  In a recent paper we described a novel approach to the detection and
parameter estimation of a non-Gaussian stochastic background of gravitational
waves. In this work we propose an improved version of the detection procedure,
preserving robustness against imperfect noise knowledge at no cost of detection
performance: in the previous approach, the solution proposed to ensure
robustness reduced the performances of the detection statistics, which in some
cases (namely, mild non-Gaussianity) could be outperformed by Gaussian ones
established in literature. We show, through a simple toy model, that the new
detection statistic performs better than the previous one (and than the
Gaussian statistic) everywhere in the parameter space. It approaches the
optimal Neyman-Pearson statistics monotonically with increasing non-Gaussianity
and/or number of detectors. In this study we discuss in detail its efficiency.
This is a second, important step towards the implementation of a
nearly--optimal detection procedure for a realistic non-Gaussian stochastic
background. We discuss the relevance of results obtained in the context of the
toy model used, and their importance for understanding a more realistic
scenario.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Astrophysical Phenomena