Super-efficiency in blind signal separation of symmetric heavy-tailed sources

This paper addresses the blind source separation (BSS) problem in the context of "heavy-tailed", or "impulsive" source signals, characterized by the nonexistence of finite second (or higher) order moments. We consider Pham's (1997) quasi-maximum likelihood (QML) approach, a modification of the maximum likelihood (ML) approach, applied using some presumed distributions of the sources. We introduce a related family of suboptimal estimators, termed restricted QML (RQML). A theoretical analysis of the asymptotic performance of RQML is presented. The analysis is used for showing that the variance of the optimal (non-RQML) estimator's error must decrease at a rate faster than 1/T (where T is the number of independent observations). This surprising property, sometimes called super-efficiency, has been observed before (in the BSS context) only for finite-support source distributions. Simulation results illustrate the good agreement with theory.