Scale mixtures of skew-normal-Cauchy distributions

Skewness-Based Projection Pursuit as an Eigenvector Problem in Scale Mixtures of Skew-Normal Distributions

This paper addresses the projection pursuit problem assuming that the distribution of the input vector belongs to the flexible and wide family of multivariate scale mixtures of skew normal distributions. Under this assumption, skewness-based projection pursuit is set out as an eigenvector problem, described in terms of the third order cumulant matrix, as well as an eigenvector problem that involves the simultaneous diagonalization of the scatter matrices of the model. Both approaches lead to dominant eigenvectors proportional to the shape parametric vector, which accounts for the multivariate asymmetry of the model; they also shed light on the parametric interpretability of the invariant coordinate selection method and point out some alternatives for estimating the projection pursuit direction. The theoretical findings are further investigated through a simulation study whose results provide insights about the usefulness of skewness model-based projection pursuit in the statistical practice.

On the scale mixtures of multivariate skew slash distributions

In this paper, the scale mixtures of multivariate skew slash distributions is introduced. The probability density function with some additional properties are discussed. The first four order moments, skewness and kurtosis of this distribution are calculated. Furthermore, the first two moments of its quadratic forms are obtained. In particular, the linear transformation, stochastic representation and hierarchical representation are studied. In the end, the EM algorithm is proposed.