Rank-based Inference for Independent Component Analysis

Abstract

The focus of this dissertation is statistical inference for Independent Component Analysis (ICA). ICA is a method for representing the joint distribution of multivariate data as the product of statistically independent, univariate component distributions. It was initially developed for the problem of blind source separation where observations of mixed signals are separated into latent source signals. ICA then seeks to represent the observed mixed signals in terms of independent source signals. The difficulty of this problem cannot be understated because there exists little, if any, knowledge about those sources.

To be practically useful, any method of statistical inference in ICA must be robust over a wide range of settings for the source signals. This dissertation explores rank-based inference for ICA that incorporates nonparametric statistics, such as ranks, which possess favorable distribution-free qualities to construct consistent estimators and tests meeting level constraints under a large range of settings for the independent component distributions. By following a semiparametric approach to inference, explicit asymptotic distributions for estimators and asymptotic powers of tests under alternative distributions can be derived from these rank-based methods, which yield a notion of optimal inference.

The first contribution of this dissertation is a R-estimator for the mixing matrix in ICA that governs how the observations of the mixed signals depend on source signals. After deriving explicit asymptotic properties for the R-estimator, simulation experiments and an empirical example illustrate the efficacy of the R-estimator in finite sample settings. This dissertation also proposes tests for multiple-output linear regression with independent component errors: these tests can remove the effect of known covariates from mixed signals before applying existing methods of mixing matrix estimation in ICA, many of which require that the very strong assumption of source signals be independent hold.