Assuming a general linear model with unknown and possibly unequal normal error variances, the interest is to develop a one-sample procedure to handle the hypothesis testing on all, partial, or a subset of regression parameters. The sampling procedure is to split up each single sample of size ni at a controllable regressor's data point into two portions, the first consisting of the ni - 1 observations for initial estimation and the second consisting of the remaining one for overall use in the final estimation in order to define a weighted sample mean based on all sample observations at each data point. Then, the weighted sample mean is used to serve as a basis for parameter estimates and test statistics for a general linear regression model. It is found that the distributions of the test statistics based on the weighted sample means are completely independent of the unknown variances. This method can be applied to analysis of variance under various designs of experiments with unequal variances.
On Testing a Subset of Regression parameters Under Heteroscedasticity