%0 Journal Article %1 LETAC20081393 %A Letac, Gérard %A Massam, Hélène %D 2008 %J Journal of Multivariate Analysis %K distributions multivariate statistics wishart %N 7 %P 1393 - 1417 %R https://doi.org/10.1016/j.jmva.2008.04.006 %T The noncentral Wishart as an exponential family, and its moments %U http://www.sciencedirect.com/science/article/pii/S0047259X08001139 %V 99 %X While the noncentral Wishart distribution is generally introduced as the distribution of the random symmetric matrix Y1∗Y1+⋯+Yn∗Yn where Y1,…,Yn are independent Gaussian rows in Rk with the same covariance, the present paper starts from a slightly more general definition, following the extension of the chi-square distribution to the gamma distribution. We denote by γ(p,a;σ) this general noncentral Wishart distribution: the real number p is called the shape parameter, the positive definite matrix σ of order k is called the shape parameter and the semi-positive definite matrix a of order k is such that the matrix ω=σaσ is called the noncentrality parameter. This paper considers three problems: the derivation of an explicit formula for the expectation of tr(Xh1)…tr(Xhm) when X∼γ(p,a,σ) and h1,…,hm are arbitrary symmetric matrices of order k, the estimation of the parameters (a,σ) by a method different from that of Alam and Mitra K. Alam, A. Mitra, On estimated the scale and noncentrality matrices of a Wishart distribution, Sankhyā, Series B 52 (1990) 133–143 and the determination of the set of acceptable p’s as already done by Gindikin and Shanbag for the ordinary Wishart distribution γ(p,0,σ).

@article{LETAC20081393, abstract = {While the noncentral Wishart distribution is generally introduced as the distribution of the random symmetric matrix Y1∗Y1+⋯+Yn∗Yn where Y1,…,Yn are independent Gaussian rows in Rk with the same covariance, the present paper starts from a slightly more general definition, following the extension of the chi-square distribution to the gamma distribution. We denote by γ(p,a;σ) this general noncentral Wishart distribution: the real number p is called the shape parameter, the positive definite matrix σ of order k is called the shape parameter and the semi-positive definite matrix a of order k is such that the matrix ω=σaσ is called the noncentrality parameter. This paper considers three problems: the derivation of an explicit formula for the expectation of tr(Xh1)…tr(Xhm) when X∼γ(p,a,σ) and h1,…,hm are arbitrary symmetric matrices of order k, the estimation of the parameters (a,σ) by a method different from that of Alam and Mitra [K. Alam, A. Mitra, On estimated the scale and noncentrality matrices of a Wishart distribution, Sankhyā, Series B 52 (1990) 133–143] and the determination of the set of acceptable p’s as already done by Gindikin and Shanbag for the ordinary Wishart distribution γ(p,0,σ).}, added-at = {2018-03-18T05:14:59.000+0100}, author = {Letac, Gérard and Massam, Hélène}, biburl = {https://www.bibsonomy.org/bibtex/2a2cf0534281636575786532de57488b0/shabbychef}, description = {The noncentral Wishart as an exponential family, and its moments - ScienceDirect}, doi = {https://doi.org/10.1016/j.jmva.2008.04.006}, interhash = {db41a6691f3593df99a06869f71d33b5}, intrahash = {a2cf0534281636575786532de57488b0}, issn = {0047-259X}, journal = {Journal of Multivariate Analysis}, keywords = {distributions multivariate statistics wishart}, note = {Special Issue: Multivariate Distributions, Inference and Applications in Memory of Norman L. Johnson}, number = 7, pages = {1393 - 1417}, timestamp = {2018-03-18T05:14:59.000+0100}, title = {The noncentral Wishart as an exponential family, and its moments}, url = {http://www.sciencedirect.com/science/article/pii/S0047259X08001139}, volume = 99, year = 2008 }