Simulating Normal Rectangle Probabilities and Their Derivatives: Effects of Vectorization

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Abstract
This paper studies the computationally difficult prob lem 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 proba bility, implied by the multinomial probit model used in econometrics and biometrics. Classical parametric esti mation of this type of model requires repeated evalu ation of such probabilities and their derivatives, thus making such estimation computationally intractable. Monte Carlo simulators have been developed to ap proximate the orthant probability integral and its deriv atives that limit computation while possessing proper ties that facilitate their use in statistical inference. I dis cuss Gauss and Fortran implementations of 13 simulation algorithms and show that these differ greatly with respect to the degree of vectorization: in some cases vectorization achieves a speedup factor in excess of 10, while in others the gains are negligible. Judged in terms of simulation root-mean-squared-er ror for a given investment in computation time, I find that the importance sampling recursive triangulariza tion simulator GHK remains the best method for simu lating probabilities irrespective of vectorization; the crude Monte Carlo simulator CFS offers the greatest benefits from vectorization; and the Gibbs resampling algorithm GSS emerges as one of the preferred meth ods for simulating logarithmic derivatives, especially in the absence of vectorization.