Estimating the parameter of selected uniform population under the squared log error loss function

Let π1, …, πk be k (⩾ 2) independent populations, where πi denotes the uniform distribution over the interval (0, θi) and θi > 0 (i = 1, …, k) is an unknown scale parameter. The population associated with the largest scale parameter is called the best population. For selecting the best population, We use a selection rule based on the natural estimators of θi, i = 1, …, k, for the case of unequal sample sizes. Consider the problem of estimating the scale parameter θL of the selected uniform population when sample sizes are unequal and the loss is measured by the squared log error (SLE) loss function. We derive the uniformly minimum risk unbiased (UMRU) estimator of θL under the SLE loss function and two natural estimators of θL are also studied. For k = 2, we derive a sufficient condition for inadmissibility of an estimator of θL. Using these condition, we conclude that the UMRU estimator and natural estimator are inadmissible. Finally, the risk functions of various competing estimators of θL are compared through simulation.