Page 205 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 187

2-51. Since N is large in general, we terminate the expansion at the second
order throughout this book.

Substituting (5.10) through (5.15) into (5.3), the bias term of the esti-
mate, E { Af} = E { f} - f, becomes

(5.18)

Note that the effect of N is successfully separated, and that g(N) of (5.5)
becomes l/N. This is true for any functional form off, provided f is a function
of the expected vectors and covariance matrices of two normal distributions.
Similarly, the variance can be computed from (5.4), resulting in
r

(5.19)

Note that, in order to calculate the bias and variance, we only need to compute
af/amj’), df/acy, d2f/drnj’)*, and a2f/acj;)2 for I’ = 1,2.

Non-normal cases: Even when the distributions of X are not normal,
(5.11), (5.12), and (5.13) are valid as the first and second order moments.