Page 445 - Probability and Statistical Inference

422 8. Tests of Hypotheses

normal, and gamma, the family of pmf or pdf of the associated sufficient
statistic T would be MLR in T. We leave out the verifications as exercises.

Some families of distributions may not enjoy the MLR property.
The Exercise 8.4.6 gives an example.

8.4.3 UMP Test via MLR Property

The following useful result is due to Karlin and Rubin (1956). We simply state
the result without giving its proof.
Theorem 8.4.1 (Karlin-Rubin Theorem) Suppose that we wish to
test H : θ = θ versus H : θ > θ . Consider a real valued sufficient statistic
1
0
0
0
T = T(X) for θ ∈ Θ (⊆ ℜ). Suppose that the family {g(t; θ) : θ ∈ Θ} of the
pdf’s induced by T has the MLR (non-decreasing) property. Then, the test
function

corresponds to the UMP level a test if k is so chosen that

If the null hypothesis in the Karlin-Rubin Theorem is replaced by
H : θ ≤ θ , the test given by (8.4.6) continues to be UMP level α.
0 0
Remark 8.4.2 Suppose that the null hypothesis in Theorem 8.4.1 is re-
placed by H : θ ≤ θ . Recall that the level of the test is defined by
0 0
One may, however, note that the maximum Type
I error probability is attained at the boundary point θ = θ because of the MLR
0
(non-decreasing) property in T. If the distribution of T happens to be discrete,
then the theorem continues to hold but it will become necessary to randomize
on the set {T = k} as we did in the Examples 8.3.6-8.3.7. In the Karlin-Rubin
Theorem, if the family {f(x; θ) : θ ∈ Θ} has instead the MLR non-increasing
property in T(X), then the UMP level α test function ψ(X) would instead be 1
or 0 according as T(X) < k or T(X) > k respectively.
Example 8.4.8 (Example 8.4.1 Continued) Suppose that X , ..., X are
1
n
iid N(µ, σ ) with unknown µ ∈ ℜ, but assume that σ ∈ ℜ + is known. Let
2
which is sufficient for µ and its pdf has the MLR increas-
ing property in T. One may use the representation in (8.4.5) and the remark
about the one-parameter exponential family. Now, we wish to test the null
hypothesis H : µ ≤ µ versus H : µ > µ with level α where µ is a fixed
0 0 1 0 0