Page 351 - Introduction to Statistical Pattern Recognition

7 Nonparametric Classification and Error Estimation 333

Let us assume e= 1 for example. The test sample, Xi!), was used
II
to compute p (.) as in (7.57), but never used for p2(.). Therefore,
the removal of Xi') does not change ;2(.) which is the case of the
first line of (7.58). The removal of Xi1), however, affects in
two different ways, depending on whether (.) is evaluated at Xj')
or Xy. il(Xj") is the summation of NI-1 kernels excluding
as
K~(X~')-X~')) seen in the first line of (7.57). Therefore, the
is
further removal of K~(X~')-X~')) can be computed by the second
line of (7.58). On the other hand, since i I (X12)) is the summation
of NI kernels as in the second line of (7.57), the removal of
K~(X:~)-XZI)) can be computed by the third line of (7.58). The
case with E = 2 may be discussed similarly.

(b) Calculate the likelihood ratio estimates at all samples Xy) # Xf)
based on the modified density estimates.

(7.59)

(c) Find the value oft which minimizes the error among the NI+N2-I
samples (without including Xf)), under the decision rule

(7.60)

This is best accomplished by first sorting the likelihood ratio esti-