Inferences on Linear Combinations of Normal Means with Unknown and Unequal Variances

Abstract

This paper considers the problem of making inferences on a linear combination of means from independent normal distributions with unknown variances. While standard inference procedures require the assumption that the variances of the normal distributions are all equal or have known ratios, in this paper procedures are developed which allow the variances to take any unknown values. Bounds are developed for the confidence levels of confidence intervals, and for the sizes and p-values of hypothesis tests, which are valid for any values of the variances. These bounds are sharp in the sense that they are attained for certain limiting values of the variances. Applications to the one-way linear model are illustrated, and the Behrens-Fisher problem, which is a special case with just two means and for which the popular approximate Welch method can be employed, is also discussed. Some examples and simulations of the implementation of the procedures are provided, and comparisons are made with other procedures.