Hypothesis testing and interval estimation for quantiles in a bivariate normal setup with a common mean

This article deals with hypothesis testing and interval estimation of a quantile in a bivariate normal population with a common mean setup. We consider independent and identically distributed random samples from a bivariate normal population with an unknown common mean $\mu,$ unknown variances $\sigma_{1}^2$ and $\sigma_{2}^2,$ and an unknown correlation coefficient $\rho.$ First, we derive an asymptotic confidence interval, followed by a classical confidence interval using the method of recovery of the variance estimate. We also develop approximate confidence intervals, including the bootstrap-p, bootstrap-t, and highest posterior density intervals. Additionally, we obtain two generalized confidence intervals utilizing some estimators of the common mean. Before addressing the hypothesis testing regarding the target quantile, we first tested the equality of the quantiles for two marginal distributions. We derive several tests for this hypothesis testing problem, including the likelihood ratio test, tests based on computational approaches, and tests based on the generalized variable method. An extensive simulation study has been carried out to demonstrate the performance of the proposed tests and interval estimates. The interval estimators are assessed on the basis of their coverage probability and average length, while the tests are evaluated according to their size and power values. Finally, the article concludes with two real-life examples demonstrating the potential applications of the proposed methods.