Asymptotic Distribution of Studentized Contribution Ratio in High-Dimensional Principal Component Analysis

Abstract

This article is concerned with consistent estimators of the asymptotic variances of the sample cumulative contribution ratio and the one of logit transformation. We deal with the case in which the covariance matrix has a spiked model in a high-dimensional case where the number of observations and the sample size are both large. Studentized statistics for the high-dimensional case are formulated. Our results are generalizations of Fujikoshi et al. (2008). Numerical simulations show that only the studentized statistic of the logit is reasonably accurate in high dimensions.

Keywords:

• Asymptotic distribution
• High-dimension
• Spiked population model
• Studentized cumulative contribution ratio

Mathematics Subject Classification:

• 62H10
• 62E20

Acknowledgments

The authors would like to thank the referee for suitable comments and careful reading. We are greatful to Professors Takakazu Sugiyama, Yasunori Fujikoshi, and Takashi Seo for their advice and encouragement.