ABSTRACT
As indicated earlier if N1=N2, then the classification rule R=(R1, R2) where ?>0 is the region R1, ?<0 is the region R2, has equal probabilities of misclassification. Various attempts have been made to evaluate these two probabilities of misclassification for the rule R in the general case N1?N2. This classification rule is sometimes referred to as Anderson’s rule in literature. As pointed out earlier in this section, the distribution of V, though known, is too complicated to be of any practical help in evaluating these probabilities. Let
We shall now discuss several methods for estimating P1, P2. Let us recall that when the parameters are known these probabilities are given by [taking k=0 in (9.23)]
Method 1 This method uses the sample observations a=1,…, N1, from p1, a=1,…, N2, from p2, used to estimate the unknown parameters, to assess the
performance of R based on V. Each of these N1+N2 observations a=1,…, Ni, is substituted in V and the proportions of misclassified observations from among these, using the rule R, are noted. These proportions are taken as the estimates of P1, P2. This method, which is sometimes called the resubstitution method, was suggested by Smith (1947). It is obviously very crude and often gives estimates of P1 and P2 that are too optimistic, as the same observations are used to compute the value of V and also to evaluate its performance.
