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13. DISCRIMINATION OF TWO POPULATIONS WITH COMMON UNKNOWN
COVARIANCE MATRIX. G13 -ANDERSON-FISHER STATISTICS ESTIMATOR
Let ~xi , ~yj ; i = 1, ..., n1 ; j = 1, ..., n2 be independent observations of m-dimensional
√
√
random vectors ξ~ = ~a1 + R~
µ, µ
~ T = {µi , i = 1, . . . , m} and ~η = ~a2 + R~ν , ~ν T =
{νi , i = 1, . . . , m} respectively and suppose that random variables µi , νi , i = 1, . . . , m
are independent for every n.
For the observation classification in the case of two Normal populations, the so-called
discriminant function is used:
U (~x) =
½
¾T
1
~x − (~
α1 + α
~ 2)
R−1 (~
α1 − α
~ 2 ) , ~xT = (x1 , · · · , xm ) .
2
We use empirical mean value vectors and the covariance matrix R,
ˆ1 = n−1
~a
1
n1
X
~xi ,
ˆ2 = n−1
~a
2
n2
X
1
R̂ =
n1 + n2 − 2
~yi ,
i=1
i=1
(n
)
n2 ³
1 ³
´³
´T X
´³
´T
X
ˆ1 ~xi − ~a
ˆ1
ˆ2 ~yi − ~a
ˆ2
~xi − ~a
+
~yi − ~a
.
i=1
i=1
We shall refer to the expression
½
´T
³
´¾ n + n − 2 − m
1 ³ˆ
1
2
−1 ˆ
ˆ
ˆ
G13 (~x) = ~x −
~a1 + ~a2 R̂
~a1 − ~a2
2
n1 + n2 − 2
as the G13 (~x) -estimator of the discriminant function.
Theorem 13.1. [Gir54, p.611] If in addition to the conditions of Theorem 11.1
lim
n1 ,n2 →∞
£
¤
T
−1
(~a1 + ~a2 ) R−1 (~a1 + ~a2 ) n−1
= 0,
1 + n2
then
p lim {G13 (νi ) − U (νi )} = 0,
n→∞
where ~νi is an observation of vector ξ~ or ~η which does not depend on ~xi , ~yj ; i = 1, ..., n1 ;
j = 1, ..., n2 and distributed as N {~a1 , R} or N {~a2 , R} .
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