Sankhya: The Indian Journal of Statistics

2004, Volume 66, Pt. 4,  652-677

The LINEX Risk of Maximum Likelihood Estimators of Parameters of Normal Populations Having Order Restricted Means

By

Neeraj Misra, Indian Institute of Technology Kanpur, India, Srikanth K. Iyer, Indian Institute of Science, Bangalore, India      Harshinder Singh, West Virginia University, Morgantown, USA

SUMMARY. We compare the performances of the restricted and unrestricted maximum likelihood estimators of means $\mu_1$ and $\mu_2,$ and common variance $\sg^2,$ of two normal populations under  LINEX (linear-exponential) loss functions, when it is known apriori that $\mu _1 \leq \mu _2$ . If $\del$ is any estimator of the real parameter $g(\uth),$ then the LINEX loss function is defined by $L(\uth, \del) = e^{a(\del - g(\uth))} - a(\del - g(\uth)) - 1,$ $a \ne 0.$ We show that the restricted maximum likelihood estimator (MLE) $\moh$ of $\mu_1$ is better than the unrestricted MLE $\xbo$, for $a \in [a_1,0)\cup (0,\infty),$ where $a_1 < 0$ is a constant depending on the sample sizes $n_1$ and $n_2$. For $a < a_1,$ the two estimators are shown to be not comparable. Similarly for a constant $a_{1}^{*} > 0$, depending on the sample sizes, the restricted MLE $\mth$ of $\mu_2$ is shown to be superior to the unrestricted MLE $\xbt$ for $a \in (-\infty, 0) \cup (0,a_{1}^{*}]$, and the two   stimators are shown to be not comparable for $a > a_{1}^{*}$. Similar results are obtained for the simultaneous estimation of $(\mu_1,\mu_2)$ under the sum of LINEX loss functions.  For the estimation of $\sg^2$, we show that the restricted MLE $\sgh ^2$ is superior to the unrestricted MLE $S^2$ for $ a \in (-\infty,0) \cup (0,a_2]$ and the two estimators are shown to be not comparable for $a \in (a_2,a_3),$ where $0<a_2<a_3$ are constants depending on the sample sizes. Interestingly, for $a \in [a_3, (n_1 + n_2)/2)$, it turns out that the unrestricted MLE $S^2$ is better than the restricted MLE $\sgh^2$. We also prove a conjecture of Gupta and Singh (1992) concerning the dominance of the restricted MLE $\sgh^2$ over the unrestricted MLE $S^2$ under the Pitman nearness criterion.  Finally, we generalize some of these results to the case of $k$ ($\geq 2$) normal populations.

AMS (1991) subject classification. Primary 62F10; Secondary 62A10 ;

Key words and phrases. Squared error loss, LINEX loss, maximum likelihood estimation

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