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

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.

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.

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.

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

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. $Z_{i}=X_{i}-\theta_{i},\;i=1,2$.