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

Mishra, Neeraj and Iyer, Srikanth K and Singh, Harshinder (2004) The LINEX Risk of Maximum Likelihood Esimators of Parameters of Normal Populations Having Order Restricted Means. In: Sankhya: The Indian Journal of Statistics, 66 (4). pp. 652-677.

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We compare the performance of restricted and unrestricted maximum likelihood estimators of means ${\mu}_1$ and ${\mu}_2$ , and common variance ${\sigma}^2$, of two normal populations under LINEX (liner-exponential) loss functions, when it is known aprori that ${\mu}_1 \leq {\mu}_2$. If \delta is any estimator of the real parameter g(\b{\theta}), then the LINEX loss function is defined by L(\b{\theta}, \delta) = $e^{a(\delta-g(\b{\theta})} - a (\delta - g(\b{\theta})) - 1, a \not= 0$. We show that the restricted maximum likelihood estimator (MLE) $\hat{\mu}_1$ of ${\mu}_1$ is better than the unrestricted MLE ${\={X}}_1$, 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 $\hat{\mu}_2$ of ${\mu}_2$ is shown to be superior to the unrestricted MLE $(\={X})_2$ for a \in (- \infty, 0) \cup (0, $a^*_1$], and the two estimators 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 ${\sigma}^2$, we show that the restricted MLE $\hat{\sigma}^2$ 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 $\hat{\sigma}^2$. We also prove a conjecture of Gupta and Singh (1992) concerning the dominance of the restricted MLE $\hat{\sigma}^2$ 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.