Estimation methods for regression models with unequal error variances

In this dissertation, we consider estimation procedures for the linear model when the observations are replicated but the error variances are unequal. The aim of this research was to consider several standard methods for parameter estimation, namely Ordinary Least Squares, Weighted Least Squares, and Maximum Likelihood and to compare these with newly developed procedures. These new techniques included the use of a prior likelihood function to induce "shrinkage" towards a common value among the estimators for the error variances and procedures based on preliminary tests of the hypothesis of variance equality. Both an overall test of equality and a multiple comparison method were considered. In addition, variance estimates based on MINQUE (Minimum Norm Quadratic Unbiased Estimator) were investigated. The MINQU estimators tend to "stretch out" the variances and were found to be unsatisfactory. The performance of the above-mentioned approaches was investigated both through asymptotic theoretical results and small samples simulation studies. The results from these two approaches were found to be in broad agreement. Overall, the multiple comparison and prior likelihood procedures appear to perform best, but the prior likelihood depends upon the availability of satisfactory prior information. So the multiple comparison procedure appears to be the most effective technique in general. A further study was also conducted to examine the effects of "errors-in-variables." In general it is found that maximum likelihood is superior to ordinary least squares and weighted least squares only if a large number of replicates is used.