Page 444 - Probability and Statistical Inference

8. Tests of Hypotheses 421

Remark 8.4.1 In this definition, all we need is that L(x; θ*)/L(x; θ) should
be non-increasing or non-decreasing in T(x) whenever θ* > θ. If the likeli-
hood ratio L(x; θ*)/L(x; θ) is non-increasing instead of being non-decreas-
ing, then we shall see later that its primary effect would be felt in placing the
rejection region R in the upper or lower tail of the distribution of the test
statistic under H . One should realize that the statistic T would invariably be
0
sufficient for the unknown parameter θ.
Example 8.4.5 Suppose that X , ..., X are iid N(µ, σ ) with unknown µ ∈
2
n
1
ℜ, but assume that σ ∈ ℜ is known. Consider arbitrary real numbers µ, µ*
+
(> µ), and then with let us write

which is increasing in T. Then, we have the MLR property in T. !
Example 8.4.6 Suppose that X , ..., X are iid with the common pdf f(x;b)
1 n
= (x > 0) with unknown b ∈ ℜ . Consider arbitrary real numbers b,
+
b* (> b), and with let us write

which is increasing in T. Then, we have the MLR property in T. !
Example 8.4.7 Suppose that X , ..., X are iid Uniform(0, θ) with un-
1
n
known θ ∈ ℜ . Consider arbitrary real numbers θ, θ* (> θ), and with T(x) =
+
x , the largest order statistic, let us write
n:n

which is non-decreasing in T(x). Then, we have the MLR property in T. !
Next let us reconsider the real valued sufficient statistic T = T(X) for the
unknown parameter θ, and suppose that its family of pmf or pdf is given by
{g(t; θ): θ ∈ Θ ⊆ ℜ}. Here the domain space for T is indicated by t ∈ T ⊆ ℜ.
Suppose that g(t; θ) belongs to the one-parameter exponential family, given
by the Definition 3.8.1. That is, we can express

where b(θ) is an increasing function of θ. Then, the family {g(t; θ): θ ∈ Θ
⊆ ℜ} has the MLR property in T. Here, a(θ) and b(θ) can not involve t
whereas c(t) can not involve θ. In many distributions involving only a
single real valued unknown parameter θ, including the binomial, Poisson,