Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series B, Pt. 3 , pp. 344-355

ESTIMATION OF THE AVERAGE WORTH OF THE SELECTED SUBSET OF GAMMA POPULATIONS

By

NEERAJ MISRA, Panjab University

SUMMARY. Let $X_1 ,\ldots, X_k$ be $k$ $(\geq 2)$ independent random variables representing the populations $\Pi_1 ,\ldots, \Pi_k$, respectively and suppose that the random variable $X_i$ has a gamma distribution with known shape parameter $p$ and unknown scale parameter $\theta_i, i = 1 ,\ldots, k$. For the goal of selecting a nonempty subset of $\{ \Pi_1 ,\ldots, \Pi_k \}$, containing the best population (one associated with max ($\theta_1 ,\ldots, \theta_k ))$, we consider the decision rule which selects $\Pi_i$ if and only if $X_i \geq c$ max ($X_1 ,\ldots, X_k$), $i$ = 1, $\ldots k$, where $0 < c \leq 1$ is chosen so that the probability of including the best population in the selected subset is at least $P^{ \, *} (1 / k \leq P^{ \, *} < 1$), a pre-assigned level. We consider the problem of estimating $W$, the mean of all $\theta_i$'s in the selected subset for a general $k \ (\geq 2)$. The uniformly minimum variance unbiased estimator (which is also the uniformly minimum risk unbiased estimator) is obtained and it is established that the natural estimator of $W$ is inadmissible for the squared error loss function. Better estimators are obtained.

AMS (1980) subject classification. Primary 62F07, secondary 62F10.

Key words and phrases. Subset selection, worth of the selected subset, correct selection, natural estimator, inadmissible estimator, uniformly minimum variance unbiased estimator, uniformly minimum risk unbiased estimator.

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This article in mathematical reviews.