Optimization of functions of matrices with an application in statistics
Let \(M\) be a fixed \(N\times N\) non-zero idempotent symmetric matrix and consider the totality \(\mathcal P\) of \(R\times N\) matrices \(X\) such that \(XM\neq 0\). \(U\) is taken as an open subset of \(\mathcal P\), which is itself open in \(\mathbb{R}^{R\times N}\). The author calls an \((M; U)\) function a ratio of two linear functions of \(\text{tr}(MX' X)^2)\), \((\text{tr}(MX'X))^2\) and \(\text{tr}(MX' X)\). A theorem on finding global extrema of such functions through classical techniques using matrix differential calculus is given. Statistical application is made in the setting of the classical linear model \(\underset\widetilde{} Y= X\underset\widetilde{} \beta+ \underset\widetilde{} \varepsilon\), \(\underset\widetilde{} \varepsilon\sim {\mathcal N}(0, V)\), with the covariance matrix \(V\) permitted to be singular, and whose eigenvectors (but not eigenvalues) are assumed known. The problem addressed is the best (minimal MSE) estimation of the positive eigenvalues among classes of non-negative quadratic potential estimators. The article is in the tradition of \textit{J. R. Magnus} and \textit{H. Neudecker} [Matrix differential calculus with applications in statistics and econometrics (1988; Zbl 0651.15001)].
scientific article; zbMATH DE number 847505
scientific article