Effect of nonnormality on tests for a mean vector with missing data under an elliptically contoured pattern-mixture model

Abstract

We derive the asymptotic null distribution of Hotelling’s T²-type and likelihood ratio test statistics using two-step monotone sample under an elliptically contoured pattern-mixture model up to the order $N^{-1}$ to investigate the robustness of the test statistics with respect to nonnormality, where N denotes the total sample size. Using some matrix algebra, we obtain more simple expression of the test statistic, reducing the amount of the calculation required for obtaining the main result. The main result indicates some remarkable properties with respect to their distribution up to the order $N^{-1}:$ each test statistic obtained using two-step monotone missing data exhibits a lower effect of nonnormality when compared with the statistic obtained using listwise deletion, and the upper percentiles of each test statistic exhibit lower values when the kurtosis parameters are positive and large. The main results are applied to extend the Bartlett-type correction to the test statistics under an elliptically contoured pattern-mixture model. Monte Carlo simulation demonstrates that the test statistics exhibit the aforementioned distributional properties and that our proposed tests outperform those obtained by assuming multivariate normality in aspects of controlling type I error exactly. Furthermore, the application of our proposed tests to an actual dataset is presented.

Keywords:

• Elliptically contoured distribution
• Hotelling’s T2-test
• likelihood ratio test
• pattern-mixture model
• monotone missing data

Mathematics Subject Classification:

• 62H15
• 62E20

Acknowledgments

The author would like to thank to Mr K. Takagi of Kobe University for their help in performing Monte Carlo simulations. Furthermore, the author would also like to express his sincere gratitude to the associate editor and the two anonymous referees. The comments and suggestions of the referees have considerably improved the presentation of this paper.

Additional information

Funding

This work was supported by Grant-in-Aid for Young Scientists (JP19K14595) from Japan Society for the Promotion of Science.