A one-sample location test based on weighted averaging of two test statistics when the dimension and the sample size are large

ABSTRACT

We discuss a one-sample location test that can be used when the dimension and the sample size are large. It is well-known that the power of Hotelling’s test decreases when the dimension is close to the sample size. To address this loss of power, some non exact approaches were proposed, e.g., Dempster (1958, 1960), Bai and Saranadasa (1996), and Srivastava and Du (2008). In this article, we focus on Hotelling’s test and Dempster’s test. The comparative merits and demerits of these two tests vary according to the local parameters. In particular, we consider the situation where it is difficult to determine which test should be used, that is, where the two tests are asymptotically equivalent in terms of local power. We propose a new statistic based on the weighted averaging of Hotelling’s T²-statistic and Dempster’s statistic that can be applied in such a situation. Our weight is determined on the basis of the maximum local asymptotic power on a restricted parameter space that induces local asymptotic equivalence between Hotelling’s test and Dempster’s test. Numerical results show that our test is more stable than Hotelling’s T²-statistic and Dempster’s statistic in most parameter settings.

KEYWORDS:

• Asymptotic power
• Dempster’s test
• High-dimensional data
• One-sample location test
• T²-statistic

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H12
• 62H15

Acknowledgments

The authors thank Professor Yasunori Fujikoshi and Professor Takashi Seo for extensive discussions, references, and encouragement and Ms. Manami Okuyama for the Monte Carlo simulation used to obtain the numerical results. We also would like to thank the associate editor and three reviewers for many valuable comments and helpful suggestions which led to an improved version of this article.

Funding

The research of the second author was supported in part by Grant-in-Aid for Young Scientists (B) (26730020) from Japan Society for the Promotion of Science.