Page 251 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 233

A
-
A
,.
Therefore, the statistical properties of the bias, &h = E~-&~, can be studied.
The expected value of &h is
(5.152)

where

(5.153)

A
And, the variance of &h is

Example 3: The explicit expression for p, of (5.153) can be obtained by
using the same technique used to compute (5.81), if two distributions are nor-
mal with MI 0, M2 = M, and C, = Z2 = I, and the quadratic classifier of
=
(5.54) is used. For N = N2 = N
I
&e
.
)
Ed{ c’~”‘~) l””‘x’[l+jwEd( Ah(X)+&Ah2(X) )]Ee/o”‘X’ (5.155)
2
The last approximation was made because E,, { Ah + jwAh2/2 ] is proportional
to 1/N. Then, e/w”(x’p,(X) = 1,2) are given in (5.78) and (5.79). Thus, the
(i
integration of (5.153) merely involves computing the moments of the normal
distributions of (5.78) and (5.79), resulting in

Ir, 1 .-.X’-. M’M 117 +{ (M7M)2 MTM I].
---I
2
2dGzG 16 2
(5.156)
Note that PI of (5.156) is exactly twice v, of (5.81). That is, the bias between