A new family of multivariate centrally symmetric distributions

A family of dimension-wise scaled normal mixtures (DSNMs) is proposed to model the joint distribution of a d-variate random variable with real-valued components. Each member of the family generalizes the multivariate normal (MN) distribution in two directions. Firstly, the DSNM has a more general type of symmetry with respect to the elliptical symmetry of the MN distribution and, secondly, the univariate marginals have similar heavy-tailed normal scale mixture distributions with (possibly) different tailedness parameters. As a consequence of practical interest, the DSNM allows for a different excess kurtosis on each dimension. We examine a number of properties of DSNMs and we describe two members of the DSNM family obtained in the case of components of the mixing random vector being either uniform or shifted exponential. These are examples of mixing distributions that guarantee a closed-form expression for the joint density of the DSNM. For the two DSNMs analyzed in detail, we describe algorithms, based on the expectation-maximization (EM) principle, to estimate the parameters by maximum likelihood. We use real data from the financial and biometrical fields to appreciate the advantages of our DSNMs over other symmetric heavy-tailed distributions available in the literature.