Equivariant estimation of quantile vector of two normal populations with a common mean

Abstract

The problem of estimating quantile vector $\theta=(\theta_1,\theta_2)$ of two normal populations, under the assumption that the means ($\mu_i$s) are equal has been considered. Here $\theta_i=\mu+\eta\sigma_i,$ $i=1,2,$ denotes the $p^{th}$ quantile of the $i^{th}$ population, where $\eta=\Phi^{-1}(p)$, $0<p<1,$ and $\Phi$ denotes the c.d.f. of a standard normal random variable. The loss function is taken as sum of the quadratic losses. First, a general result has been proved which helps in constructing some improved estimators for the quantile vector $\theta.$ Further, classes of equivariant estimators have been proposed and sufficient conditions for improving estimators in these classes are derived. In the process, two complete class results have been proved. A numerical comparison of these estimators are done and recommendations have been made for the use of these estimators. Finally, we conclude our results with some practical examples.