Page 440 - Probability and Statistical Inference

8. Tests of Hypotheses 417

In view of the Neyman-Pearson Lemma, we would reject H if and only if
0
L(x; µ )/L(x; µ ) is “large”, that is when X + X > k. This will be the MP level
2
0
1
1
α test when k is chosen so that P {X + X > k} = α. But, under H , the
µ0 1 2 0
statistic X + X is distributed as So, we can rephrase
1 2
the MP level a test as follows:
We leave out the power calculation as Exercise 8.3.17. !
Look at the related Exercises 8.3.16, 8.3.18 and 8.3.21.

8.4 One-Sided Composite Alternative Hypothesis

So far we have developed the Neyman-Pearson methodology to test a simple
null hypothesis versus a simple alternative hypothesis. Next suppose that we
have a simple null hypothesis H : θ = θ , but the alternative hypothesis is H :
1
0
0
θ > θ . Here, H : θ > θ represents an upper-sided composite hypothesis. It is
1
0
0
conceivable that the alternative hypothesis in another situation may be H : θ <
1
θ which is a lower-sided composite hypothesis.
0
We recall the concept of the UMP level a test of H versus H , earlier laid
0
1
out in the Definition 8.2.4. A test with its power function Q(θ) would be
called UMP level a provided that (i) Q(θ ) ≤ α and (ii) Q(θ) is maximized at
0
every point θ satisfying H .
1
In other words, a test would be UMP level a provided that it has level α
and its power Q(θ) is at least as large as the power Q*(θ) of any other level a
test, at every point θ > θ or θ < θ , as the case may be. The following
0
0
subsections lay down a couple of different approaches to derive the UMP
level θ tests.
8.4.1 UMP Test via the Neyman-Pearson Lemma
This method is very simple. We wish to obtain the UMP level θ test for H 0
: θ = θ versus H : θ > θ . We start by fixing an arbitrary value θ ∈ T such
1
0
0
1
that θ > θ . Now, we invoke the Neyman-Pearson Lemma to conclude
0
1
that there exists a MP level a test for deciding between the choices of two
simple hypotheses θ and θ . If this particular test, so determined, hap-
0
1
pens to remain unaffected by the choice of the specific value θ , then by
1
the Definition 8.2.4, we already have on hand the required UMP level