A Kotz-Type Distribution for Multivariate Statistical Inference

Abstract

In this chapter, we consider a Kotz-type distribution (of a p-variate random vector X) which has fatter tail regions than that of multivariate normal distribution, and its probability density function (pdf) is given by $$f(x,\mu ,\Sigma ) = c_p \left| \Sigma \right|^{ - \tfrac{1}{2}} \exp \{ - [(x - \mu )'\Sigma ^{ - 1} (x - \mu )]^{\tfrac{1}{2}} \} ,$$where μ∈ℜp, Σ is a positive definite matrix and $$c_p = \tfrac{{\Gamma (\tfrac{p}{2})}}{{2\pi ^{\tfrac{p}{2}} \Gamma (p)}}$$. We review various characteristics and provide a simulation algorithm to simulate samples from this distribution. Estimation of the parameters using the maximum likelihood method is discussed. An interesting fact is that the maximum likelihood estimators under this distribution are the generalized spatial median (GSM) estimators as defined by (1988). Using the asymptotic distribution of the estimates, statistical inferences on the parameters of the distribution are illustrated with an example.