Simulation of Multivariate Normal Rectangle Probabilities and
Their Derivatives: Theoretical and Computational Results

An extensive literature in econometrics and in numerical analysis has
considered the problem of evaluating the multivariate normal integral
over a rectangle. A leading case of such an integral is the negative
orthant probability. The problem is computationally difficult except
in very special cases. The multinomial probit (MNP) model used in
econometrics and biometrics has cell probabilities that are negative
orthant probabilities, with means and covariances depending on unknown
parameters (and, in general, on covariates). Estimation of this model
requires, for each trial parameter vector and each observation in a
sample, evaluation of the probability and its derivatives with respect
to mean and covariance parameters.  This paper surveys Monte Carlo
techniques that have been developed for approximations of the
probability and its linear and logarithmic derivatives, that limit
computation while possessing properties that facilitate their use in
iterative calculations for statistical inference: the Crude Frequency
Simulator (CFS), Normal Importance Sampling (NIS), a Kernel-Smoothed
Frequency Simulator (KFS). Stern's Decomposition Simulator (SDS), the
Geweke-Hajivassiliou-Keane Simulator (GHK), a Parabolic Cylinder
Function Simulator (PCF), Deak's Chi-squared Simulator (DCS), an
Acceptance/Rejection Simulator (ARS), the Gibbs Sampler Simulator
(GSS), a Sequentially Unbiased Simulator (SUS), and an Approximately
Unbiased Simulator (AUS).  The authors also discuss Gauss and FORTRAN
implementations of these algorithms and present our computational
experience with them.  The authors find that GHK is overall the most
reliable method.