Estimation after selection from bivariate normal population using LINEX loss function

TL;DR

This paper develops estimators for a characteristic of a selected bivariate normal population under LINEX loss, proposing improvements and comparing their performance through simulations and a real data example.

Contribution

It introduces new natural and Bayes estimators for the selected population characteristic and establishes conditions for their improvement under LINEX loss.

Findings

Proposed estimators are improved upon existing natural estimators.

A sufficient condition for estimator improvement is derived.

Simulation studies demonstrate the effectiveness of the proposed estimators.

Abstract

Let $\pi_1$ and $\pi_2$ be two independent populations, where the population $\pi_i$ follows a bivariate normal distribution with unknown mean vector $\boldsymbol{\theta}^{(i)}$ and common known variance-covariance matrix $\Sigma$, $i=1,2$. The present paper is focused on estimating a characteristic $\theta_{\textnormal{y}}^S$ of the selected bivariate normal population, using a LINEX loss function. A natural selection rule is used for achieving the aim of selecting the best bivariate normal population. Some natural-type estimators and Bayes estimator (using a conjugate prior) of $\theta_{\textnormal{y}}^S$ are presented. An admissible subclass of equivariant estimators, using the LINEX loss function, is obtained. Further, a sufficient condition for improving the competing estimators of $\theta_{\textnormal{y}}^S$ is derived. Using this sufficient condition, several estimators improving upon the proposed natural estimators are obtained. Further, a real data example is provided for illustration purpose. Finally, a comparative study on the competing estimators of $\theta_{\text{y}}^S$ is carried-out using simulation.