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

Abstract

In principal component analysis we use the sample cumulative contribution ratio as an estimate of the population cumulative contribution ratio which is a proportion of information condensed into the first several principal components. The main purpose of this paper is to derive an asymptotic distribution of the cumulative contribution ratio in a high-dimensional situation where both the dimension and the sample size are large. Its asymptotic distribution is derived under a spiked model in which the variances of the remainder population principal components are assumed to be equal and small. We consider its logit transformation which is useful in a situation where the population cumulative contribution ratio is high or low. The corresponding large sample results are also summarized with a generalization. Numerical simulation revealed that our new approximation is more accurate than the classical large sample approximation as the dimension increases.

Key Words and Phrases:

• Asymptotic distribution
• contribution ratio
• high-dimension
• logit transformation
• principal component analysis