Transelliptical Component Analysis

We propose a high dimensional semiparametric scale-invariant principle component analysis,named TCA, by utilize the natural connection between the elliptical distribution family and the principal component analysis. Elliptical distribution familyincludes many well-known multivariate distributions like multivariate Gaussian, t and logistic and it is extended to the meta-elliptical by Fang et.al (2002) using the copula techniques. In this paper we extend the meta-elliptical distribution family to a even larger family, called transelliptical. We prove that TCA can obtain a near-optimal s logd/n estimation consistency rate in recovering the leading eigenvector of the latent generalized correlation matrix under the transelliptical distribution family, even if the distributions are very heavy-tailed, have infinite second moments, do not have densities and possess arbitrarily continuous marginaldistributions. A feature selection result with explicit rate is also provided. TCA is further implemented in both numerical simulations and largescale stockdata to illustrate its empirical usefulness. Both theories and experiments confirmthat TCA can achieve model flexibility, estimation accuracy and robustness at almost no cost.