R-Estimation for Asymmetric Independent Component Analysis

Abstract

Independent Component Analysis (ICA) recently has attracted much attention in the statistical literature as an attractive and useful alternative to elliptical models. Whereas k-dimensional elliptical densities depend on one single unspeci ed radial density, however, k-dimensional independent component distributions involve k unspecifi ed component densities. In practice, for a given sample size n and given dimension k, this makes the statistical analysis much harder. We focus here on the estimation, from an independent sample, of the mixing/demixing matrix of the model. Traditional methods (FOBI, Kernel-ICA, FastICA) mainly originate from the engineering literature. The statistical properties of those methods are not well known, and they typically require very large samples. So does the "classical semiparametric" approach by Chen and Bickel (2006), which is based on an estimation of the k component densities (those densities being those of the unobserved independent components). The \double scatter matrix" method of Oja et al. (2006) and (2008) requires the arbitrary choice of two scatter matrices generally based on estimated higher-order moments which are likely to be poorly robust. As a reaction, an efficient (signed-)rank-based approach has been proposed by Ilmonen and Paindaveine (2011) for the case of symmetric component densities; their estimators unfortunately fail to be root-n consistent as soon as one of the component densities violates the symmetry assumption. In this paper, using ranks rather than signed ranks, we extend their approach to the asymmetric case and propose a one-step R-estimator for ICA mixing matrices. The finite-sample performances of those estimators are investigated and compared to those of existing methods under moderately large sample sizes. Particularly good performances are obtained from a version involving data-driven scores taking into account the skewness and kurtosis of residuals. Finally, we show, by an empirical exercise, that our methods also may provide excellent results in contexts such as image analysis, where the basic assumptions of ICA are quite unlikely to hold.