USING COMPLEX INTEGRATION TO COMPUTE MULTIVARIATE NORMAL PROBABILITIES

The execution of most multiple comparison methods involves, at least i n part, the computation of the probability that a multivariate normal or multivarite t random vector is in a hyper-rectangle. In multiple co mparison with a control as well as multiple comparison with the best ( of normal populations or multinomial cell probabilities), the correlat ion matrix R of the random vector is nonsingular and of the form R = D + lambda lambda' where D is a diagonal matrix and lambda is a known v ector. It is well known that, in this case, the multivariate normal re ctangular probability can be expressed as a one-dimensional integral a nd successfully computed using Gaussian quadrature techniques. However , in multiple comparison with the mean (sometimes called analysis of m eans) of normal distributions, all-pairwise comparisons of three norma l distributions, as well as simultaneous inference on multinomial cell probabilities themselves, the correlation matrix is singular and of t he form R = D - eta eta'. It is not well known that, in this latter ca se, the multivariate normal rectangular probability can still be expre ssed as a single integral, albeit one with complex variables in its in tegrand. Previously published proofs of the validity of this expressio n either contained a gap or relied on a numerical demonstration, and t his article will provide an analytic proof. Furthermore, we explain ho w this complex integral can be computed accurately, using Romberg inte gration of complex variables when the dimension is low, and using Sida k's inequality as an approximation when the dimension is at least mode rate.