The multivariate leptokurtic-normal distribution and its application in model-based clustering

This article proposes the elliptical multivariate leptokurtic-normal (MLN) distribution to fit data with excess kurtosis. The MLN distribution is a multivariate Gram–Charlier expansion of the multivariate normal (MN) distribution and has a closed form representation characterized by one additional parameter denoting the excess kurtosis. It is obtained from the elliptical representation of the MN distribution, by reshaping its generating variate with the associated orthogonal polynomials. The strength of this approach for obtaining the MLN distribution lies in its general applicability as it can be applied to any multivariate elliptical law to get a suitable distribution to fit data. Maximum likelihood is discussed as a parameter estimation technique for the MLN distribution. Mixtures of MLN distributions are also proposed for robust model-based clustering. An EM algorithm is presented to specifically obtain maximum likelihood estimates of the mixture parameters. Benchmark real data are used to show the usefulness of mixtures of MLN distributions.