Moments of generalized Wishart distributions
Moments of a distribution are of interest partially because they can be used to represent certain statistical models, to carry out statistical inference, and to approximate the given distribution. Moment calculation has, however, been influenced by results on matrix algebra and derivatives. In fact, many results on matrix algebra and derivatives have been motivated by the problem of finding moments and for other statistical results. In Sections 2 and 3, we relate moments of a random matrix \(Y\) to tensor differential forms and permutations without assuming that \(Y\) is normal or left elliptically contoured. Examples are given for two large classes of distributions. The first class consists of the generalized Wishart distributions induced by an \(s\times t\) random matrix \(Y\) with covariance \(I_s \otimes \Sigma\). The second class consists of the generalized complex Wishart distributions induced by a \((2n) \times p\) random matrix \(Y\) with covariance \(I_n\otimes \left(\begin{smallmatrix} \Sigma_1 & - \Sigma_2 \\ \Sigma_2 & \Sigma_3\end{smallmatrix} \right)\). \textit{N. R. Goodman} [see Ann. Math. Stat. 34, 152-177 (1963; Zbl 0122.36903)] and many other authors considered the case where \(Y\) is normal, the covariance is nonsingular, and \(\Sigma_1 =\Sigma_3\). Our generalized complex Wishart distributions combine real Wishart distributions and the complex Wishart distributions of Goodman into a much larger class. In Section 4, we find the \(q\)th order moments of noncentral generalized Wishart distributions induced by left elliptically contoured distributions. The formula contains \(\sum^q_{v=0} (2q)!/[v! 2^v(2q-2v)!]\) terms and a certain degree of complication is, therefore, inevitable. We shall derive this formula in a simple way and represent it in a transparent form.
generalized real and complex Wishart distributions
multivariate left elliptically contoured distribution
scientific article; zbMATH DE number 739446