Numerical and Simulation Studies

Assume that Z_j for $j=1,\ldots ,m$ are independently and normally distributed with mean 0 and variance $(1+\alpha_j^2)/\alpha_j^2$. Let $\tilde{Z} =\sum_{j=1}^m Z_j\alpha_j^2/(1+\alpha_j^2)$. Kwong (1995) showed that the standardized multivariate normal random variables with singular correlation structure given in (3) can be generated by the transformation $X_j=\alpha_j (Z_j-\tilde{Z})$for $j=1,\ldots ,m$, where $\sum_{j=1}^m \alpha_j^2/(1+\alpha_j^2) =1$. Therefore, for any given b_j and $\alpha_j$ for $j=1,\ldots ,m$, we generate all the Z_j and transform each of them to X_j based on (4). Then, we observe whether absolute value of each X_j is less than its corresponding b_j for $j=1,\dots,m$, 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 $m=4,\ldots, 12$ 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.

 

Table 1: Bounds and Estimated Values for MVN Probabilities

bj's			f Values	f Values
	Upper	Lower	Simulated	F Estimate
$\alpha_j^2/(1+\alpha_j^2)$'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)