Page 259 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 24 1

density function of mi by collection of Ni impulses which are located at the
existing sample points, X?), . . . ,X$l. That is,

(5.163)

where * indicates something related to the bootstrap operation. In the bootstrap
operation, the density function of (5.163) is treated as the true density from
which samples are generated. Therefore, in this section, Xy) is considered a
given fixed vector and is not random as it was in the previous sections.
When samples are drawn from p:(X) randomly, we select only the exist-
ing sample points with random frequencies. Thus, the N, samples drawn from
pr(X) form a random density function

Within each class, the wy)’s are identically distributed under the condition
E::, wj‘) = 1. Their statistical properties are known as

1
,’
E{w‘”] = - (5.165)
,
Ni
(5.166)

E{AwY)Aw(”] = 0 fori # k , (5.167)

where Awj‘) = w$’-l/N,.
..x
The H error in the bootstrap procedure, E~,, is obtained by generating
AX
..-
samples, designing a classifier based on p, (X), and testing p;(X) of (5.163).
A*
On the other hand, the R error, E~, computed by testing p, (X). The bias
is
between them can be expressed by