# Numerical and Simulation Studies

Assume that Zj for are independently and normally distributed with mean 0 and variance . Let . Kwong (1995) showed that the standardized multivariate normal random variables with singular correlation structure given in (3) can be generated by the transformation for , where . Therefore, for any given bj and for , we generate all the Zj and transform each of them to Xj based on (4). Then, we observe whether absolute value of each Xj is less than its corresponding bj for , respectively. The process is repeated N times, and the nominal probabilities from the simulation and a standard error are calculated. Those simulated probabilities are compared with two bounds obtained numerically according to Section 2.1, and with the numerical evaluation of the F integrals described in Section 2.2. Randomized lattice rules (see Cranley and Patterson, 1976), were used for the numerical integration of F, and the absolute accuracy requested was 0.001. For this method, the amount of work required was measured as the number N of integrand values (f values) required to estimate F with error less than 0.001. The error estimates used for the randomized lattice rules were three times the standard errors for these randomized rules. In order to compare these values with values from the simulation method, we used the same Nfor the simulation method, and report an error estimate for the simulation that is three times the standard error for the simulation method.

Some selected cases for are presented in Table 1. It is obvious that the differences among the two bounds are negligible in all the considered cases. However, the computational time of evaluating the bounds increases rapidly as m increases. It is impractical to compute the bounds for m> 12. The computational time of new approach described in Section 2.2 also increases with m, but the estimate values of all the cases considered in this study were obtained in a short period of computational time. The error estimates for the simulation method, using the same number of function values, were in all cases significantly larger than the error estimates for the new method. The new method can also be applied to the multivariate normal distributions with any arbitrary singular correlation structures. Therefore, we conclude that the proposed approach provides an efficient and accurate way to estimate the F integrals with any singular correlation matrices.

 bj's f Values f Values Upper Lower Simulated F Estimate 's Bound Bound Error Est. Error Est. (2.3, 2.2, 2.1, 2.0) 4224 4224 .887369 .887310 .888968 .887541 (.2, .1, .4, .3) .014504 .000374 (.5, 2.4, 1.0, 2.0, 1.6) 496 496 .232658 .232373 .286290 .232567 (.1, .2, .2, .2, .3) .060951 .000642 (2.2, 2.4, 2.5, 2.0, 2.1) 6992 6992 .880775 .880773 .879720 .878440 (.3, .1, .05, .5, .05) .011671 .000682 (2.4, .5, 1.2, .4, 1.9, 2.0) 496 496 .089252 .089103 .066532 .089192 (.1, .1, .2, .2, .2, .2) .033603 .000479 (1.6, 1.7, 1.8, 1.4, 2.1, 2.5, 6692 6692 1.6) .554366 .554366 .560212 .554429 (.1, .1, .2, .2, .2, .1, .1) .017809 .000979 (2.0, 2.1, 1.9, 1.8, 2.0, 2.1, 6692 6692 2.2, 2.3) .714231 .714231 .698227 .713891 (.1, .1, .1, .1, .15, .05, .016470 .000985 .2, .2) (.4, 2.2, 2.5, 3.1, .9, 1.8, 496 496 .8, 2.3, 2.9) .102861 .102861 .098790 .102832 (.01, .02, .07, .1, .15, .05, .040234 .000269 .3, .2, .1) (2.8, 2.9, 2.8, 2.7, 2.4, 3.3, 1248 1248 3.4, 2.5, 2.6, 2.7) .935023 .935023 .927885 .934968 (.1, .05, .05, .04, .06, .1, .021976 .000658 .15, .15, .1, .2) (3.0, 2.8, 2.4, 2.5, 1.9, 2.2, 6992 6992 2.1, 2.0, 2.4, .9, 1.8) .475903 .475903 .496281 .475583 (.02, .08, .04, .06, .1, .1, .017939 .000827 .16, .14, .15, .1, .05) (2.5, 2.7, 3.4, .9, 2.4, 1.7, 496 496 1.8, 2.3, 2.4, 2.6, .9, .8) .185877 .185877 .181452 .185936 (.01, .03, .06, .05, .05, .1, .051966 .000689 .15, .05, .1, .14, .16, .1)

Alan C Genz
2001-02-09