Matrix variate Kummer-gamma distribution (Q2755166)

The authors propose the matrix variate Kummer-gamma distribution and study its properties. A \(p\times p\) random symmetric positive definite matrix \(V\) is said to have a matrix variate Kummer-gamma distribution with parameter \(\alpha, \gamma\) and \(\Xi\), denoted as \(V\sim KG_{p}(\alpha,\gamma,\Xi)\), if its p.d.f. is given by NEWLINE\[NEWLINEK(\alpha,\gamma,\Xi) {\text{etr}}(-\Xi V) {\text{det}}(V)^{\alpha-{1\over 2}(p+1)} {\text{det}}(I_{p}+V)^{-\gamma}, V>0,NEWLINE\]NEWLINE where \(\alpha>{1\over 2}(p-1), -\infty<\gamma<\infty, \Xi(p\times p)>0\) and \(K(\alpha,\gamma,\Xi)\) is the normalizing constant given as \(K^{-1}(\alpha,\gamma,\Xi)=\Gamma_{p}(\alpha)\Psi(\alpha,\alpha-\gamma+{1\over 2}(p+1);\Xi)\), \({\text{Re}}(\Xi)>0\), where \(\Gamma_{p}(a)=\pi^{{1\over 4}p(p-1)}\prod\limits_{j=1}^{p}\Gamma\left(a-{1\over 2}(j-1)\right)\), \({\text{Re}}(a)>{1\over 2}(p-1)\) and \(\Psi\) is the confluent hypergeometric function of matrix argument. The following results, in particular, are proved.NEWLINENEWLINENEWLINELet \(V\sim K(\alpha,\gamma,\Xi)\). Then the characteristic function of \(V=(v_{ij})\), i.e. the joint characteristic function of \(v_{11}, v_{12},\ldots,v_{pp}\) is \(\phi_{V}(Z)=\Psi(\alpha,\alpha-\gamma+{1\over 2}(p+1); \Xi-iZ)/\Psi(\alpha,\alpha-\gamma+{1\over 2}(p+1);\Xi)\), where \(Z=Z'(p\times p)=({1\over 2}(1+\delta_{ij})z_{ij}), i=\sqrt{-1}\), and \({\text{Re}}(\Xi-iZ)>0\). If \(V\sim K(\alpha,\gamma,\Xi)\), then NEWLINE\[NEWLINE{\mathbf E}[{\text{det}}(V)^{h}]={\Gamma_{p}(\alpha+h)\over \Gamma_{p}(\alpha)}\cdot{\Psi(\alpha+h,\alpha-\gamma+{1\over 2}(p+1)+h;\Xi)\over\Psi(\alpha,\alpha-\gamma+{1\over 2}(p+1);\Xi)}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\mathbf E}[{\text{det}}(I_{p}+V)^{h}]={\Psi(\alpha,\alpha-\gamma+{1\over 2}(p+1)+h;\Xi)\over\Psi(\alpha,\alpha-\gamma+{1\over 2}(p+1);\Xi)}, \quad{\text{Re}}(h)>-\alpha+{1\over 2}(p-1).NEWLINE\]NEWLINE The distributions of some random quadratic forms are obtained also.