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R-Estimation for Asymmetric Independent Component Analysis

Independent Component Analysis (ICA) recently has attracted attention in the statistical literature as an alternative to elliptical models. Whereas k-dimensional elliptical densities depend on one single unspecified radial density, however, k-dimensional independent component distributions involve k unspecified component densities that for given sample size n and dimension k making statistical analysis harder. We focus here on estimating the model's mixing matrix. Traditional methods (FOBI, Kernel-ICA, FastICA) originating from the engineering literature have consistency that requires moment conditions without achieving any type of asymptotic efficiency. When based on robust scatter matrices, the two-scatter methods developed by Oja, et al. (2006) and Nordhausen, et al. (2008) enjoy better robustness features but have unclear optimality properties. The semiparametric approach by Chen and Bickel (2006) achieves semiparametric efficiency but requires estimating the k unobserved independent component densities. As a reaction, an efficient (signed-)rank-based approach has been proposed by Ilmonen and Paindaveine (2011) for the case of symmetric component densities that fail to be root-n consistent as soon as one of the component densities is asymmetric. 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 when using 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 a context such as image analysis, where the basic assumptions of ICA are quite unlikely to hold.