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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, usinga LINEX loss function. A natural selection rule is used for achieving the aimof selecting the best bivariate normal population. Some natural-type estimatorsand Bayes estimator (using a conjugate prior) of $\theta_{\textnormal{y}}^S$are presented. An admissible subclass of equivariant estimators, using theLINEX loss function, is obtained. Further, a sufficient condition for improvingthe competing estimators of $\theta_{\textnormal{y}}^S$ is derived. Using thissufficient condition, several estimators improving upon the proposed naturalestimators are obtained. Further, a real data example is provided forillustration purpose. Finally, a comparative study on the competing estimatorsof $\theta_{\text{y}}^S$ is carried-out using simulation.

Estimation after selection from bivariate normal population using LINEX loss function