Page 277 - Introduction to Statistical Pattern Recognition

6 Nonparametric Density Estimation 259

w =jl?(Y)dY. (6.15)
Then, $(.)/w becomes a density function. Therefore, (6.14) becomes

(6.16)

where

(6.17)

and r-'B is the covariance matrix of ?(X)/w.

Substituting (6.12) and (6.16) into (6.7) and (6.9), the moments of p(X)
are approximated by

1
E (&X) } 'I- p (X)[1 + ya(X)r2] 2nd order approximation

P (X) 1st order approximation , (6.18)

2nd order approximation

1
E -[wp(X) - p2(X)] 1st order approximation . (6.19)
N

Note that the variance is proportional to 1/N and thus can be reduced by
increasing the sample size. On the other hand, the bias is independent of N,
and is determined by V2p (X), A, and r 2.

Normal kernel: When the kernel function is normal with zero expected
vector and covariance matrix r2A, Nx(0,r2A), $(X) becomes normal as
cNx(0,r2A 12) where c = 2-"'2(2~)-"/2 IA l-"2r-". Therefore,