Estimation Of Regression Parameters In Linear Models With Uniform Covariance Structure

This paper aims to study the estimation problem of regression parameters in linear models with uniform covariance structure. According to Gauss-Markov Theorem, the generalized least squares estimation (GLSE) is the best linear unbiased estimation (BLUE) and is superior to the least squares estimation (LSE). However, the correlation coefficient is included in the expression of GLSE, and in most cases it is unknown. Therefore, we first study the necessary and sufficient conditions of LSE being equal to GLSE. We prove that LSE is equal to the best linear unbiased estimation when the conditions are satisfied. Secondly, when the conditions are not satisfied, we first estimate the correlation coefficient, and then define a two-stage estimator of regression parameters. We prove that the two-stage estimation is an unbiased estimate and is superior to LSE when some conditions on the estimation of correlation coefficient are satisfied. Finally, based on the method of moments we propose a corrective estimation of correlation coefficient and then obtain a two-stage estimator of regression parameters which is shown to be unbiased. Furthermore, a simulation study and an illustrative example are carried out to compare the performances of the two-stage estimation with those of LSE. The simulation results show that in most cases the two-stage estimation is superior to LSE and the performances of the two-stage estimation are close to those of BLUE with the increase of the number of samples.