Numerical Computation Of Multivariate Normal Probabilities

Normal distribution, as the most common distribution in nature and science, hasbeen widely used in various fields of engineering technology. Nowadays, the theoryand numerical calculation algorithms of normal probabilities integration have beenmature. In recent years, due to the theoretical and practical demands, calculatingintegration of multivariate normal probabilities is becoming another hot researchdirection. However, calculation amount of traditional numerical algorithms alwaysincreases by geometric times. High dimensions make it impossible to solve theproblem using the usual algorithms. Monte Carlo and Quasi-Monte Carlo methods,which have been applied in solving high-dimensional cases, arrive people’s concern.Firstly, article introduces current research status for computing multivariatenormal probabilities, and then presents two problems: conversion of multivariatenormal probabilities and the algorithm to calculate it. Three different formats areintroduced for conversion and Genz’s method is determined in this paper. For methods,article proposes Monte Carlo and Quasi-Monte Carlo methods in detail. Simulationresults shows that the later method is superior to the first one both in time and accuracy.In the end, this paper proposes improvements in two aspects against existingQuasi-Monte Carlo algorithms. On one hand, when the correlation coefficient is verylarge, then the covariance matrix becomes nearly singular, and the result will be gotwith low accuracy. Through sorting integration regions effectively, accuracy can beraised to a certain extent; On the other hand, when we select the determined points, inaddition to the identified points, their antithetic points in this space will also beselected, which ensures the reliability of the results.