Semiparametric Principal Component Analysis

We propose two new principal component analysis methods in this paper utilizing a semiparametric model. The according methods are named Copula Component Analysis (COCA) and Copula PCA. The semiparametric model assumes that, af- ter unspecified marginally monotone transformations, the distributions are multi- variate Gaussian. The robust nonparametric rank-based correlation coefficient estimator, Spearman's rho, is exploited in estimation. We prove that, under suitable condi- tions, although the marginal distributions can be arbitrarily continuous, the COCA and Copula PCA estimators obtain fast estimation rates and are feature selection consistent in the setting where the dimension is nearly exponentially large relative to the sample size.