On the conditional test of order restricted multivariate normal mean vectors: simulation study and application

Abstract

Multivariate isotonic regression theory plays an important role in the field of testing statistical hypotheses under order restriction for several vector valued parameters and this problem may arise in several situations. Sasabuchi et al. derived a likelihood ratio type test for homogeneity of order restricted mean vectors under a multivariate normality assumption with unknown and common covariance matrices; however, its null distribution depends on the unknown covariance matrix. In order to overcome this difficulty, in the present paper we propose a test that conditions on the sufficient statistic for the covariance matrix. We remove the dependence of the null distribution of test statistic proposed by Sasabuchi et al. on the unknown value of covariance matrix and present a condition on a sufficient statistic for covariance matrix under the null hypothesis. A method based on Markov Chain Monte Carlo sampling is used to calculate the $p-$value of the test. We evaluate the operating characteristics of the proposed test through simulation studies. Our extensive simulation studies demonstrate that the power of this new test is generally higher than the test proposed by Sasabuchi et al. and also by Sasabuchi. A real data example is presented.

Keywords:

• Likelihood ratio type test
• Markov Chain Monte Carlo sampling
• Multivariate isotonic regression
• Power of test

AMS (2000) SUBJECT CLASSIFICATION:

• Primary 62F30
• Secondary 62F03
• 62H15

Acknowledgements

The author would like to thank the insightful comments from the reviewers, which greatly improved the presentation of this manuscript. This paper is dedicated to my beautiful and kind wife, Soheila, for all her patience.