On the performance of a discriminant function

In two-class discriminant problems, objects are allocated to one of the two classes by means of threshold rules based on discriminant functions. In this paper we propose to examine the quality of a discriminant functiong in terms of its performance curve. This curve is the plot of the two misclassification probabilities as the thresholdt assumes various real values. The role of such performance curves in evaluating and ordering discriminant functions and solving discriminant problems is presented. In particular, it is shown that: (i) the convexity of such a curve is a sufficient condition for optimal use of the information contained in the data reduced byg, and (ii)g with non-convex performance curve should be corrected by an explicitly obtained transformation.