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

Year 2019, Volume: 48 Issue: 1, 255 - 273, 01.02.2019

Manas Ranjan Tripathy Adarsha Kumar Jena Somesh Kumar

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.

Keywords

Equivariant estimator, Estimation of quantiles, Complete class results, Common mean, Inadmissibility, Relative risk comparison

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