Shrinkage Estimators Dominating Some Naive Estimators of the Selected Entropy

Consider two populations characterized by independent random variables
$X_{1}$
and
$X_{2}$
such that
$X_{i}, i=1,2$
, follows a gamma distribution with an unknown scale parameter
$\theta _{i}>0$
, and known shape parameter
$\alpha >0$
(the same shape parameter for both the populations). Here
$(X_{1},X_{2})$
may be an appropriate minimal sufficient statistic based on independent random samples from the two populations. The population associated with the larger (smaller) Shannon entropy is referred to as the “worse” (“better”) population. For the goal of selecting the worse (better) population, a natural selection rule is the one that selects the population corresponding to
$\max \{X_{1},X_{2}\} (\min \{X_{1},X_{2}\})$
as the worse (better) population. This natural selection rule is known to possess several optimum properties. We consider the problem of estimating the Shannon entropy of the population selected using the natural selection rule (to be referred to as the selected entropy) under the mean squared error criterion. In order to improve upon various naive estimators of the selected entropy, we derive a class of shrinkage estimators that shrink various naive estimators towards the central entropy. For this purpose, we first consider a class of naive estimators comprising linear, scale and permutation equivariant estimators and identify optimum estimators within this class. The class of naive estimators considered by us contains three natural plug-in estimators. To further improve upon the optimum naive estimators, we consider a general class of equivariant estimators and obtain dominating shrinkage estimators. We also present a simulation study on the performances of various competing estimators. Illustrative examples have been given using real data sets.