Numerical Computation of Multivariate Normal and Multivariate-t Probabilities over Convex Regions

Abstract

A methodology has been developed and Fortran 90 programs have been written to evaluate multivariate normal and multivariate-t integrals over convex regions. The Cholesky transformation is used to transform the integrand into a product of standard normal or spherically symmetric t variables. For any random direction from the origin, an unbiased estimate of the value of the integral is Pr[X ² ≥ r²] (multivariate normal) or Pr[F ≥ r²/k] (multivariate-t), where r is the distance from the origin to the boundary in a randomly chosen direction, and k is the dimension of the integral. Two Fortran 90 programs have been written. MVI uses the average of many estimates. MVIB uses a binning procedure to obtain an empirical distribution of the distance from the origin to the boundary. Gauss-Legendre quadrature is then used to estimate the value of the integral. The running time for MVIB is modestly smaller than that for MVI. However, in solving certain integral equations (e.g., using an iterative procedure to find the percentage point of a statistic), using MVIB usually requires no Monte Carlo sampling after the first iteration, and is considerably more efficient. MVIB and MVI are highly efficient for the evaluation of integrals whose value is large. “Naive” Monte Carlo (MC) may be competitive with MVI or MVIB only if the value of the probability integral is small or the shape of the region is “extreme.”

Key Words:

• Convex region
• Integral equation
• Monte Carlo
• Multiple comparisons
• Multivariate normal integral
• Multivariate-t integral