Simulating Normal Rectangle Probabilities and Their Derivatives: The Effects of Vectorization

An extensive literature in econometrics and in numerical analysis has considered the computationally difficult problem of evaluating the multiple integral representing the probability of a multivariate normal random vector constrained to lie in a rectangular region. A leading case of such an integral is the negative orthant probability, implied by the multinomial probit (MNP) model used in econometrics and biometrics. Classical parametric estimation of this model requires, for each trial parameter vector and each observation in a sample, evaluation of a normal orthant probability and its derivatives with respect to the mean vector and the variance-covariance matrix. Several Monte Carlo simulators have been developed to approximate the orthant probability integral and its linear and logarithmic derivatives that limit computation while possessing properties that facilitate their use in iterative calculations for statistical inference. In this paper, I discuss Gauss and FORTRAN implementations of 13 simulation algorithms, and I present results on the impact of vectorization on the relative computational performance of the simulation algorithms. I show that the 13 simulators differ greatly with respect to the degree of vectorizability: in some cases activating the CRAY-Y/MP4 vector facility achieves a speed-up factor in excess of 10 times, while in others the gains in speed are negligible. Evaluating the algorithms in terms of lowest simulation root-mean-squared-error for given computation time, I find that (1) GHK, an importance sampling recursive triangularization simulator, remains the best method for simulating probabilities irrespective of vectorization; (2) the crude Monte Carlo simulator CFS offers the greatest benefits from vectorization; and (3) the GSS algorithm, based on "Gibbs resampling," emerges as one of the preferred methods for simulating logarithmic derivatives, especially in the absence of vectorization.