Page 245 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 221

Proof of g 2 0: The perturbation term, g, of (5.130) is always positive
-2
no matter what N,, d, (Xt)), and n are. The proof is given as follows.
Assuming N, > 2, I& I of (5.127) should be positive because ilk is a
sample covariance matrix and should be a positive definite matrix. Therefore,
N, A2
1- d; (Xi!)) > 0 (5.131)
(N, -
On the other hand, from (5.130)
A2
ag =1 (N:-3N, + l)/(N, - 1 )+2N, d,
-
-2
32; (N,-I)’ - N,d,
1 [ (N;-3N,+ 1 )if/(N, - 1 )+N, i:]N,
+-
2 [(N,-l)’ - N,2f12

1
+- -Nj/(Nj-1)’
-2
l-[Nj/(Nj-1)’]d;
-4 -2
1 -N’d; +N; (2N:-3N, +2)dj -(Nj- 1)(2N;- 1 )
- (5.132)
--
2 [(N;-l)’ - NI$l2
A2
The term &lad; is equal to zero when
A2 1 3N: - 3Nj! + Ni
dj = - or (5.133)
Ni N;
The second solution of (5.133) does not satisfy the condition of (5.131). Since
-2 -2
g and aglad; for dj = 0 are positive and negative, respectively, the first solu-
tion of (5.133) gives the minimum g, which is

1 (Nj!-3Nj+I)/(N;-I)+l n Nj-1 1 1
- +-In- + - In[] - 1
2 Nj[(Ni-1)2-1] 2 Nj-2 2 ~ (N;- 1 )2
-1
N,-1
n-1
1
-_ 1 +-In[ (N1-1Y-l N,-1 + - N,-2
2
In-
-
1
2 N,(N,-l) 2 (N,-l)’ N,-2