Improved bounds and simulation procedures on the value of the multivariate normal probability distribution function

Improved bounds and simulation procedures on the value of the multivariate normal probability distribution function value are given in the paper. The author's variance reduction technique was based on the Bonferroni bounds involving the first two binomial moments only. The new variance reduction technique is adapted to the most refined new bounds developed in the last decade for the estimation of the probability of union or intersection of events. Numerical test results prove the efficiency of the simulation procedures described in the paper.