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400 8. Tests of Hypotheses
In defining a size α or level α test, what we are doing is quite simple. For
every θ in the null space Θ , we look at the associated Type I error probability
0
which coincides with Q(θ), and then consider the largest among all Type I
error probabilities. may be viewed as the worst possible Type I
error probability. A test is called size α if = α whereas it is
called level α if ≤ α. We may equivalently restrict our attention
to a class of test functions such that or
≤ α. It should be obvious that a size a test is also a level α
test.
It is important to be able to compare all tests for H versus H such that
1
0
each has some common basic property to begin with. In Chapter 7, for ex-
ample, we wanted to find the best estimator among all unbiased estimators of
θ. The common basic property of each estimator was its unbiasedness. Now,
in defining the best test for H versus H , we compare only the level a tests
1
0
among themselves.
Definition 8.2.4 Consider the collection C of all level α tests for H : θ ∈
0
Θ versus H : θ ∈ Θ where A test belonging to C
0
1
1
with its power function Q(θ) is called the best or the uniformly most powerful
(UMP) level α test if and only if Q (θ) ≥ Q* (θ) for all θ ∈ Θ where Q* (θ)
is the power function of any other test belonging to C. If the alternative
hypothesis is simple, that is if Θ is a singleton set, then the best test is called
1
the most powerful (MP) level a test.
Among all level a tests our goal is to find the one whose power hits the
maximum at each point θ ∈ Θ . Such a test would be UMP level α. In the
1
simple null versus simple alternative case first, Neyman and Pearson (1933a,b)
gave an explicit method to determine the MP level α test. This approach is
described in the next section. We end this section with an example.
Example 8.2.4 (Example 8.2.1 Continued) Consider a population with the
pdf N(θ, 1) where θ ∈ ℜ is unknown. Again an experimenter postulates two
hypotheses, H : θ = 5.5 and H : θ = 8. A random sample X = (X , ..., X ) is
1
9
1
0
collected. Let us look at the following tests:
Let us write Q (θ) for the power function of Test #i, i = 1, 2, 3. Using
i
MAPLE, we verified that Q (5.5) = Q (5.5) = Q (5.5) = .049995. In other
1
3
2
words, these are all level a tests where α = .049995. In the Figure 8.2.1,
we have plotted these three power functions. It is clear from this plot that