We propose a family of tests of the validity of the assumptions underlying independent component analysis methods. The tests are formulated as L2-type procedures based on characteristic functions and involve weights; a proper choice of these weights and the estimation method for the mixing matrix yields consistent and affine-invariant tests. Due to the complexity of the asymptotic null distribution of the resulting test statistics, implementation is based on permutational and resampling strategies. This leads to distribution-free procedures regardless of whether these procedures are performed on the estimated independent components themselves or the componentwise ranks of their components. A Monte Carlo study involving various estimation methods for the mixing matrix, various weights, and a competing test based on distance covariance is conducted under the null hypothesis as well as under alternatives. A real-data application demonstrates the practical utility and effectiveness of the method.

翻译：我们提出了一族检验独立成分分析方法中潜在假设有效性的方法。该检验方法基于特征函数构建L2型程序，并引入权重参数；通过恰当选择权重及混合矩阵估计方法，可获得一致性且仿射不变的检验。由于检验统计量的渐近零分布具有复杂性，实际实施中采用置换与重抽样策略。无论是对估计独立成分本身进行操作，还是对其分量的秩进行成分级运算，所获程序均具有无分布特性。我们采用蒙特卡洛方法，在零假设及备择假设条件下，对多种混合矩阵估计方法、不同权重设置以及基于距离协方差的竞争性检验进行了系统研究。实际数据应用验证了该方法的实用性与有效性。