4.1 Known densities ⊙ standard deviations as well as the correlation coefficients will be reported in what follow s. First Simulation: First we compare procedures 1 and 2 using two groups with the following parameters of normal distributions: h=(0,0)'and (16) as well as 2=(2,0)'and 22 (17) The true expected error is approximately 14.97%calculated by exact integration to be able to judge the following results. By means of the naive Monte Carlo estimator we obtain Euc(error)=(15.00±2.53)% (18) where 2.53%is the estimated standard deviation of Evc(error).Obviously,the bias is negligible. With procedure 2 one obtains E(error-errorcu)=(-0.92+1.65)% (19) and E(errore)equals 15.87%(exact integration).Summing up for the right hand side of equation(5)we arrive at(14.95+1.65)%.This expression shows a distinctly lower variance than(18).The mean estimated correlation coefficient ist o=0.79.The lowest standard deviation we can get by (8)is therefore 1.55%.This corresponds to a variance reduction of more than 60%in relation to the naive Monte Carlo. Moreover,with procedure 1 one obtains E(error-errorc)=(-0.02+0.61)% (20) and E(errorco)equals 15.08%(exact integration).Summing up for the right hand side of equation (5)we arrive at (15.0610.61)%.This expression shows an even much lower variance than with procedure 2.This indicates that LDA is a very adequate method for this example.Indeed,the mean estimated correlation coefficient is o= 0.97. Second Simulation: Now we compare procedures 1 and 2 by an example with three different groups which do not separate that nicely as in the previous simulation.In addition to the4.1 Known densities 8 standard deviations as well as the correlation coecients will be reported in what follows. First Simulation: First we compare procedures 1 and 2 using two groups with the following parameters of normal distributions: 1 = (0 ;0)0 and 1 =  1 0 0 1 (16) as well as 2 = (2 ;0)0 and 2 =  1 0 :5 0:5 1  : (17) The true expected error is approximately 14:97% calculated by exact integration to be able to judge the following results. By means of the naive Monte Carlo estimator we obtain E^ MC (error) = (15:00  2:53)% (18) where 2:53% is the estimated standard deviation of E^ MC (error). Obviously, the bias is negligible. With procedure 2 one obtains E( ^ error ￾ errorcv)=( ￾0:92 1:65)% (19) and E(errorcv) equals 15:87% (exact integration). Summing up for the right hand side of equation (5) we arrive at (14:95  1:65)%. This expression shows a distinctly lower variance than (18).The mean estimated correlation coecient ist % = 0 :79. The lowest standard deviation we can get by (8) is therefore 1:55%. This corresponds to a variance reduction of more than 60% in relation to the naive Monte Carlo. Moreover, with procedure 1 one obtains E( ^ error ￾ errorcv)=( ￾0:02 0:61)% (20) and E(errorcv) equals 15:08% (exact integration). Summing up for the right hand side of equation (5) we arrive at (15:060:61)%. This expression shows an even much lower variance than with procedure 2. This indicates that LDA is a very adequate method for this example. Indeed, the mean estimated correlation coecient is % = 0:97. Second Simulation: Now we compare procedures 1 and 2 by an example with three dierent groups which do not separate that nicely as in the previous simulation. In addition to the