The numerical evaluation of a class of integrals. II

Abstract

Consider the integralwhere x1, x2, …, xN are jointly distributed in a multivariate normal distribution f(x1, x2, …, xN) with (pij) as the correlation matrix. The integral has been expressed in an infinite series of tetrachoric functions for N≥2. The infinite series is not only complicated, but also is very slowly convergent and is consequently not of much practical use. Plackett (8) obtains a reduction formula for expressing normal integrals in four variates as a finite sum of single integrals of tabulated functions. These integrals have then to be evaluated by a rather awkward numerical quadrature.

On the Distributions of the Range and Mean Range for Samples from a Normal Distribution

“…The range for the sample is defined as R* = xxl. If the variance of the normal distribution is o2 then the range Rn for a standardized normal distribution is related to R* as follows: Rn = R*/(r. The distribution function of Rn is given by This translation from the known integral form of the range, to the form involving probabilities, is accomplished by first expanding the integral form using the binomial theorem and then comparing each term with the expression obtained by Das (1956) on the middle of p. 445 of his article. In general, the density becomes fRJ(W) = n(n '1) G()…”

Representations of multivariate normal distributions with special correlation structures

“…where X1,X2, ..., Xn,Y are now normal variates with mean zero and covariance matrix If we specialize Theorem 2 to the case where C = D is a diagonal matrix with positive elements [I -a;], we obtain the following result used by Das (1956). where C is a covariance matrix with associated correlation matrix p(C) and A is such that with the matrix is a correlation matrix.…”

An asymptotic expansion for the multivariate normal distribution and Mills' ratio

“…(S ee also Bartholomew [1] for a further application.) 3 ft is of some int e res t that thi s prope rt y pl ays an im port a nt role in oth er t heo re ti cal ap pli cati ons in stati sti cs. e.g.…”

“…Formula (3.1) is a special case of a more general formula, due (independently) to Dunnett and Sobel [4], Das [3], and Stuart [15], in which F(a, M) is expressed as a univariate integral involving the standardized normal density and distribution functions when Pij, the correlation between Xi and Xj, is of the form Pii = CiiCij(j =;1= i), and a is arbitrary. Formula (3.1), specialized further by a = 0, was proved (again independently) by Ruben [7] and Moran [6].…”