On linear functions of ordered correlated normal random variables

Many problems of statistical inference, notably the ones in multiple decisions and life testing involve the use of ordered Xi's. In an earlier paper Gupta, Pillai & Steck (1964) considered the distribution of linear functions of X(j)'s and gave closed form results for the case when the random variables are equally correlated. Also in the same paper closed form expressions for the distribution of the range W = X(l) - X(1> were obtained for n = 3 and 4 for the general case. In this paper we study the characteristic functions of individual order statistics and also the linear functions for the bivariate and trivariate cases for the general correlation matrix. Formulae for the expected values of X(f), X&2i) and X(j) X(j) and the first and second moments of a linear function of the X(j)'s are obtained. The joint distribution of the range and the mid-range is given in a closed form in the trivariate case, from which the distributions of the mid-range and the mid-range/range ratio are derived, again in closed forms. Best linear unbiased estimators of the common mean of three correlated normal variables have been obtained and tabulation of the coefficients made for different sets of values of pij's. Applications in the fields of life testing and time series analysis are discussed.