d) $\sigma_{1}=1$, $\sigma_{2}=10$ and $\rho=0.2$.

(e) $\sigma_{1}=10$, $\sigma_{2}=1$ and $\rho=0.5$.

(f) $\sigma_{1}=0.5$, $\sigma_{2}=0.4$ and $\rho=0.9$.

Figure 1. Risk plots of the BLEE $(X_{1},X_{2})$, the IREs with
$\alpha=\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}$ and the
restricted MLE.

The simulated values of risks of various estimators are plotted in Figure 1.
The following observations are evident from Figure 1:

(i) The IREs with
$\alpha=\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}$ and the
restricted MLE always dominate the BLEE $(X_{1},X_{2})$.

(ii) The Restricted MLE always perform better than the other estimators.

Under the sum of squared error loss functions (4.2), in Example 4.1.2, we
considered estimation of the location parameters $(\theta_{1},\theta_{2})$ of
two independent exponential distributions having unknown order restricted
location parameters (i.e., $\theta_{1}\leq\theta_{2}$) and known scale
parameters ($\sigma_{1}>0$ and $\sigma_{2}>0$). We have shown that the class
of isotonic regression estimators (IREs)
$\\{\underline{\delta}_{\alpha}^{M}:\alpha\in(-\infty,\frac{p_{1}}{p_{1}+p_{2}}]\\}$
is admissible within the class
$\mathcal{D}=\\{\underline{\delta}_{\alpha}^{M}:-\infty<\alpha<\infty\\}$.
Also, the class of estimators
$\\{\underline{\delta}_{\alpha}^{M}:\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\}\leq\alpha<1\\}$
dominates the BLEE
$\underline{\delta}_{0}(\underline{X})=(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$.
To further evaluate the performances of various estimators under the loss
function (4.2) with $p_{1}=p_{2}=1$, in this section, we compare the risk
performances of the isotonic regression estimators (IREs)
$\underline{\delta}_{1}^{M}(\underline{X})=(\delta_{1,1}^{M}(\underline{X}),\delta_{2,1}^{M}(\underline{X}))=(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$,
$\underline{\delta}_{0.75}^{M}(\underline{X})=(\delta_{1,0.75}^{M}(\underline{X}),\delta_{2,0.75}^{M}(\underline{X}))$,
$\underline{\delta}_{0.5}^{M}(\underline{X})=(\delta_{1,0.5}^{M}(\underline{X}),\delta_{2,0.5}^{M}(\underline{X}))$,
$\underline{\delta}_{0}^{M}(\underline{X})=(\delta_{1,0}^{M}(\underline{X}),\delta_{2,0}^{M}(\underline{X}))$
and the restricted MLE
$\underline{\delta}_{R}(\underline{X})=(\min\\{X_{1},X_{2}\\},X_{2})$,
numerically, through the Monte Carlo simulations; here

$\delta_{1,\alpha}^{M}(\underline{X})=\begin{cases}X_{1}-\sigma_{1},&\text{
if}\;\;X_{1}-\sigma_{1}\leq X_{2}-\sigma_{2}\\\
\alpha(X_{1}-\sigma_{1})+(1-\alpha)(X_{2}-\sigma_{2}),&\text{
if}\;\;X_{1}-\sigma_{1}>X_{2}-\sigma_{2}\end{cases},$ $\text{and
}\delta_{2,\alpha}^{M}(\underline{X})=\begin{cases}X_{2}-\sigma_{2},&\text{
if}\;\;X_{1}-\sigma_{1}\leq X_{2}-\sigma_{2}\\\
(1-\alpha)(X_{1}-\sigma_{1})+\alpha(X_{2}-\sigma_{2}),&\text{
if}\;\;X_{1}-\sigma_{1}>X_{2}-\sigma_{2}\end{cases}.$

For simulations, we generated 50000 samples of size 1 each from relevant
exponential distributions and computed the simulated risks of estimators
$\underline{\delta}_{1}^{M}(\underline{X})=(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$,
$\underline{\delta}_{0.75}^{M}(\underline{X})$,
$\underline{\delta}_{0.5}^{M}(\underline{X})$,
$\underline{\delta}_{0}(\underline{X})$ and
$\underline{\delta}_{R}(\underline{X})$.

The simulated values of risks of various estimators are plotted in Figure 2.
The following observations are evident from Figure 2:

(i) The risk function values of the IREs
$\underline{\delta}_{0.75}^{M}(\underline{X})$,
$\underline{\delta}_{0.5}^{M}(\underline{X})$,
$\underline{\delta}_{0.25}^{M}(\underline{X})$ and
$\underline{\delta}_{0}^{M}(\underline{X})$ are nowhere larger than the risk
function values of the BLEE $(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$, which is in
conformity with theoretical findings of Example 4.1.2.

(ii) From Figure 2, we can observe that the IREs
$\underline{\delta}_{0.5}^{M}(\underline{X})$,
$\underline{\delta}_{0}^{M}(\underline{X})$ and
$\underline{\delta}_{R}^{M}(\underline{X})$ are not comparable and the IREs
$\underline{\delta}_{1}^{M}(\underline{X})$ and
$\underline{\delta}_{0.75}^{M}(\underline{X})$ are inadmissible. This is in
conformity with theoretical findings of Example 4.1.2.

(iii) When $\sigma_{1}>>\sigma_{2}$, for small and moderate values of
$\theta_{2}-\theta_{1}$, the restricted MLE outperforms the other estimators.

(iv) There is no clear cut winner between various estimators, but performance
of the BLEE
$\underline{\delta}_{1}^{M}(\underline{X})=(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$
and $\underline{\delta}_{0.75}^{M}(\underline{X})$ are worse than other
estimators. Also, when $\sigma_{1}\leq\sigma_{2}$, the performance of the
restricted MLE is the worst among other estimators.

(a) $\sigma_{1}=0.5$ and $\sigma_{2}=0.5$.

(b) $\sigma_{1}=1$ and $\sigma_{2}=0.2$.

(c) $\sigma_{1}=1$ and $\sigma_{2}=1$.

(d) $\sigma_{1}=2$ and $\sigma_{2}=10$.

(e) $\sigma_{1}=5$ and $\sigma_{2}=2$.

(f) $\sigma_{1}=5$ and $\sigma_{2}=8$.

Figure 2. Risk plots of estimators
$\underline{\delta}_{1}^{M}(\underline{X})=(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$,
$\underline{\delta}_{0.75}^{M}(\underline{X})$,
$\underline{\delta}_{0.5}^{M}(\underline{X})$,
$\underline{\delta}_{0}^{M}(\underline{X})$ and
$\underline{\delta}_{R}^{M}(\underline{X})$ against the values of
$\lambda=\theta_{2}-\theta_{1}$.

##  5\. Admissibility of Isotonic regression estimators of Scale Parameters
under weighted sum of squared errors loss

Let the random vector $\underline{X}=(X_{1},X_{2})$ have the Lebesgue p.d.f.
belonging to the scale family

(5.1)
$f_{\underline{\theta}}(x_{1},x_{2})=\frac{1}{\theta_{1}\theta_{2}}f\\!\left(\frac{x_{1}}{\theta_{1}},\frac{x_{2}}{\theta_{2}}\right),\;(x_{1},x_{2})\in\Re^{2},\;\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0},$

where
$\Theta_{0}=\\{(\theta_{1},\theta_{2})\in\Re_{++}^{2}:\theta_{1}\leq\theta_{2}\\}$.
Throughout, we will assume that the distributional support of
$\underline{X}=(X_{1},X_{2})$ is a subset of $\Re_{++}^{2}$. Generally,
$\underline{X}=(X_{1},X_{2})$ would be a minimal-sufficient statistic based on
a bivariate random sample or two independent random samples.

For estimating $\underline{\theta}=(\theta_{1},\theta_{2})$, consider the loss
function

(5.2)
$L(\underline{\theta},\underline{a})=p_{1}(a_{1}-\theta_{1})^{2}+p_{2}(a_{2}-\theta_{2})^{2},\;\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0},\;\underline{a}=(a_{1},a_{2})\in\mathcal{A}=\Re_{++}^{2},$

where $p_{1}>0$ and $p_{2}>0$ are pre-specified constants.

Under the (unrestricted) parameter space $\Theta=\Re_{++}^{2}$, the problem of
simultaneously estimating $\theta_{1}$ and $\theta_{2}$, with the loss
function (5.2), is invariant under the multiplicative group of transformations
$\mathcal{G}=\\{g_{c_{1},c_{2}}:(c_{1},c_{2})\in\Re_{++}^{2}\\}$, where
$g_{c_{1},c_{2}}(x_{1},x_{2})=(c_{1}x_{1},c_{2}x_{2})$,
$(x_{1},x_{2})\in\Re_{++}^{2},\;(c_{1},c_{2})\in\Re_{++}^{2}$. The group
$\mathcal{G}$ induces the groups of transformations
$\overline{\mathcal{G}}=\\{\overline{g}_{c_{1},c_{2}}:(c_{1},c_{2})\in\Re_{++}^{2}\\}$
and
$\tilde{\mathcal{G}}=\\{\tilde{g}_{c_{1},c_{2}}:(c_{1},c_{2})\in\Re_{++}^{2}\\}$
on the parameter space $\Theta$ and the action space $\mathcal{A}$,
respectively, where
$\overline{g}_{c_{1},c_{2}}(\theta_{1},\theta_{2})=(c_{1}\theta_{1},c_{2}\theta_{2})$,
$\tilde{g}_{c_{1},c_{2}}(a_{1},a_{2})=(c_{1}a_{1},c_{2}a_{2})$,
$(\theta_{1},\theta_{2})\in\Theta,\;(a_{1},a_{2})\in\mathcal{A}=\Re_{++}^{2},\;(c_{1},c_{2})\in\Re_{++}^{2}$.
Under the group of transformations $\mathcal{G}$, an estimator
$\underline{\delta}=(\delta_{1},\delta_{2})$ is invariant (scale invariant)
for estimating $\underline{\theta}=(\theta_{1},\theta_{2})$ if, and only if

$\underline{\delta}=(\delta_{1}(\underline{X}),\delta_{2}(\underline{X}))=(c_{1}X_{1},c_{2}X_{2}),$

for some $\underline{c}=(c_{1},c_{2})\in\Re_{++}^{2}$. The best scale
equivariant estimator (BSEE) of $(\theta_{1},\theta_{2})$ is

$\underline{\delta}_{0}(\underline{X})=(\delta_{0,1}(\underline{X}),\delta_{0.2}(\underline{X}))=(c_{0,1}X_{1},c_{0,2}X_{2}),$

where, for $Z_{i}=\frac{X_{i}}{\theta_{i}},$
$c_{0,i}=\frac{E[Z_{i}]}{E[Z_{i}^{2}]},\;i=1,2.$

Based on BSEEs $(c_{0,1}X_{1},c_{0,2}X_{2})$, define isotonic regression
estimators
$\underline{\delta}^{M}_{\alpha}(\underline{X})=(\delta_{1,\alpha}^{M}(\underline{X}),\delta_{2,\alpha}^{M}(\underline{X}))$,
where

(5.3)
$\displaystyle\delta_{1,\alpha}^{M}(\underline{X})=\begin{cases}c_{0,1}X_{1},&\text{
if}\quad c_{0,1}X_{1}\leq c_{0,2}X_{2}\\\ \alpha
c_{0,1}X_{1}+(1-\alpha)c_{0,2}X_{2},&\text{ if}\quad
c_{0,1}X_{1}>c_{0,2}X_{2}\end{cases}$

and

(5.4)
$\displaystyle\delta_{2,\alpha}^{M}(\underline{X})=\begin{cases}c_{0,2}X_{2},&\text{
if}\quad c_{0,1}X_{1}\leq c_{0,2}X_{2}\\\
\frac{p_{1}}{p_{2}}(1-\alpha)c_{0,1}X_{1}+(1-\frac{p_{1}}{p_{2}}(1-\alpha))c_{0,2}X_{2},&\text{
if}\quad c_{0,1}X_{1}>c_{0,2}X_{2}\end{cases},\;\;\alpha\in\Re.$

Let
$\mathcal{D}=\\{\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M}(\underline{X}),\delta_{2,\alpha}^{M}(\underline{X})):\alpha\in\Re\\}$
be the class of isotonic regression estimators (IREs) based on BSEEs
$(c_{0,1}X_{1},c_{0,2}X_{2})$. The risk function of the estimator
$\underline{\delta}_{\alpha}^{M}$, $\alpha\in\Re$, is given by

$\displaystyle R(\underline{\theta},\underline{\delta}_{\alpha}^{M})$
$\displaystyle=p_{1}E_{\underline{\theta}}\left[(c_{0,1}\theta_{1}Z_{1}-\theta_{1})^{2}\,I_{\left(\frac{c_{0,1}}{c_{0,2}\lambda},\infty\right)}(Z)\right]$
$\displaystyle\quad+p_{1}E_{\underline{\theta}}\left[\\{\alpha(c_{0,1}\theta_{1}Z_{1}-c_{0,2}\theta_{2}Z_{2})+c_{0,2}\theta_{2}Z_{2}-\theta_{1}\\}^{2}\,I_{\left(0,\frac{c_{0,1}}{c_{0,2}\lambda}\right)}(Z)\right]$
$\displaystyle\quad+p_{2}E_{\underline{\theta}}\left[(c_{0,2}\theta_{2}Z_{2}-\theta_{2})^{2}\,I_{\left(\frac{c_{0,1}}{c_{0,2}\lambda},\infty\right)}(Z)\right]$
$\displaystyle\quad+p_{2}E_{\underline{\theta}}\left[\bigg{\\{}\alpha\frac{p_{1}}{p_{2}}(c_{0,2}\theta_{2}Z_{2}-c_{0,1}\theta_{1}Z_{1})-\frac{p_{1}-p_{2}}{p_{2}}c_{0,2}\theta_{2}Z_{2}+\frac{p_{1}}{p_{2}}c_{0,1}\theta_{1}Z_{1}-\theta_{2}\bigg{\\}}^{2}\,I_{\left(0,\frac{c_{0,1}}{c_{0,2}\lambda}\right)}(Z)\right],\;\underline{\theta}\in\Theta_{0},$

where $Z_{1}=\frac{X_{1}}{\theta_{1}}$, $Z_{2}=\frac{X_{2}}{\theta_{2}}$ and
$Z=\frac{Z_{2}}{Z_{1}}$. Define $\lambda=\frac{\theta_{2}}{\theta_{1}}$
($\lambda\geq 1$) and let $f_{Z}(\cdot)$ denote the p.d.f. of
$Z=\frac{Z_{2}}{Z_{1}}$, so that
$f_{Z}(z)=\int_{-\infty}^{\infty}|s|f(s,sz)\,ds,\;-\infty<z<\infty$. Clearly,
for any fixed $\underline{\theta}\in\Theta_{0}$ (or $\lambda\geq 1)$,
$R(\underline{\theta},\underline{\delta}_{\alpha}^{M})$ is minimized at

$\displaystyle\alpha=\alpha(\lambda)$
$\displaystyle=\frac{p_{1}}{p_{1}+p_{2}}-\frac{p_{2}}{p_{1}+p_{2}}\frac{(\lambda-1)\,E_{\underline{\theta}}[Z_{1}(c_{0,1}-c_{0,2}\lambda
Z)I_{(0,\frac{c_{0,1}}{c_{0,2}\lambda})}(Z)]}{E_{\underline{\theta}}[Z_{1}^{2}(c_{0,1}-c_{0,2}\lambda
Z)^{2}I_{(0,\frac{c_{0,1}}{c_{0,2}\lambda})}(Z)]}$ (5.5)
$\displaystyle=\frac{p_{1}}{p_{1}+p_{2}}-\frac{p_{2}}{p_{1}+p_{2}}\alpha_{1}(\lambda),\;\;\;(say),$

where, for $h_{1}(z)=E[Z_{1}|Z=z]$ and $h_{2}(z)=E[Z_{1}^{{}^{2}}|Z=z]$,

$\displaystyle\alpha_{1}(\lambda)$
$\displaystyle=\frac{(\lambda-1)\,E_{\underline{\theta}}[h_{1}(Z)(c_{0,1}-c_{0,2}\lambda
Z)I_{(0,\frac{c_{0,1}}{c_{0,2}\lambda})}(Z)]}{E_{\underline{\theta}}[h_{2}(Z)(c_{0,1}-c_{0,2}\lambda
Z)^{2}I_{(0,\frac{c_{0,1}}{c_{0,2}\lambda})}(Z)]}\qquad\qquad$
$\displaystyle=\frac{(\lambda-1)\,\int_{0}^{\frac{c_{0,1}}{c_{0,2}\lambda}}h_{1}(z)\,(c_{0,1}-c_{0,2}\lambda
z)\,f_{Z}\left(z\right)dz}{\int_{0}^{\frac{c_{0,1}}{c_{0,2}\lambda}}h_{2}\left(z\right)(c_{0,1}-c_{0,2}\lambda
z)^{2}\,f_{Z}\left(z\right)dz}$
$\displaystyle=\frac{(\lambda-1)\,\int_{0}^{1}h_{1}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)\,(1-t)\,f_{Z}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)dt}{c_{0,1}\int_{0}^{1}h_{2}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)(1-t)^{2}\,f_{Z}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)dt}$
$\displaystyle=\frac{1}{c_{0,1}}E_{\lambda}[k(S_{\lambda},\lambda)],\;\;\lambda\geq
1;$

here
$k(t,\lambda)=\frac{(\lambda-1)h_{1}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)}{(1-t)\,h_{2}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)},\;0<t<1,\;\lambda\geq
1,$ and $S_{\lambda}$ is a r.v. having the p.d.f.

(5.6)
$h_{\lambda}(z)=\begin{cases}\frac{(1-t)^{2}h_{2}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)f_{Z}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)}{\int_{0}^{1}(1-s)^{2}h_{2}\left(\frac{c_{0,1}s}{c_{0,2}\lambda}\right)f_{Z}\left(\frac{c_{0,1}s}{c_{0,2}\lambda}\right)\,ds},&\text{if}\;\;0<t<1\\\
0,&\text{otherwise}\end{cases},\;\;\lambda\geq 1.$

Clearly, if, for each $\theta\in(0,1)$, $\frac{h_{2}(\theta z)f_{Z}(\theta
z)}{h_{2}(z)f_{Z}(z)}$ is increasing (decreasing) in
$z\in\left(0,\frac{c_{0,1}}{c_{0,2}}\right)$, then
$S_{\lambda_{1}}\leq_{lr}S_{\lambda_{2}}$
($S_{\lambda_{2}}\leq_{lr}S_{\lambda_{1}}$) and, consequently,
$S_{\lambda_{1}}\leq_{st}S_{\lambda_{2}}$
($S_{\lambda_{2}}\leq_{st}S_{\lambda_{1}}$), whenever
$1\leq\lambda_{1}<\lambda_{2}<\infty$.

The following lemma, whose proof is immediate from Proposition 2.1, will be
useful in proving the main result of this section.

Lemma 5.1.(a) Suppose that, for each $\theta\in(0,1)$, $\frac{h_{2}(\theta
z)f_{Z}(\theta z)}{h_{2}(z)f_{Z}(z)}$ is increasing (decreasing) in
$z\in\left(0,\frac{c_{0,1}}{c_{0,2}}\right)$. Also, suppose that for every
fixed $z\in(0,1)$, $k(z,\lambda)$ is increasing in $\lambda\in[1,\infty)$ and,
for every fixed $\lambda\geq 1$, $k(z,\lambda)$ is increasing (decreasing) in
$z\in(0,1)$. Then $\alpha_{1}(\lambda)$ is an increasing function (and hence
$\alpha(\lambda)$ is a decreasing function) of $\lambda\in[1,\infty)$,
$\inf_{\lambda\geq
1}\alpha(\lambda)=\frac{p_{1}}{p_{1}+p_{2}}-\frac{p_{2}}{p_{1}+p_{2}}\lim_{\lambda\to\infty}\alpha_{1}(\lambda)=\alpha_{\infty},\text{
say, and }\sup_{\lambda\geq
1}\alpha(\lambda)=\frac{p_{1}}{p_{1}+p_{2}}-\frac{p_{2}}{p_{1}+p_{2}}\alpha_{1}(1)=\frac{p_{1}}{p_{1}+p_{2}}.$

(b) Suppose that, for each $\theta\in(0,1)$, $\frac{h_{2}(\theta
z)f_{Z}(\theta z)}{h_{2}(z)f_{Z}(z)}$ is increasing (decreasing) in
$z\in\left(0,\frac{c_{0,1}}{c_{0,2}}\right)$. Also, suppose that, for every
fixed $z\in(0,1)$, $k(z,\lambda)$ is decreasing in $\lambda\in[1,\infty)$ and,
for every fixed $\lambda\geq 1$, $k(z,\lambda)$ is decreasing (increasing) in
$z\in(0,1)$. Then $\alpha_{1}(\lambda)$ is a decreasing function (and hence
$\alpha(\lambda)$ is an increasing function) of $\lambda\in[1,\infty)$,
$\inf_{\lambda\geq
1}\alpha(\lambda)=\frac{p_{1}}{p_{1}+p_{2}}-\frac{p_{2}}{p_{1}+p_{2}}\alpha_{1}(1)=\frac{p_{1}}{p_{1}+p_{2}},\text{
say, and }\sup_{\lambda\geq
1}\alpha(\lambda)=\frac{p_{1}}{p_{1}+p_{2}}-\frac{p_{2}}{p_{1}+p_{2}}\lim_{\lambda\to\infty}\alpha_{1}(\lambda)=\alpha_{\infty}.$

Now we present the main result of this section. We use the convention that
$[\frac{p_{1}}{p_{1}+p_{2}},\alpha_{\infty}]=[\frac{p_{1}}{p_{1}+p_{2}},\infty),$
if $\alpha_{\infty}=\infty$. The proof of the theorem, being similar to that
of Theorem 4.1, is being omitted.

Theorem 5.1. (a) Suppose that assumptions of Lemma 5.1 (a) hold. Then the
estimators that are admissible within the class
$\mathcal{D}=\\{\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M}):\alpha\in\Re\\}$
are
$\\{\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M}):\alpha\in[\alpha_{\infty},\frac{p_{1}}{p_{1}+p_{2}}]\\}$.
Moreover, for $-\infty<\alpha_{1}<\alpha_{2}\leq\alpha_{\infty}$ or
$\frac{p_{1}}{p_{1}+p_{2}}\leq\alpha_{2}<\alpha_{1}<\infty$, the estimator
$\underline{\delta}_{\alpha_{2}}^{M}(\underline{X})$ dominates the estimator
$\underline{\delta}_{\alpha_{1}}^{M}(\underline{X})$, for any
$\underline{\theta}\in\Theta_{0}$.

(b) Suppose that assumptions of Lemma 5.1 (b) hold. Then the estimators that
are admissible within the class
$\mathcal{D}=\\{\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M}):\alpha\in\Re\\}$
are
$\\{\underline{\delta}_{\alpha}^{M}=(\delta_{1,\alpha}^{M},\delta_{2,\alpha}^{M}):\alpha\in[\frac{p_{1}}{p_{1}+p_{2}},\alpha_{\infty}]\\}$.
Moreover, for $-\infty<\alpha_{1}<\alpha_{2}\leq\frac{p_{1}}{p_{1}+p_{2}}$ or
$\alpha_{\infty}\leq\alpha_{2}<\alpha_{1}<\infty$, the estimator
$\underline{\delta}_{\alpha_{2}}^{M}(\underline{X})$ dominates the estimator
$\underline{\delta}_{\alpha_{1}}^{M}(\underline{X})$, for any
$\underline{\theta}\in\Theta_{0}$.

###  5.1. Applications

Now we illustrate some applications of Theorem 5.1.

Example 5.1.1. Let $\underline{X}=(X_{1},X_{2})$ be a random vector having the
p.d.f. (5.1), with
$f(z_{1},z_{2})=f_{1}(z_{1})f_{2}(z_{2}),\;\;\underline{z}=(z_{1},z_{2})\in\Re^{2}$
and

$f_{i}(z)=\begin{cases}\frac{e^{-z}z^{\alpha_{i}-1}}{\Gamma{\alpha_{i}}},&\text{
if}\;\;z>0\\\ 0,&\text{ otherwise}\end{cases},\;\;i=1,2.$

Here $\alpha_{1}>0$ and $\alpha_{2}>0$ are known shape parameters. We have
$c_{0,i}=\frac{E[Z_{i}]}{E[Z_{i}^{2}]}=\frac{1}{\alpha_{i}+1},\;i=1,2,$ and
the BSEE of $\theta_{i}$ is
$\delta_{i,0}(\underline{X})=\frac{X_{i}}{\alpha_{i}+1},\;i=1,2$. The p.d.f.
of $Z=\frac{Z_{2}}{Z_{1}}$ is

$f_{Z}(z)=\begin{cases}\frac{\Gamma{(\alpha_{1}+\alpha_{2})}}{\Gamma{\alpha_{1}}\Gamma{\alpha_{2}}}\frac{z^{\alpha_{2}-1}}{(1+z)^{\alpha_{1}+\alpha_{2}}},&\text{
if}\;\;0<z<\infty\\\ 0,&\text{ otherwise}\end{cases},$

and the conditional p.d.f. of $Z_{1}$ given $Z=z$ ($0<z<\infty$) is

$f_{Z_{1}|Z}(z_{1}|z)=\begin{cases}\frac{(1+z)^{\alpha_{1}+\alpha_{2}}\,z_{1}^{\alpha_{1}+\alpha_{2}-1}\,e^{-(1+z)z_{1}}}{\Gamma{(\alpha_{1}+\alpha_{2})}},&\text{
if}\;\;0<z_{1}<\infty\\\ 0,&\text{ otherwise}\end{cases}.$

Clearly, $h_{1}(z)=E[Z_{1}|Z=z]=\frac{\alpha_{1}+\alpha_{2}}{1+z},\;z>0,$
$h_{2}(z)=E[Z_{1}^{2}|Z=z]=\frac{(\alpha_{1}+\alpha_{2}+1)(\alpha_{1}+\alpha_{2})}{(1+z)^{2}},\;z>0,$
and, for $0<\theta<1$, $\frac{h_{2}(\theta z)f_{Z}(\theta
z)}{h_{2}(z)f_{Z}(z)}$ is increasing in
$z\in\left(0,\frac{c_{0,1}}{c_{0,2}}\right]$. For any fixed $\lambda\geq 1$,

$k(z,\lambda)=\frac{(\lambda-1)h_{1}\left(\frac{c_{0,1}z}{c_{0,2}\lambda}\right)}{(1-z)\,h_{2}\left(\frac{c_{0,1}z}{c_{0,2}\lambda}\right)}=\frac{(\lambda-1)}{(1-z)(\alpha_{1}+\alpha_{2}+1)}\left(1+\frac{c_{0,1}z}{c_{0,2}\lambda}\right)$

is increasing in $z\in(0,1)$ and, for any fixed $z\in(0,1)$, $k(z,\lambda)$ is
increasing in $\lambda\geq 1$. Using Lemma 5.1 (a), we have

$\displaystyle\sup_{\lambda\geq 1}\alpha(\lambda)$
$\displaystyle=\frac{p_{1}}{p_{1}+p_{2}},$
$\displaystyle\text{and}\quad\inf_{\lambda\geq 1}\alpha(\lambda)$
$\displaystyle=\frac{p_{1}}{p_{1}+p_{2}}-\frac{p_{2}}{p_{1}+p_{2}}\lim_{\lambda\to\infty}\alpha_{1}(\lambda)$
$\displaystyle=\frac{p_{1}}{p_{1}+p_{2}}-\frac{p_{2}}{p_{1}+p_{2}}\lim_{\lambda\to\infty}\frac{(\lambda-1)\,\int_{0}^{1}h_{1}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)\,(1-t)\,f_{Z}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)dt}{c_{0,1}\int_{0}^{1}h_{2}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)(1-t)^{2}\,f_{Z}\left(\frac{c_{0,1}t}{c_{0,2}\lambda}\right)dt}$
$\displaystyle=-\infty.$

Using Theorem 5.1 (a), it follows that the estimators
$\\{\underline{\delta}_{\alpha}^{M}:\alpha\in(-\infty,\frac{p_{1}}{p_{1}+p_{2}}]\\}$
are admissible within the class
$\\{\underline{\delta}_{\alpha}^{M}:-\infty<\alpha<\infty\\}$ of isotonic
regression estimators of $(\theta_{1},\theta_{2})$. Moreover the estimators
$\\{\underline{\delta}_{\alpha}^{M}:\alpha>\frac{p_{1}}{p_{1}+p_{2}}\\}$ are
inadmissible and, for $\frac{p_{1}}{p_{1}+p_{2}}\leq\alpha_{2}<\alpha_{1}$,
the estimator $\underline{\delta}_{\alpha_{2}}^{M}$ dominates the estimator
$\underline{\delta}_{\alpha_{1}}^{M}$. Using Theorem 3.1, we conclude that
estimators
$\\{\underline{\delta}_{\alpha}^{M}:\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\}\leq\alpha<1\\}$
dominate the BSEE $(\frac{1}{\alpha_{1}+1}X_{1},\frac{1}{\alpha_{2}+1}X_{2})$.

Example 5.1.2. Let $\underline{X}=(X_{1},X_{2})$ be a random vector with
p.d.f. (5.1), where
$f(z_{1},z_{2})=f_{1}(z_{1})f_{2}(z_{2}),\;\underline{z}=(z_{1},z_{2})\in\Re_{++}^{2}$
and, for positive constants $\alpha_{1}$ and $\alpha_{2}$,

$f_{i}(z)=\begin{cases}\alpha_{i}z^{\alpha_{i}-1},&\text{ if}\;\;0<z<1\\\
0,&\text{ otherwise}\end{cases},\;\;i=1,2.$

Here
$c_{0,i}=\frac{E[Z_{i}]}{E[Z_{i}^{2}]}=\frac{\alpha_{i}+2}{\alpha_{i}+1},\;i=1,2,$
and the BSEE of $\theta_{i}$ is
$\delta_{i,0}(\underline{X})=\frac{\alpha_{1}+2}{\alpha_{i}+1}X_{i},\;i=1,2$.
The p.d.f. of $Z=\frac{Z_{2}}{Z_{1}}$ is

$f_{Z}(z)=\frac{\alpha_{1}\alpha_{2}}{\alpha_{1}+\alpha_{2}}z^{\alpha_{2}-1}\left(\min\bigg{\\{}1,\frac{1}{z}\bigg{\\}}\right)^{\alpha_{1}+\alpha_{2}},\;\;z>0,$

the conditional p.d.f. of $Z_{1}$ given $Z=z$ ($0<z<\infty$) is

$f_{Z_{1}|Z}(z_{1}|z)=\begin{cases}\frac{(\alpha_{1}+\alpha_{2})\,z_{1}^{\alpha_{1}+\alpha_{2}-1}}{\left(\min\big{\\{}1,\frac{1}{z}\big{\\}}\right)^{\alpha_{1}+\alpha_{2}}},&\text{
if}\;\;0<z_{1}<\min\big{\\{}1,\frac{1}{z}\big{\\}}\\\ 0,&\text{
otherwise}\end{cases},$

$h_{1}(z)=E[Z_{1}|Z=z]=\frac{\alpha_{1}+\alpha_{2}}{\alpha_{1}+\alpha_{2}+1}\min\big{\\{}1,\frac{1}{z}\big{\\}},\,z>0,\text{
and
}h_{2}(z)=E[Z_{1}^{2}|Z=z]=\frac{\alpha_{1}+\alpha_{2}}{\alpha_{1}+\alpha_{2}+2}(\min\big{\\{}1,\frac{1}{z}\big{\\}})^{2},$
$\,z>0.$ Clearly, for $0<\theta<1$, $\frac{h_{2}(\theta z)f_{Z}(\theta
z)}{h_{2}(z)f_{Z}(z)}$ is increasing in
$z\in\left(0,\frac{c_{0,1}}{c_{0,2}}\right]$. For any fixed $\lambda\geq 1$,

$k(z,\lambda)=\frac{(\lambda-1)h_{1}\left(\frac{c_{0,1}z}{c_{0,2}\lambda}\right)}{(1-z)\,h_{2}\left(\frac{c_{0,1}z}{c_{0,2}\lambda}\right)}=\frac{(\lambda-1)(\alpha_{1}+\alpha_{1}+2)}{(1-z)(\alpha_{1}+\alpha_{2}+1)}\max\bigg{\\{}1,\frac{c_{0,1}z}{c_{0,2}\lambda}\bigg{\\}}$

is increasing in $z$ and, for any fixed $z\in(0,1)$, $k(z,\lambda)$ is
increasing in $\lambda\geq 1$. Using Lemma 5.1 (a), we get

$\displaystyle\sup_{\lambda\geq
1}\alpha(\lambda)=\frac{p_{1}}{p_{1}+p_{2}},\qquad\text{and}\qquad\inf_{\lambda\geq
1}\alpha(\lambda)=-\infty.$

Now, using Theorem 5.1 (a), it follows that the estimators
$\\{\underline{\delta}_{\alpha}^{M}:\alpha\in(-\infty,\frac{p_{1}}{p_{1}+p_{2}}]\\}$
are admissible within the class
$\\{\underline{\delta}_{\alpha}^{M}:-\infty<\alpha<\infty\\}$ of isotonic
regression estimators of $(\theta_{1},\theta_{2})$. Moreover the estimators
$\\{\underline{\delta}_{\alpha}^{M}:\alpha>\frac{p_{1}}{p_{1}+p_{2}}\\}$ are
inadmissible and, for $\frac{p_{1}}{p_{1}+p_{2}}\leq\alpha_{2}<\alpha_{1}$,
the estimator $\underline{\delta}_{\alpha_{2}}^{M}$ dominates the estimator
$\underline{\delta}_{\alpha_{1}}^{M}$. Also, using Theorem 3.1, we conclude
that the class of estimators
$\\{\underline{\delta}_{\alpha}^{M}:\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\}\leq\alpha<1\\}$
dominate the BSEE
$(\frac{\alpha_{1}+2}{\alpha_{1}+1}X_{1},\frac{\alpha_{2}+2}{\alpha_{2}+1}X_{2})$.

###  5.2. Simulation Study For Estimation of Scale Parameters
$(\theta_{1},\theta_{2})$

In Example 5.1.1, we have considered two independent gamma distributions with
unknown order restricted scale parameters (i.e., $\theta_{1}\leq\theta_{2}$)
and known shape parameters ($\alpha_{1}>0$ and $\alpha_{2}>0$). For
simultaneous estimation of scale parameters ($\theta_{1}$,$\theta_{2}$), under
the sum of the squared error loss functions (5.2), we have shown that the
isotonic regression estimators (IREs)
$\\{\underline{\delta}_{\alpha}^{M}:\alpha\in(-\infty,\frac{p_{1}}{p_{1}+p_{2}}]\\}$
are admissible within the class
$\mathcal{D}=\\{\underline{\delta}_{\alpha}^{M}:-\infty<\alpha<\infty\\}$.
Also, the estimators
$\\{\underline{\delta}_{\alpha}^{M}:\max\\{0,\frac{p_{1}-p_{2}}{p_{1}}\\}\leq\alpha<1\\}$
dominate the BSEE
$\underline{\delta}_{0}(\underline{X})=(c_{0,1}X_{1},c_{0,2}X_{2})$, where
$c_{0,i}=\frac{1}{\alpha_{i}+1},\;i=1,2$. In this section, for simplicity, we
take $p_{1}=1$ and $p_{2}=1$. To further evaluate the performances of various
estimators under the loss function
$L(\underline{\theta},\underline{a})=(a_{1}-\theta_{1})^{2}+(a_{2}-\theta_{2})^{2},\;\underline{\theta}=(\theta_{1},\theta_{2})\in\Theta_{0},\;\underline{a}=(a_{1},a_{2})\in\Re_{++}^{2}$,
in this section, we compare the risk performances of isotonic regression
estimators (IREs)
$\underline{\delta}_{1}^{M}(\underline{X})=(\delta_{1,1}^{M}(\underline{X}),\delta_{2,1}^{M}(\underline{X}))=(c_{0,1}X_{1},c_{0,2}X_{2})$,
$\underline{\delta}_{0.75}^{M}(\underline{X})=(\delta_{1,0.75}^{M}(\underline{X}),\delta_{2,0.75}^{M}(\underline{X}))$,
$\underline{\delta}_{0.5}^{M}(\underline{X})=(\delta_{1,0.5}^{M}(\underline{X}),\delta_{2,0.5}^{M}(\underline{X}))$,
$\underline{\delta}_{0}^{M}(\underline{X})=(\delta_{1,0}^{M}(\underline{X}),\delta_{2,0}^{M}(\underline{X}))$
and the restricted MLE
$\underline{\delta}_{R}(\underline{X})=\left(\min\Big{\\{}\frac{X_{1}}{\alpha_{1}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}}\Big{\\}},\max\Big{\\{}\frac{X_{2}}{\alpha_{2}},\frac{X_{1}+X_{2}}{\alpha_{1}+\alpha_{2}}\Big{\\}}\right)$,
numerically, through the Monte Carlo simulations.

For simulations, we generated 50000 samples of size 1 each from relevant gamma
distributions and computed the simulated risks of estimators
$\underline{\delta}_{1}^{M}(\underline{X})=(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$,
$\underline{\delta}_{0.75}^{M}(\underline{X})$,
$\underline{\delta}_{0.5}^{M}(\underline{X})$,
$\underline{\delta}_{0}^{M}(\underline{X})$ and
$\underline{\delta}_{R}^{M}(\underline{X})$ for different values of shape
parameters $(\alpha_{1},\alpha_{2})$.

(a) $\alpha_{1}=0.5$ and $\alpha_{2}=0.5$.

(b) $\alpha_{1}=0.8$ and $\alpha_{2}=0.5$.

(c) $\alpha_{1}=5$ and $\alpha_{2}=2$.

(d) $\alpha_{1}=1$ and $\alpha_{2}=10$.

(e) $\alpha_{1}=25$ and $\alpha_{2}=10$.

(f) $\alpha_{1}=5$ and $\alpha_{2}=10$.

Figure 3. Risk plots of estimators
$\underline{\delta}_{1}^{M}(\underline{X})=(c_{0,1}X_{1},c_{0,2}X_{2})$,
$\underline{\delta}_{0.75}^{M}(\underline{X})$,
$\underline{\delta}_{0.5}^{M}(\underline{X})$,
$\underline{\delta}_{0}^{M}(\underline{X})$ and
$\underline{\delta}_{R}^{M}(\underline{X})$ against the values of
$\lambda=\theta_{2}/\theta_{1}$.

The simulated values of risks of various estimators are plotted in Figure 3.
The following observations are evident from Figure 3:

(i) The risk function values of estimators
$\underline{\delta}_{0.75}^{M}(\underline{X})$,
$\underline{\delta}_{0.5}^{M}(\underline{X})$ and
$\underline{\delta}_{0}^{M}(\underline{X})$ are less than the risk function
values of the BSEE $(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$, which is in
conformity with theoretical findings of Example 5.1.1 and Theorem 5.1.

(ii) From Figure 3, we can observe that the IREs
$\underline{\delta}_{0.5}^{M}(\underline{X})$,
$\underline{\delta}_{0}^{M}(\underline{X})$ and
$\underline{\delta}_{R}^{M}(\underline{X})$ are not comparable and the IREs
$\underline{\delta}_{1}^{M}(\underline{X})$ and
$\underline{\delta}_{0.75}^{M}(\underline{X})$ are inadmissible. This is in
conformity with theoretical findings of Example 5.1.1.

(iii) When values of $\alpha_{1}$ and $\alpha_{2}$ are large and
$\alpha_{1}<<\alpha_{2}$, for small values of $\theta_{2}/\theta_{1}$, the
restricted MLE outperforms the other estimators.

(iv) There is no clear cut winner between various estimators, but performance
of the BLEE
$\underline{\delta}_{1}^{M}(\underline{X})=(X_{1}-\sigma_{1},X_{2}-\sigma_{2})$
and $\underline{\delta}_{0.75}^{M}(\underline{X})$ are worse than other
estimators. Also, when $\alpha_{1}\leq 1$ and $\alpha_{2}\leq 1$, the
performance of the restricted MLE is the worst among other estimators.

## Disclosure statement

There is no conflict of interest by authors.

## 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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