Some theoretical properties of two kurtosis matrices, with application to invariant coordinate selection

Invariant coordinate selection (ICS) is a multivariate statistical method aimed at detecting data structures by means of the simultaneous diagonalization of two scatter matrices. Statistical applications of ICS include cluster analysis, independent component analysis, outlier detection, regression analysis and projection pursuit. Scatter matrices based on fourth-order moments often appear in ICS, partly due to their known asymptotic behaviour. This paper focuses on their theoretical properties, with special emphasis on symmetric distributions, finite mixtures and stochastic processes. Theoretical results highlight both appealing properties and limitations of kurtosis-based ICS as a tool for detecting data structures.