490 Optimal Tests of Hypotheses
The solutions forθ 1 ,θ 2 ,andθ 3 are, respectively,
u 1 = n−^1
∑n
1
xi
u 2 = m−^1
∑m
1
yi
w′ =(n+m)−^1
[n
∑
1
(xi−u 1 )^2 +
∑m
1
(yi−u 2 )^2
]
,
and, further,u 1 ,u 2 ,andw′maximizeL(Ω). The maximum is
L(Ω) =ˆ
(
e−^1
2 πw′
)(n+m)/ 2
,
so that
Λ(x 1 ,...,xn,y 1 ,...,ym)=Λ=
L(ˆω)
L(Ω)ˆ
=
(
w′
w
)(n+m)/ 2
.
The random variable defined by Λ^2 /(n+m)is
∑n
1
(Xi−X)^2 +
∑m
1
(Yi−Y)^2
∑n
1
{Xi−[(nX+mY)/(n+m)]}^2 +
∑n
1
{Yi−[(nX+mY)/(n+m)]}^2
.
Now
∑n
1
(
Xi−
nX+mY
n+m
) 2
=
∑n
1
[
(Xi−X)+
(
X−
nX+mY
n+m
)] 2
=
∑n
1
(Xi−X)^2 +n
(
X−
nX+mY
n+m
) 2
and
∑m
1
(
Yi−
nX+mY
n+m
) 2
=
∑m
1
[
(Yi−Y)+
(
Y−
nX+mY
n+m
)] 2
=
∑m
1
(Yi−Y)^2 +m
(
Y−
nX+mY
n+m
) 2
.
But
n
(
X−
nX+mY
n+m
) 2
=
m^2 n
(n+m)^2
(X−Y)^2