characterized by independent random variables $X_1$ and $X_2$ such that $X_i, i=1,2,$ follows a gamma distribution with unknown scale parameter $\theta_i>0$ and known shape parameter $\alpha Consider two populations attached. >0$ (same shape parameter in both populations). where $(X_1,X_2)$ may be a good least sufficient statistic based on independent random samples from two populations. A population associated with a larger (smaller) Shannon entropy is called a “worse” (“better”) population. For the goal of selecting the worse (better) population, the natural selection rule selects the population corresponding to $\max\{X_1,X_2\} ~(\min\{X_1,X_2\})$ Worse (better) population. This natural selection rule is known to have some optimal properties. Consider the problem of estimating the Shannon entropy (called selection entropy) of a population selected using natural selection rules under the mean squared error criterion. To improve the various simple estimators of the chosen entropy, we derive a class of contraction estimators that shrink the various simple estimators towards the central entropy. For this purpose, we first consider a class of simple estimators, including linear, scale, and permutation equivariant estimators, and identify the best estimator within this class. The class of simple estimators we considered includes three natural plug-in estimators. To further improve the best naive estimator, we consider the general class equivariate estimator and obtain the dominant shrinkage estimator. We also present a simulation study on the performance of various competing estimators. Real data analyzes are also reported to illustrate the applicability of the proposed estimator.
