A Linear Model with a Special Class of Covariance Structure

M.S. Haq and V.M. Ng (1994). A Linear Model with a Special Class of Covariance Structure . Journal of Statistical Research, Vol. 28, No. 1-2, pp. 189-198.

Abstract

A linear model with a special class of covariance matrix for the error variable is considered. The covariance matrix $\Omega$ is such that $\Omega^{-1}$ is of the from or can be approximated as $\alpha I + \gamma H$ , where I is the identity matrix, H is a symmetric matrix of known values; $\alpha$ and $\gamma$ are unknown scalar quantities. The model is treated as a conditional structural model. The analysis provides the structural distribution of the regression parameter $\beta$ and the scale parameter $\sigma$ (for known $\alpha$ and $\gamma$ ), and the marginal likelihood function for $\alpha$ and $\gamma$ . The results have been specialized to cover the case of normal errors with serial correlation coefficient. Then using the likelihood modulation technique the non-null distribution of the upper bound of the Durbin-Watson statistic has been derived. Some examples of the marginal likelihood function of $\rho$ are provided.