On Certain Inequalities for Normal Distributions and their Applications to Simultaneous Confidence Bounds

Abstract

In practical situations, one is generally faced with multivariate problems in the form of testing the hypotheses or obtaining a set of simultaneous confidence bounds on certain parameters of interest. We shall consider here the variates under study to be normally distributed. A lot of work on the univariate and multivariate normal populations for the simultaneous confidence bounds on the location and scale parameters has been done, (see references, not necessarily exhaustive). Establishing certain inequalities for normal variates, we try to give shorter confidence bounds on variances and on a given set of linear functions of location parameters when this set is previously chosen for study. For the univariate case, Dunn [6], [8] using the Bonferroni inequality, obtained shorter confidence bounds when the number of linear functions is not too large. We may note that Nair [12], David [5], Dunn [6], [7], [8] and Siotani [22], [24] have studied the closeness of the Bonferroni inequality while deriving the percentage points of certain statistics in univariate and multivariate normal cases. In this paper, we improve the Bonferroni inequality in all the situations considered by Siotani [22], [23], [24] and Dunn [6], [7], [8], and point out various uses of these results in obtaining simultaneous confidence bounds on variances and on linear functions of means (or location parameters) with confidence greater than or equal to $1 - \alpha$ where $\alpha$ is the size of the test. We mention our main results in Section 2 for those who are interested in results and not in proofs. Since our results are extensions of Dunn [6], [8], Siotani [22], [24] and Banerjee [2], [3], their comments on the shortness of the confidence bounds apply to our cases too.