Page 290 - Introduction to Statistical Pattern Recognition

272 Introduction to Statistical Pattern Recognition

T(k-m)T(N +1)
E{u-"] = for k-m > 0 (6.85)
T( k)T(N + 1 -m )
Therefore,
N(N-1)
E(u-1) = - and E(u-~} = (6.86)
k-1 (k - 1 )(k -2)
and

r(k-l+&)r(N+l) N T(k-1+6) T(N)
E{&'} = - (6.87)
T(k)T(N +6) k-I T(k-1) T(N+6) '

62 - T(k-2+6)T(N+l) - N(N-1) T(k-2+6) T(N-1) (6.88)
-
E'u I-- T(k)T(N-1+6) (k-l)(k-2) T(k-2) T(N-1+6)
where T(x+l) = xT(x) is used. It is known that

(6.89)

is a good approximation for large x and small 6. Therefore, applying this
approximation,
k-1 N k-2
E(u61} E(-)&' and E(&'] z -(-)&I (6.90)
N k-1 N-2
Combining (6.83), (6.84), (6.86), and (6.90),
k-1 1 k-1
E{fj(X)) = -E(v-' 1 E ~(X)[1+-~(X)(cp(X))-2"'(-)2i'~l
N 2 N

(6.91)

k-1 k-2
+ cL(cp)-2"-)(-)2",-1
N N-1