Improved estimators for the selected location parameters

Let pi(i), i = 1, 2,..., k be k independent exponential populations with different unknown location parameters theta(i), i = 1, 2,..., k and common known scale parameter sigma. Let Y-i denote the smallest observation based on a random sample of size n from the i-th population. Suppose a subset of the given k populations is selected using the subset selection procedure according to which the population pi(i) is selected iff Y-i greater than or equal to Y-(1) - d, where Y-(1) is the largest of the Y-i's and d is some suitable constant. The estimation of the location parameters associated with the selected populations is considered for the squared error loss. It is observed that the natural estimator dominates the unbiased estimator. It is also shown that the natural estimator itself is inadmissible and a class of improved estimators that dominate the natural estimator is obtained. The improved estimators are consistent and their risks are shown to be O(kn(-2)). As a special case, we obtain the corresponding results for the estimation of theta((1)), the parameter associated with Y-(1).