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426 8. Tests of Hypotheses

test function corresponding to the UMP level α test for deciding between H 0
: µ = µ versus can be written as
0

But, ψ* is also a UMP level a test for deciding between H versus H” . Again,
0
1
the two test functions ψ* and ψ must also then coincide whenever ψ is zero
2
2
or one. Now, we argue as follows.
Suppose that the observed data X is such that the test statistic’s cal-
culated value does not exceed –z . Then, for such X, we
α
must have ψ (x) = 0, ψ (x) = 1. That is, on the part of the sample space
1
2
where the test function ψ*(X) will fail to coincide
with both ψ (x), ψ (x). So, we have arrived at a contradiction. In other
2
1
words, there is no UMP level a test for deciding between H : µ = µ 0
0
against H : µ ≠ µ .
1 0
8.5.2 An Example Where UMP Test Exists
Suppose that X , ..., X are iid Uniform(0, θ) with unknown θ ∈ ℜ . Consider
+
n
1
T(X) = X , the largest order statistic, which is sufficient for θ. We wish to
n:n
test the simple null hypothesis H : θ = θ against the two-sided alternative
0
0
hypothesis H : θ ≠ θ with level α where θ is a fixed positive number. We will
0
0
1
show that there exists a UMP level α test in this situation. This is one of many
celebrated exercises from Lehmann (1986, p. 111).
As a follow-up of the earlier Example 8.4.11, let us first show that any test
function ψ*(X) such that
corresponds to a UMP level a test for deciding between H’ : θ ≤ θ and H’ :
1
0
0
θ > θ .
0
Using the MLR property, from the Example 8.4.11, one may write down
the test function of the UMP level a test for H’ versus H’ as follows:
0 1

Now, for any ψ > ψ , the power function associated with the test function
0