Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

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