4 Simulations 7 Figure 1:The error rate of the left group is gray shaded. the first intersections points of its density with the other densities on the left and on the right.For simplicity let us now assume that the relevant intersection points are 12,determining the error of group 1 and the 'left part'of the error of group 2, and x23,determining the error of group 3 and the'right part'of the error of group 2.This then leads to the following formula for the exact error rate corresponding to the 3 groups: E(erToreu)=(1-Φ(c12)+(Φ2(x12)+(1-④2(x23)+Φ3(x23)/3) (14) As an example consider 3 groups with group means =-3,u2=-2,and u3 0,and with standard deviations o1 2.037,02=0.9406,ando3 1.These parameters lead to intersection points 12 =-3.17 and x23 =-1,as well as an exact error rate E(erroree)=31.45%. For procedure 1 we only succeeded to find a general analytic formula for the exact expected error rate in the case of 2 groups.Procedure 1 assumes equal covariance matrices for all groups.This covariance matrix is estimated by the pooled covariance matrix∑=（∑1+∑2）/2，where:is the estimated covariance matrix of group i, i=1 2.For normal group distributions with means and u2 and a common covariance matrix one can show (see [6,p.12)that the exact error rate is E(erroTq)=Φ(-0.5d2）where 012=V(u2-h)T∑-1(u2-】 (15)） and is the distribution function of the standard normal. 4 Simulations 4.1 Known densities In this subsection we assume that the group densities are fully known so that pa- rameter estimation is superfluous.This means in particular that QDA as well as LDA is carried out with the correct densities.In this way the outcome does not depend on the goodness of parameter estimation.In the next subsection,we will discuss the case when density parameters have to be estimated. In all examples sample size N=100 is used for each group.Also,W =100 is used.In order to be independent of the drawn random vectors,this experiment was repeated V=100 times and the means of the meanerror rates and the corresponding4 Simulations 7 Figure 1: The error rate of the left group is gray shaded. the rst intersections points of its density with the other densities on the left and on the right. For simplicity let us now assume that the relevant intersection points are x12, determining the error of group 1 and the 'left part' of the error of group 2, and x23, determining the error of group 3 and the 'right part' of the error of group 2. This then leads to the following formula for the exact error rate corresponding to the 3 groups: E(errorcv) = ((1 ￾ 1(x12)) + (2(x12) + (1 ￾ 2(x23)) + 3(x23))=3) (14) As an example consider 3 groups with group means 1 = ￾3, 2 = ￾2, and 3 = 0, and with standard deviations 1 = 2 :037,2 = 0 :9406, and3 = 1. These parameters lead to intersection points x12 = ￾3:17 and x23 = ￾1, as well as an exact error rate E(errorcv) = 31 :45%. For procedure 1 we only succeeded to nd a general analytic formula for the exact expected error rate in the case of 2 groups. Procedure 1 assumes equal covariance matrices for all groups. This covariance matrix is estimated by the pooled covariance matrix =( ^ ^ 1 + ^ 2)=2 , where ^ i is the estimated covariance matrix of group i, i = 1 2. For normal group distributions with means ; 1 and 2 and a common covariance matrix  one can show (see [6], p. 12) that the exact error rate is E(errorcv) = ( ￾0:512) where 12 = p (2 ￾ 1)T ￾1(2 ￾ 1) (15) and  is the distribution function of the standard normal. 4 Simulations 4.1 Known densities In this subsection we assume that the group densities are fully known so that pa￾rameter estimation is super uous. This means in particular that QDA as well as LDA is carried out with the correct densities. In this way the outcome does not depend on the goodness of parameter estimation. In the next subsection, we will discuss the case when density parameters have to be estimated. In all examples sample size N = 100 is used for each group. Also, W = 100 is used. In order to be independent of the drawn random vectors, this experiment was repeated V = 100 times and the means of the mean error rates and the corresponding