On estimating the scale parameter of the selected uniform population under the entropy loss function

Abstract

Let $\pi_{1},\ldots,\pi_{k}$ be $k$ ($\geq2$) independent populations, where $\pi_{i}$ denotes the uniform distribution over the interval $(0,\theta_{i})$ and $\theta_{i}>0$ ($i=1,\ldots,k$) is an unknown scale parameter. Let $\theta_{[1]}\leq\cdots\leq\theta_{[k]}$ be the ordered values of $\theta_{1},\ldots,\theta_{k}$. The population $\pi_{(k)}$ ($\pi_{(1)}$) associated with the unknown parameter $\theta_{[k]}$ ($\theta_{[1]}$) is called the best (worst) population. For selecting the best population, we consider a general class of selection rules based on the natural estimators of $\theta_{i},i=1,\ldots,k$. Under the entropy loss function, we consider the problem of estimating the scale parameter $\theta_{S}$ of the population selected using a fixed selection rule from this class. We derive the uniformly minimum risk unbiased estimator of $\theta_{S}$ and two natural estimators of $\theta_{S}$ are also considered. We derive a general result for improving a scale invariant estimator of $\theta_{S}$ under the entropy loss function. A simulation study on the performances of various competing estimators of $\theta_{S}$ is also reported. Finally, we provide similar results for the problem of estimating the scale parameter of selected population when the selection goal is that of selecting the worst uniform population.