On Probabilities of Rectangles in Multivariate Student Distributions: Their Dependence on Correlations
where X = (X1, *, Xj) is as before, while Si = (;P= 1 Z,)+, i = 1 '*, k, where ZV = (Z1l , Zkv), v = 1, *',p, is a random sample of p vectors, which are mutually independent and independent of X, and each of which has, in the simplest case, the same normal distribution as X. More generally, the Zv's have some normal distributions with mean values 0 and with some covariance matrices which need not coincide with that of X and even need not be identical. A certain proof of (2) was presented by A. Scott [6] but we shall give here a counterexample showing that, unfortunately, this proof is incorrect. However, if the correlations between Xi and Xj have the form A A p i (i, j =1,* , k; i 1j) where J<il 1 (i = 1, **,k) and where {pij} is any fixed correlation matrix, and if the correlations between Ziv and Zjv have the form Tzv Tjv (i, j = 1, * * *, k; i + j; v= 1 , p) where ITiv I< 1(i= I, ,k;v= 1,''. ,p), we shall prove here that the left-hand side probability in (2) is a non-decreasing function of each jfil and each ITivl; therefore, in this case of a special correlation structure, (2) is indeed true. The general validity of (2) still remains an open question.