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Mixture of the Riesz distribution with respect to the generalized multivariate gamma distribution
Louati, Mahdi 한국통계학회 2013 Journal of the Korean Statistical Society Vol.42 No.1
Wishart natural exponential families (NEFs) characterized by Letac (1989) are extended to the Riesz NEFs on symmetric matrices. These families are characterized by their variance functions defined in Hassairi and Lajmi (2001). This work uses a particular basis of these NEFs to describe the class of the generalized multivariate gamma distributions and then to study the statistical model obtained by the mixture of this distribution with the Riesz one on the space of symmetric matrices.
Mixture of the Riesz distribution with respect to the generalized multivariate gamma distribution
Mahdi Louati 한국통계학회 2013 Journal of the Korean Statistical Society Vol.42 No.1
Wishart natural exponential families (NEFs) characterized by Letac (1989) are extended to the Riesz NEFs on symmetric matrices. These families are characterized by their variance functions defined in Hassairi and Lajmi (2001). This work uses a particular basis of these NEFs to describe the class of the generalized multivariate gamma distributions and then to study the statistical model obtained by the mixture of this distribution with the Riesz one on the space of symmetric matrices.
Estimation of the parameters of a Wishart extension on symmetric matrices
Ghorbel Emna,Kammoun Kaouthar,Louati Mahdi,Sallem Akram 한국통계학회 2022 Journal of the Korean Statistical Society Vol.51 No.4
This paper deals with the parameters of a natural extension of the Wishart distribution, that is the Riesz distribution on the space of symmetric matrices. We estimate the shape parameter using two different approaches. The first one is based on the method of moments, we give its expression and investigate some of its properties. The second represents the maximum likelihood estimator. Unfortunately, in this case we do not have an explicit formula for this estimator. This latter is expressed in terms of the digamma function and sample mean of log-gamma variables. However, we derive the strong consistency and asymptotic normality properties of this estimator. A numerical comparative study between the two estimators is carried out in order to test the performance of the proposed approaches. For the second parameter, that is the scale parameter, we prove that the distribution of the maximum likelihood estimator given by Kammoun et al. (J Statist Prob Lett 126:127–131, 2017) is related to the Riesz distribution. We examine some properties concerning this estimator and we assess its performance by a numerical study.