On the central moments of the linear combinations of multidimensional random variables: a contribution to maximum and minimum values determination

Abstract

In the statistical analysis of multivariate distributions, the linear combinations for which the standardized third and fourth moments have their maximum or minimum value may by utilized, for instance, to inquire on robustness, test on multinormality or identify possible outliers or clusters. In this note some theoretical results are shown widening the concept of a matrix eigenvalue and eigenvector which are obtained by the author in determining such combinations. There are given explicit solutions for bivariate distributions, while an iterative method is suggested for those of greater dimensions.