Page 536 - Probability and Statistical Inference

11. Likehood Ratio and Other Tests 513

θ = (µ, σ ), Θ = {(µ, ) : µ ∈ ℜ, σ is fixed} and Θ = {(µ, σ ) : µ ∈ ℜ, σ ∈
2
2
0 0
ℜ }.
+
The likelihood function is again given by

Now, observe that

On the other hand, one has
from (11.2.11) where Now, we combine this with
(11.2.17) to obtain the likelihood ratio

Now, one rejects H if and only if ? is small. Thus, we decide as follows:
0

where k(> 0) is a generic constant.

Figure 11.2.3. Plot of the g(u) Function

In order to express the LR test in an implementable form, we proceed
as follows: Consider the function g(u) = ue for u > 0 and investigate
1-u