Page 281 - Introduction to Statistical Pattern Recognition

6 Nonparametric Density Estimation 263

p(X) accurately, it is futile to seek the more accurate but more complex
expression for the variance. After all, what we can hope for is to get a rough
estimate of I' to be used.
Therefore, using the second order approximation of (6.18) and the first
order approximation of (6.29) for simplicity,

(6.35)

Note that the first and second terms correspond to the variance and squared
bias of p(X), respectively.

Minimization of MSE: Solving 3MSE/& = 0 [5], the resulting optimal
I-, I'*, is

n +2 I
nu-) I" N
2 l1+4 (6.36)
= [ 7~"~(n+2)"'~p '12a2 -7
I
IA
where 1' = CI'" and

(6.37)

The resulting mean-square error is obtained by substituting (6.36) into (6.35).

When the integral mean-square error of (6.33) is computed, 1' and I' are
supposed to be constant, being independent of X. Therefore, from (6.35)

1 1
IMSE = -jp (X)dX + -r4ja2(X)p2(X)dX
Nv 4