An approach to upper bound problems for risks of generalized least squares estimators (Q1086936)

The linear model \(y=X\beta +u\) is considered with \(E(u)=0\), \(E(u'u)=\sigma^ 2\Sigma (\theta)\). \(\beta\) belongs to \({\mathbb{R}}^ k\). The unknown parameters are \(\beta\), \(\sigma^ 2\) and \(\theta\). The authors are particularly interested in the following case: \[ \Sigma^{- 1}(\theta)=I_ n+\lambda_ n(\theta)C \] where \(\lambda_ n\) is a real continuous function. First the ordinary least squares estimator of \(\theta\), say \({\hat \theta}\), is calculated; then the estimator of \(\beta\) is the generalized least squares estimator with the metric defined by the matrix \(\Sigma\) (\({\hat \theta}\)) and the quadratic risk, say R, is the \(k\times k\) matrix \(R=E[({\hat \beta}-\beta)({\hat \beta}-\beta)'].\) The main result concerns an upper bound and a lower bound for R, in terms of nonnegative definiteness of matrices. The autoregressive models where the errors are i.i.d. \(N(0,\sigma^ 2)\) and the heteroscedastic regression models play the role of examples.