Robust Higher Order Statistics

Sample estimates of moments and cumulants are known to be unstable in the presence of outliers. This problem is especially severe for higher order statistics, like kurtosis, which are used in algorithms for independent components analysis and projection pursuit. In this paper we propose robust generalizations of moments and cumulants that are more insensitive to outliers but at the same time retain many of their desirable properties. We show how they can be combined into series expansions to provide estimates of probability density functions. This in turn is directly relevant for the design of new robust algorithms for ICA. We study the improved statistical properties such as B-robustness, bias and variance while in experiments we demonstrate their improved behavior.