非对称独立成分分析的R-估计
R-Estimation for Asymmetric Independent Component Analysis
摘要：
独立成分分析（ICA）作为椭圆模型的替代方法，近年来在统计学文献中引起了广泛的关注。然而，k维椭圆密度依赖于一个单一的未指定的径向密度，然而，k维独立分量分布涉及k个未指定的分量密度，对于给定的样本量n和维度k，使得统计分析更困难。我们在这里集中于估计模型的混合矩阵。源自工程文献的传统方法（FOBI、Kernel-ICA、FastICA）具有一致性，需要矩条件，但无法实现任何类型的渐近有效性。当基于鲁棒分散矩阵时，Oja等人（2006）和Nordhausen等人（2008）开发的双分散方法具有更好的鲁棒性特征，但最优性不明确。Chen和Bickel（2006）的半参数方法实现了半参数效率，但需要估计k个未观察到的独立分量密度。作为一种反应，Ilmonen和Paindaveine（2011）提出了一种有效的（带符号）基于秩的方法，用于对称组分密度的情况，一旦其中一个组分密度是不对称的，这种情况就不能是根-n一致的。在本文中，使用秩而不是符号秩，我们将他们的方法扩展到非对称的情况下，并提出了一个一步的ICA混合矩阵的R-估计。这些估计的有限样本性能进行了研究，并与现有的方法相比，适度大的样本量。当使用数据驱动的分数考虑到残差的偏度和峰度时，获得了特别好的性能。最后，我们表明，通过实证练习，我们的方法也可以提供出色的结果，如图像分析的背景下，ICA的基本假设是不太可能举行。
英文摘要：
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.
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