  This paper introduces a novel statistical framework for independent component
analysis (ICA) of multivariate data. We propose methodology for estimating and
testing the existence of mutually independent components for a given dataset,
and a versatile resampling-based procedure for inference. Independent
components are estimated by combining a nonparametric probability integral
transformation with a generalized nonparametric whitening method that
simultaneously minimizes all forms of dependence among the components.
U-statistics of certain Euclidean distances between sample elements are
combined in succession to construct a statistic for testing the existence of
mutually independent components. The proposed measures and tests are based on
both necessary and sufficient conditions for mutual independence. When
independent components exist, one may apply univariate analysis to study or
model each component separately. Univariate models may then be combined to
obtain a multivariate model for the original observations. We prove the
consistency of our estimator under minimal regularity conditions without
assuming the existence of independent components a priori, and all assumptions
are placed on the observations directly, not on the latent components. We
demonstrate the improvements of the proposed method over competing methods in
simulation studies. We apply the proposed ICA approach to two real examples and
contrast it with principal component analysis.
