Independent Component Analysis (ICA) recently has attracted much attention in the statistical literature as an appealing 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. In practice, for given sample size n and 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. Their consistency requires moment conditions, they are poorly robust, and do not achieve 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 their optimality properties remain unclear. The “classical semiparametric ” approach by Chen and Bickel (2006), quite on the contrary, achieves semiparametric efficiency, but requires the estimation of the densities of the k unobserved independent compo-

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Abstract This work is concerned with testing the population mean vector of nonnormal high-dimensional multivariate data. Several tests for high-dimensional mean vector have been proposed in the literature, but they may not perform well for high-dimensional continuous, nonnormal multivariate data, which frequently arise in genomics studies and quantitative finance. This paper aims to develop a novel high-dimensional nonparametric test for the population mean vector so that multivariate normality assumption becomes unnecessary. With the aid of new tools in modern probability theory, we proved that the limiting null distribution of the proposed test is normal under mild conditions for p > n. We further study the local power of the proposed test and compare its relative efficiency with a modified Hotelling T 2 test for high-dimensional data. Our theoretical results indicate that the newly proposed test can have even more substantial power gain than the traditional nonparametric multivariate test does with finite fixed p. We assess the finite sample performance of the proposed test by examining its size and power via Monte Carlo studies. We illustrate the application of the proposed test by an empirical analysis of a genomics data set.

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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 unspecified radial density, however, k-dimensional independent component distributions involve k unspecified 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