Page 543 - Probability and Statistical Inference

520 11. Likehood Ratio and Other Tests

On the other hand, one has

Now, we combine (11.3.16)-(11.3.17) to express the likelihood ratio as

where a ≡ a(n , n ), b = b(n , n ) are positive numbers which depend on n ,
1
2
1
1
2
n only. Now, one rejects H if and only if ∧ is small. Thus, we decide as
2
0
follows:
or equivalentl y,

where k(> 0) is a generic constant.
In order to express the LR test in an implementable form, we proceed as
follows: Consider the function g(u) = u n1/2 (u + b) –n1+n2)/2 for u > 0 and inves-
tigate its behavior in order to check when it is small (< k). Note that g′(u) =
½u( n–2)/2 (u +b) –(n1+n2+2)/2 {n b – n u} which is positive (negative) when u <
1
2
(>)n b/n . Hence, g(u) is strictly increasing (decreasing) on the left (right)
1
2
hand side of u = n b/n . Thus, g(u) is going to be “small” for both very small
1
2
or very large values of u(> 0).
Next, we rewrite the LR test (11.3.19) as follows: