Document Zbl 1425.62037 - zbMATH Open

Estimation after selection from gamma populations with unequal known shape parameters. (English) Zbl 1425.62037

Summary: Let \(\pi_1,\dots,\pi_k\) be \(k\) (\(\geq 2\)) independent gamma populations, where the population \(\pi_i\) has an unknown scale parameter \(\theta_i>0\) and known shape parameter \(\nu_i > 0\), \(i = 1,{\dots}, k\). We call the population associated with \(\mu_{[k]} = \max\{\mu_1,\dots,\mu_k\}\), \(\mu_i = \nu_i\theta_i\) the best population. For the goal of selecting the best population, the first two authors [Stat. Methodol. 18, 41–63 (2014; Zbl 1486.62059)] proposed a class \(\mathcal{D}_{0}\) of selection rules for the case of (possibly) unequal shape parameters. In this article, we consider the problem of estimating the mean \(\mu_S\) of the population selected by a fixed selection rule \(\underline{\delta}^{\overline{a}} \in \mathcal{D}_0\), under a scale-invariant loss function. We derive the uniformly minimum variance unbiased estimator (UMVUE). Two other natural estimators \(\varphi_{N,1}\) and \(\varphi_{N,2}\), which are respectively the analogs of the UMVUE and the best scale invariant estimators of \(\mu_i\)’s for the component problem, are studied. We show that \(\varphi_{N,2}\) is generalized Bayes with respect to a noninformative prior distribution, and is also minimax when \(k = 2\). The UMVUE and the natural estimator \(\varphi_{N,1}\) are shown to be inadmissible, and better estimators are obtained. A numerical study on the performance of various estimators indicates that the natural estimator \(\varphi_{N,2}\) outperforms the other natural estimators.