Page 201 - Introduction to Statistical Pattern Recognition
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5 Parameter Estimation 183
If the estimates are unbiased,
E(AY) = 0
and subsequently the expecred value of T is
Similarly, the variance of T can be derived as
I-
where the approximation from the first line to the second line is made by dis-
carding terms higher than second order.
Equation (5.3) shows that f is a biased estimate in general and that the
bias depends on a2 flay2 and E { AYAYT ), where a2 flay2 is determined by the
functional form off and E(AYAYT) is determined by the distribution of ?,
,.
,.
p(Y), and the number of samples, N, used to compute Y. Likewise, the vari-
ance depends on af/aY and E 1 AYAYT ).
,.
Estimation of f: For many estimates, the effects of p(Y) and N on
E (AYAYT) can be separated as
E~AYAY~) g(~) K@(?)), (5.5)
=
,.
where the scalar g and the matrix K are functions determined by how Y is
computed. Substituting (5.5) into (5.3),
