Gauss-Markov estimation with an incorrect dispersion matrix

Abstract

We consider the general Gauss-Markov model $( Y, X\boldsymbol\beta, V)$, where $E(Y)= X\boldsymbol\beta$, $D(Y)= V$, and we provide solutions to the following questions: (1)When is a specific linear representation of the BLUE of $ X\boldsymbol\beta$ under $( Y, X\boldsymbol\beta,\sigma^2 V_0)$ also a BLUE under $( Y, X\boldsymbol\beta, V)$? (2) When does $ X\boldsymbol\beta$ have a common BLUE under $( Y, X\boldsymbol\beta,\sigmaˆ 2 V_0)$ and $( Y, X\boldsymbol\beta, V)$? (3) When is the BLUE of $ X\boldsymbol\beta$, under $( Y, X\boldsymbol\beta,\sigmaˆ 2 V_0)$, irrespective of the linear representation used in its expression, also a BLUE under $( Y, X\boldsymbol\beta, V)$? We also study the problem of Gauss-Markov estimation in a Variance Components and Covariance Components model.