Wishart and chi-square distributions associated with matrix quadratic forms
The authors provide necessary and sufficient conditions for the distributional form of certain matrix quadratic forms. Specifically, they considered an \(n\times p\) matrix \(\mathbf Y\), which follows the normal distribution with parameters 0, \(\Sigma_{\mathbf Y}\) and assuming that \(\mathbf Q\) is a nonnegative definite \(n\times n\) matrix, they obtained necessary and sufficient conditions under which (i) the matrix \({\mathbf Y}'{\mathbf{QY}}\) follows some Wishart distribution (without requiring knowledge of the specific values of its parameters, say \(m\) and \(\Sigma\)) and (ii) the distribution of the random variable \(\ell'{\mathbf Y}'{\mathbf{QY}}\ell\) is a multiple of a chi-square distribution for every \(p\times 1\) vector \(\ell\). For the first problem, they have proved that \({\mathbf Y}'{\mathbf{QY}}\) follows a Wishart distribution if and only if one of the following equivalent conditions holds: (i) \(({\mathbf T}'\otimes {\mathbf I}_{\mathbf p})\Sigma_{\mathbf Y}({\mathbf T}\otimes{\mathbf I}_{\mathbf p})={\mathbf C}\otimes \Lambda\), where \({\mathbf T}\) is an \(n\times r\) matrix of rank \(r\), \(m{\mathbf C}\) is a symmetric and idempotent \(r\times r\) matrix with \(m= \text{rank}({\mathbf C})\) and \(\Lambda= E({\mathbf Y}'{\mathbf {QY}})\), (ii) \(({\mathbf Q}\otimes{\mathbf I}_{\mathbf p}) \Sigma_{\mathbf Y} ({\mathbf Q}\otimes{\mathbf I}_{\mathbf p})=m^{-1} {\mathbf{QAQ}}\otimes \Lambda\), where \({\mathbf A}\) is a nonnegative definite \(n\times n\) matrix such that \({\mathbf {QA}}\) is idempotent and \(m=\text{rank}({\mathbf{QA}})\). If one of these conditions holds then the distribution of \({\mathbf Y}'{\mathbf{QY}}\) is \(W_p (m,\Sigma)\), with \(\Sigma= m^{-1}\Lambda\). As the authors point out, these conditions can be verified effortlessly and they do not require a predetermination of the parameters of the Wishart distribution -- something which was essential for the verification of some similar conditions that were proposed earlier in the literature. Besides, the proof for the necessity and sufficiency of the new conditions is much simpler than that of the earlier conditions, while it is based only on some elementary properties of the Wishart distribution. Based on these conditions a complete characterizition of the class of possible covariance structures \(\Sigma_{\mathbf Y}\) under which the distribution of \({\mathbf Y}'{\mathbf{QY}}\) is Wishart is provided. For the second problem, it is shown that the distribution of \(\ell'{\mathbf Y}'{\mathbf{QY}} \ell\) is a multiple of a chi-square distribution if and only if \(\Delta_{11}=\dots=\Delta_{rr}\), and the \(\Delta_{ij}\)'s are skew-symmetric matrices for \(i\neq j\), \(i,j=1,\dots,r\). The matrix \(\Delta_{ij}\) is a \(p\times p\) matrix and it is the \((i,j)\) block of the matrix \(\Delta=({\mathbf T}'\otimes{\mathbf I}_{\mathbf p})\Sigma_{\mathbf Y}({\mathbf T}\otimes{\mathbf I}_{\mathbf p})\). Under the assumption that \({\mathbf Q}= {\mathbf I}_{\mathbf n}\) they prove that a necessary and sufficient condition for a multiple of a chi-square distribution of \(\ell'{\mathbf Y}'{\mathbf {QY}}\ell\) can also be found according to the structure of the matrix \(\Sigma_{\mathbf Y}\). Finally, an interesting corollary that connects somewhat the previously described theorems is stated. In particular, if \(\text{Cov}({\mathbf y}_i,{\mathbf y}_j)\) is a symmetric matrix for all \(i,j= 1,\dots, n\) (where \({\mathbf y}_i\) denotes the \(i\)th row of the matrix \({\mathbf Y}\)), then \({\mathbf Y}'{\mathbf {QY}}\) follows a Wishart distribution if and only if \(\ell'{\mathbf Y}'{\mathbf {QY}}\ell\) is distributed as a multiple of a chi-square random variable for every vector \(\ell\).
multivariate components of variance model
scientific article; zbMATH DE number 1053246
scientific article