New precise model of studentized principal components

Abstract

If the sample Mahalanobis distance (SMD) is atypically large, it is essential to statistically estimate or test the contribution of each studentized principal component (SPC) decomposed from the SMD to consider its cause. However, there are no appropriate probability models for the SPCs of small samples. This study proposes a precise probability model for the SPCs of small samples without estimating the population eigenvalues or eigenvectors. The proposed model for an SPC comprises an elementary formula of sample size and the SPC’s index multiplied by one random variable following the t-distribution, which is simpler and requires no further computing compared with previous models. Numerical experiments demonstrated that the proposed model performs well under the weak condition that population eigenvalues are closely distinct with various dimensions and sample size. For practical implementation, the proposed model was applied for correcting the SMD to the population Mahalanobis distance, demonstrating better performance than other models. Additionally, the proposed model enables precise statistical testing of SPCs for discriminant analysis, cluster analysis, and projection pursuit, and the model improves the expectation–maximization algorithm for Gaussian mixture models.