4.1 Known densities 9 Sim naiveMC procl proc2 varl var2 cor2 minl min2 mvar1 mvar2 1 2.53 0.61 1.65 94% 57% 0.97 0.79 0.61 1.55 94% 62% 2 2.55 2.31 1.96 18% 41% 0.59 0.70 2.06 1.82 35% 49% 2.55 1.93 2.45 43% 8% 0.74 0.43 1.72 2.30 55% 9% Table 1:Monte Carlo standard deviations,variance reductions,and correlations for known densities two groups in the first simulation we use a third group with the follow ing parameters of a normal distribution: 0.3 s=(3,0)'and (21) The true expected error rate is approximately 28.44% The results of naive Monte Carlo and the two control variate procedures are sum- marized in Table 1.Note that 'Sim'indicates the simulation number,'naiveMC'the estimated mean standard deviation of the naive Monte Carlo,'procl'and proc2' the corresponding standard deviations of the control variate procedures,'varl'and var2'the corresponding percentages of variance reduction,'corl'and 'cor2'the mean correlation coefficients,'minl'and 'min2'the corresponding minimal standard devi- ations of the control variate procedures,and'mvarl'and'mvar2'the corresponding maximum percentages of variance reduction. Analysing Table 1 note particularly that for simulation 2 procedure 2 leads to a big- ger variance reduction than procedure 1,but that the maximum variance reduction is nevertheless smaller than for simulation 1 since the univariate projected constel- lation of the groups is more complicated in this example.The bad performance of procedure 1 indicates that in this example the covariance matrices of the different groups can not be assumed to be approximately equal. Third Simulation: Upto now,the examples were mainly one dimensional in that the groups were shifted in the first component only.Since this might lead to an overoptimistic judgement of procedure 2,the third example is the same as the second,but with 3=(1,1) (22) i.e.with a mean shifted in both directions The corresponding Monte Carlo results can also be found in Table 1.Note in partic- ular that now again procedure 1 is very adequate,but procedure 2,unfortunately, does not lead to a substantial variance reduction,and might thus even cause an increase in computer time.The problems of procedure 2 also become clear noting that the exact expected error rate is 45%for this procedure in contrast to a true expected error rate of around 33%.4.1 Known densities 9 Sim naiveMC proc1 proc2 var1 var2 cor1 cor2 min1 min2 mvar1 mvar2 1 2.53 0.61 1.65 94% 57% 0.97 0.79 0.61 1.55 94% 62% 2 2.55 2.31 1.96 18% 41% 0.59 0.70 2.06 1.82 35% 49% 3 2.55 1.93 2.45 43% 8% 0.74 0.43 1.72 2.30 55% 9% Table 1: Monte Carlo standard deviations, variance reductions, and correlations for known densities two groups in the rst simulation we use a third group with the following parameters of a normal distribution: 3 = (3 ;0)0 and 3 =  2 ￾0:3 ￾0:3 2 :4 : (21) The true expected error rate is approximately 28:44%. The results of naive Monte Carlo and the two control variate procedures are sum￾marized in Table 1. Note that 'Sim' indicates the simulation number, 'naiveMC' the estimated mean standard deviation of the naive Monte Carlo, 'proc1' and 'proc2' the corresponding standard deviations of the control variate procedures, 'var1' and 'var2' the corresponding percentages of variance reduction, 'cor1' and 'cor2' the mean correlation coecients, 'min1' and 'min2' the corresponding minimal standard devi￾ations of the control variate procedures, and 'mvar1' and 'mvar2' the corresponding maximum percentages of variance reduction. Analysing Table 1 note particularly that for simulation 2 procedure 2 leads to a big￾ger variance reduction than procedure 1, but that the maximum variance reduction is nevertheless smaller than for simulation 1 since the univariate pro jected constel￾lation of the groups is more complicated in this example. The bad performance of procedure 1 indicates that in this example the covariance matrices of the dierent groups can not be assumed to be approximately equal. Third Simulation: Up to now, the examples were mainly one dimensional in that the groups were shifted in the rst component only. Since this might lead to an overoptimistic judgement of procedure 2, the third example is the same as the second, but with 3 = (1 ;1)0 (22) i.e. with a mean shifted in both directions. The corresponding Monte Carlo results can also be found in Table 1. Note in partic￾ular that now again procedure 1 is very adequate, but procedure 2, unfortunately, does not lead to a substantial variance reduction, and might thus even cause an increase in computer time. The problems of procedure 2 also become clear noting that the exact expected error rate is 45% for this procedure in contrast to a true expected error rate of around 33%