On the distribution of linear functions and ratios of linear functions of ordered correlated normal random variables with emphasis on range

Abstract

SUMMARY This paper deals with the distribution of linear functions and ratios of linear functions of order statistics from an equally correlated set of normal random variables. Some special cases of this family of linear functions are considered. The case of range is studied in great detail both for the case of equal correlations as well as in the general case. For the general case, formulae for the distribution of range are obtained for sample sizes of two, three and four. In the equally correlated case, expressions are obtained for the probability integral, percentage points and moments of the linear functions in terms of the corresponding expressions for the uncorrelated cases. It is also shown that ratios of certain linear functions of the order statistics have a distribution independent of this common correlation.