Let $π_1$ and $π_2$ be two independent populations, where the population $π_i$ follows a bivariate normal distribution with unknown mean vector $\boldsymbolθ^{(i)}$ and common known variance-covariance matrix $Σ$, $i=1,2$. The present paper is focused on estimating a characteristic $θ_{\textnormal{y}}^S$ 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 $θ_{\textnormal{y}}^S$ 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 $θ_{\textnormal{y}}^S$ 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 $θ_{\text{y}}^S$ is carried-out using simulation.