Simulation of multivariate normal rectangle probabilities and their derivatives theoretical and computational results

By V. Hajivassiliou, Daniel McFadden, P. Ruud. 1996

Abstract An extensive literature in econometrics and in numerical analysis has considered the problem of evaluating the multiple integral P(B; μ, Ω) = ∝ a b n(v − μ, Ω)dv ≡ E v 1(V ϵ B) , where V is a m-dimensional normal vector with mean μ, covariance matrix Ω, and density n(v − μ, Ω) , and 1(V ϵ B) is an indicator for the event B = (V¦a . A leading case of such an integral is the negative orthant probability, where B = (V ¦V . 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 μ and Ω 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 P(B; μ, Ω) and of its derivatives with respect to μ and Ω. This paper surveys Monte Carlo techniques that have been developed for approximations of P(B; μ, Ω) 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). We also discuss Gauss and FORTRAN implementations of these algorithms and present our computational experience with them. We find that GHK is overall the most reliable method.

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Citations in the corpus

Structural Workshop Paper - Discrete-Choice Models of Consumer Demand in Marketing (2011, methodology): “Kim et al. propose an MCMC algorithm to solve the problem, employing a GHK (Keane 1994; Hajivassiliou et al. 1996) algorithm to e ciently simulate from a truncated multivariate normal.”
Estimation of Errors in the Inverse Modeling of Accidental Release of Atmospheric Pollutant: Application to the Reconstruction of the Cesium-137 and Iodine-131 Source Terms from the Fukushima Daiichi Power Plant (2012, methodology): “An extensive review of different methods has been performed by Hajivassiliou et al. [1996]. We chose to use the Geweke-Hajivassiliou-Keane (GHK) simulator, which is found by these authors to be the most reliable method. We have found that, in our context, is was more efficient than Gibbs sampling, w”
Heterogeneity in R&D cooperation strategies (2004, background): “The GHK simulator (Geweke et al. 1997; Hajivassiliou et al. 1996) has been a particularly popular choice.”
A Censored Random Coefficients Model for Pooled Survey Data With Application to the Estimation of Power Outage Costs (2002, methodology): “Hajivassiliou et al. (1996) test thirteen of these techniques, and find the Geweke-Hajivassiliou-Keane (GHK) simulator to provide for the overall best estimation results. As discussed by Börsch-Supan and Hajivassiliou (1993), the simulated joint probabilities generated by the GHK algorithm have the ”