On risk unbiased estimation after selection

Abstract

In many practical situations, it is desired to compare several populations, find the best one and estimate some parametric functions associated with the selected population. This has been recognized as an important problem that arises in various applications in agricultural, industrial and medical studies. This paper concerns unbiased estimation of a general parametric function, say $\gamma(\theta)$, of selected populations under the squared error loss (SEL) function. Examples of $\gamma(\cdot)$ include reliability function, odds ratio and variance, among others. Also, we obtain the uniformly minimum risk unbiased estimators of the parameters of selected populations under some general class of loss functions other than the commonly used SEL function. Furthermore, we characterize some loss functions for which the risk unbiased estimators of parameters of selected populations do not exist. Theoretical results are augmented with various illustrations and examples.