Simultaneous estimation following subset selection of binomial populations

Summary

Let π₁, …, π_p be p (p ≥2) independent populations with π_i being binomial bin(l, θ) with an unknown parameter θ_i i = 1, …, p. Suppose independent random samples of sizes n₁, …, n _p are drawn from the populations π₁, …, π_p, respectively, and let $${erline X}_i=X_i/n_i$$, where $$X_{i}={Sigma^{ni}_{j=1}}X_{ij},i=1,$$ …, P. We call the population associated with the largest of θ_i ’s the best population. Suppose a population is selected using the Gupta’s (Gupta, S. S. (1965). On some multiple decision(selection and ranking) rules. Technometrics 7, 225–245) subset selection procedure. In this paper, we consider simultaneous estimation of the parameters of the selected populations. It is shown that neither the unbiased estimator nor the riskunbiased estimator (corresponding to the normalized squared error loss function) exists based on a single-stage sample. When additional observations are available from the selected populations, we derive an unbiased and risk-unbiased estimators for the selected subset and also prove that the natural estimators are positively biased. Finally, the bias and the risk of the natural, unbiased and risk-unbiased estimators are computed using Monte-Carlo simulation method.