Family of extended mean mixtures of multivariate normal distributions: Properties, inference and applications

A new class of skewed distributions, with a matrix skewness parameter, called extended mean mixtures of multivariate normal (EMMN) distributions, is constructed. The family of EMMN distributions includes the SN and MMN distributions as special cases. Some basic properties of this family, such as characteristic function, moment generating function, affine transformation and canonical forms of the distributions are derived. An EM-type algorithm is developed to carry out the maximum likelihood estimation of the parameters. Two special cases of this family are studied in detail. A simulation is carried out to examine the performance of the estimation method, and the flexibility is illustrated by fitting a special case of this family to a real data. Finally, the theoretical formula of the multivariate tail conditional expectation of the EMMN distribution is derived.