A New Kurtosis Matrix, with Statistical Applications

The number of fourth-order moments which can be obtained from a random vector rapidly increases with the vector's dimension. Scalar measures of multivariate kurtosis may not satisfactorily capture the fourth-order structure, and matrix measures of multivariate kurtosis are called for. In this paper, we propose a kurtosis matrix derived from the dominant eigenpair of the fourth standardized moment. We show that it is the best symmetric, positive semidefinite Kronecker square root approximation to the fourth standardized moment. Additional properties are derived for realizations from GARCH and reversible random processes. Statistical applications include independent component analysis and projection pursuit. The star product of matrices highlights the connection between the proposed kurtosis matrix and other kurtosis matrices which appeared in the statistical literature. A simulation study assesses the practical relevance of theoretical results in the paper.