Hypothesis testing and interval estimation for quantiles of two normal populations with a common mean

Abstract

The problems of interval estimation of, and testing a hypothesis on the quantile $\theta =\mu +\eta {\sigma }_1$ (for given η) have been considered when independent random samples are available from two normal populations with a common mean μ and possibly unknown and unequal variances. The asymptotic confidence interval (ACI) for the quantile has been derived using the Fisher information matrix. Further, parametric bootstrap approaches such as boot-p, boot-t as well as the generalized p-value method have been adopted to obtain the confidence intervals numerically. For hypothesis testing several tests such as the one based on the Computational Approach Test (CAT), the likelihood ratio test (LRT), a test using an estimator of quantile, and tests based on generalized p-value approach have been proposed. Finally, the sizes (powers) of all the proposed tests have been computed using Monte-Carlo simulation procedure. Also the confidence intervals have been compared through average length (AL), coverage probability (CP), and a new measure called - the probability coverage density (PCD).

Keywords:

• Asymptotic confidence interval (ACI)
• Boot-p and Boot-t confidence intervals
• computational approach test (CAT)
• generalized p-value approach
• Likelihood ratio test (LRT)
• numerical comparison
• power analysis

Acknowledgments

The authors would like to express their sincere thanks to the two anonymous referees and an Associate Editor for their constructive suggestions and comments which have helped significantly in improving the presentation of this work. The Second author (Manas Ranjan Tripathy) would like to thank Science and Engineering Research Board (SERB), [EMR/2017/003078 dated 30th May 2018], Department of Science and Technology (DST), New Delhi, India for providing some financial support.