Semiparametrically efficient inference based on signed ranks in symmetric independent component models

TL;DR

This paper develops semiparametrically efficient signed-rank inference methods for symmetric independent component models, applicable in blind source separation and ICA, with broad robustness to density assumptions.

Contribution

It introduces signed-rank based testing and estimation procedures for the mixing matrix in symmetric independent component models, achieving efficiency and robustness.

Findings

Procedures are semiparametrically efficient under correct density specification.

Methods remain valid under broad density conditions.

Finite-sample performance is demonstrated through simulations.

Abstract

We consider semiparametric location-scatter models for which the $p$-variate observation is obtained as $X=\Lambda Z+\mu$, where $\mu$ is a $p$-vector, $\Lambda$ is a full-rank $p\times p$ matrix and the (unobserved) random $p$-vector $Z$ has marginals that are centered and mutually independent but are otherwise unspecified. As in blind source separation and independent component analysis (ICA), the parameter of interest throughout the paper is $\Lambda$. On the basis of $n$ i.i.d. copies of $X$, we develop, under a symmetry assumption on $Z$, signed-rank one-sample testing and estimation procedures for $\Lambda$. We exploit the uniform local and asymptotic normality (ULAN) of the model to define signed-rank procedures that are semiparametrically efficient under correctly specified densities. Yet, as is usual in rank-based inference, the proposed procedures remain valid (correct asymptotic size under the null, for hypothesis testing, and root-$n$ consistency, for point estimation) under a very broad range of densities. We derive the asymptotic properties of the proposed procedures and investigate their finite-sample behavior through simulations.