Page 532 - Probability and Statistical Inference

11. Likehood Ratio and Other Tests 509

a specified population variance against the two-sided alternative hypothesis,
and come up with the customary two-sided ? -test assuming that the popula-
2
tion mean is unknown.

11.2.1 LR Test for the Mean
Suppose that X , ..., X are iid observations from the N(µ, σ ) population
2
n
1
where µ ∈ ℜ, σ ∈ ℜ . We assume that µ is unknown. Given α ∈ (0,1),
+
consider choosing between a null hypothesis H : µ = µ and a two-sided
0
0
alternative hypothesis H : µ ≠ µ with level α where µ is a fixed real number.
1
0
0
We address the cases involving known σ or unknown σ separately. As usual,
respectively denote the
sample mean and variance.
Variance Known
Since σ is known, we have θ = µ, Θ = {µ } and Θ = ℜ. In this case, H 0
0
0
is a simple null hypothesis. The likelihood function is given by
Observe that

since Θ has the single element µ . On the other hand, one has
0 0

Note that for any real number c, we can write

and hence by combining (11.2.2)-(11.2.3) we obtain the likelihood ratio

Intuitively speaking, one may be inclined to reject H if and only if
0
L(µ) is small or is large. But for some data, and