3.3 Exact expected error rates 6 Procedure 1 additionally needs to compute the mean of the estimated covari- ance matrices of the groups,and to evaluate group densities for each observa- tion corresponding to the pooled covariance matrix in each group. Procedure 2 additionally computes the'difference matrix'M,its first eigenvec- tor v,and the corresponding projections of the group means and covariance matrices,and evaluates the corresponding 1D normal densities in each pro- jected observation. Therefore,in procedures 1 and 2 the preparation'of the density evaluation does not depend on the number of observations,resulting in a much smaller additional preparation time'than the preparation time for naive Monte Carlo for big numbers of observations.Moreover,in procedure 2 also the additional density evaluations are much quicker than the density evaluations in naive Monte Carlo,since they are in 1D. In procedure 1 the exact expected error rates have to be calculated numerically,in general.For the exact expected error rates in procedure 2,however,an analytic formula can be derived,even.This will be done in the next subsection.In section 4 we will demonstrate the differences between procedures 1 and 2 by some examples. 3.3 Exact expected error rates In procedure 2 exact expected error rates have to be calculated for univariate normal projected distributions.In this case a general formula for the exact expected error rate could be given depending on the intersection points of the univariate normal densities corresponding to the projected group means and variances.In order to illustrate the idea,let us discuss the 2 and 3 groups cases.Moreover,let us assume equal a-priori probabilities 1/g for all g groups for simplicity.In the simulations in the following sections,we also will discuss these cases only. In the case of 2 groups let the intersection point of the two normal densities be 12. Then,obviously,the exact expected error rate corresponding to these densities is (cp.figure 1) E(erTorc)=(1-Φ1(x12）+④2(x12)/2) (13) where is the normal distribution with mean to the left of z12,and 2 the distri- bution with mean to the right. In the case of 3 groups let the distribution indices again be chosen so that a lower index indicates a lower mean.We are now interested in the relative location of the intersection points of the 3 densities.The error rate of the leftmost group 1 is determined by the first intersection on the right hand side with one of the densities of the other groups.For the rightmost group 3 the same is true for the densities on the left.The error rate of the middle group 2 is,correspondingly,determined by3.3 Exact expected error rates 6  Procedure 1 additionally needs to compute the mean of the estimated covari￾ance matrices of the groups, and to evaluate group densities for each observa￾tion corresponding to the pooled covariance matrix in each group.  Procedure 2 additionally computes the 'dierence matrix' M, its rst eigenvec￾tor v1, and the corresponding pro jections of the group means and covariance matrices, and evaluates the corresponding 1D normal densities in each pro￾jected observation. Therefore, in procedures 1 and 2 the 'preparation' of the density evaluation does not depend on the number of observations, resulting in a much smaller additional 'preparation time' than the preparation time for naive Monte Carlo for big numbers of observations. Moreover, in procedure 2 also the additional density evaluations are much quicker than the density evaluations in naive Monte Carlo, since they are in 1D. In procedure 1 the exact expected error rates have to be calculated numerically, in general. For the exact expected error rates in procedure 2, however, an analytic formula can be derived, even. This will be done in the next subsection. In section 4 we will demonstrate the dierences between procedures 1 and 2 by some examples. 3.3 Exact expected error rates In procedure 2 exact expected error rates have to be calculated for univariate normal pro jected distributions. In this case a general formula for the exact expected error rate could be given depending on the intersection points of the univariate normal densities corresponding to the pro jected group means and variances. In order to illustrate the idea, let us discuss the 2 and 3 groups cases. Moreover, let us assume equal a-priori probabilities 1=g for all g groups for simplicity. In the simulations in the following sections, we also will discuss these cases only. In the case of 2 groups let the intersection point of the two normal densities be x12. Then, obviously, the exact expected error rate corresponding to these densities is (cp. gure 1) E(errorcv) = ((1 ￾ 1(x12)+2(x12))=2) (13) where 1 is the normal distribution with mean to the left of x12, and 2 the distri￾bution with mean to the right. In the case of 3 groups let the distribution indices again be chosen so that a lower index indicates a lower mean. We are now interested in the relative location of the intersection points of the 3 densities. The error rate of the leftmost group 1 is determined by the rst intersection on the right hand side with one of the densities of the other groups. For the rightmost group 3 the same is true for the densities on the left. The error rate of the middle group 2 is, correspondingly, determined by