Error bounds for asymptotic expansions of the distribution of the MLE in a GMANOVA model
For a generalized MANOVA model \(Y=A\Xi B+\epsilon\), where Y is an \(N\times p\) matrix, A is a known \(N\times k\) matrix of rank \(k\leq N\), B is a known \(q\times p\) matrix of rank \(q\leq p\) and \(\Xi\) is a \(p\times q\) matrix of unknown parameters, \(\epsilon\) is an \(N\times p\) disturbance matrix whose rows have independent p-variate normal distribution with unknown covariance matrix \(\Sigma\). Denote by \({\hat \Xi}\) the MLE of \(\Xi\). The author considers the distributions of \[ {\hat \Theta}=(A'A)^{1/2}({\hat \Xi}-\Xi)(B\Sigma^{-1}B')^{1/2}\quad and\quad {\hat \theta}=a'({\hat \Xi}-\Xi)b/\sigma \] where \(a=(a_ 1,...,a_ k)'\), \(b=(b_ 1,...,b_ g)'\) are fixed vectors and \(\sigma\) \(2=a'(A'A)^{-1}ab'(B\Sigma^{-1}B')^{-1}b\). Asymptotic expansions of the distribution function and of the density function of \({\hat \theta}\) and error bounds for these expansions are obtained. The results on the density function of \({\hat \theta}\) are generalized for the density function of the matrix variates \({\hat \Theta}\).
scientific article; zbMATH DE number 4039004