Page 261 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 243

E*

EX

(5.174)

where Ex indicates the expectation with respect to the w's.
AX ,*
f(X, M, ,X, can be expanded around f(X,k,,i,) by the Taylor series as
)

A* A
where AM; = My - k;, A$ = Z; -Xi. and df*/&; is defined in (5.141).
Since h(~) =f(~,M~,i,) -f(~,ii*,i*),

Ah(X) = h*(X) - h(X)

(5.176)

The partial derivatives of (5.176) can be obtained by (5.144) and (5.145).

Bootstrap Expectation

Using the approximation of (5.60), (5.169) can be approximated as

(5.177)

The third term is a third order variation with the combination of AwY) and
-2
Ah , and can be ignored. Thus, our analysis will focus on the first and second
terms only. With this in mind, substituting (5.171), (5.172), and (5.176) into
(5.177) produces