The inefficiency of least squares: Extensions of the Kantorovich inequality (Q1077116)

The usual Gauss-Markov model \(Y=X\beta +\epsilon\), with V as the dispersion matrix of \(\epsilon\), is considered. Comparisons are made between the cases when V is known (and the BLUE estimator can be used) and when V is not known (and the simple least squares estimator is used). The comparison is made in terms of the difference between dispersion matrices of BLUE of \(X\beta\) and least squares estimate of \(X\beta\). In particular, the trace of this difference is suggested as a measure of inefficiency and lower and upper bounds of the trace are given.