Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

8.3. Likelihood Ratio Tests 489

independent random variables having the likelihood functions

L(ω)=

(

1

2 πθ 3

)(n+m)/ 2

exp

{

−

1

2 θ 3

[n

∑

1

(xi−θ 1 )^2 +

∑m

1

(yi−θ 1 )^2

]}

and

L(Ω) =

(

1

2 πθ 3

)(n+m)/ 2

exp

{

−

1

2 θ 3

[n

∑

1

(xi−θ 1 )^2 +

∑m

1

(yi−θ 2 )^2

]}

.

If∂logL(ω)/∂θ 1 and∂logL(ω)/∂θ 3 are equated to zero, then (Exercise 8.3.2)

∑n

1

(xi−θ 1 )+

∑m

1

(yi−θ 1 )=0

1

θ 3

[n

∑

1

(xi−θ 1 )^2 +

∑m

1

(yi−θ 1 )^2

]

= n+m. (8.3.2)

The solutions forθ 1 andθ 3 are, respectively,

u =(n+m)−^1

{n

∑

1

xi+

∑m

1

yi

}

w =(n+m)−^1

{n

∑

1

(xi−u)^2 +

∑m

1

(yi−u)^2

}

.

Further,uandwmaximizeL(ω). The maximum is

L(ˆω)=

(

e−^1

2 πw

)(n+m)/ 2

.

In a like manner, if

∂logL(Ω)

∂θ 1

,

∂logL(Ω)

∂θ 2

,

∂logL(Ω)

∂θ 3

are equated to zero, then (Exercise 8.3.3)

∑n

1

(xi−θ 1 )=0

∑m

1

(yi−θ 2 ) = 0 (8.3.3)

−(n+m)+

1

θ 3

[n

∑

1

(xi−θ 1 )^2 +

∑m

1

(yi−θ 2 )^2

]

=0.