R-Estimation for Asymmetric Independent Component Analysis
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Date
2013-04-12
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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.
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Keywords
independent component analysis (ICA), local asymptotic normality (LAN), ranks, R-estimation, robustness