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8. Tests of Hypotheses 425

1/n
so that k = (1 – α) θ . One may also check directly that the Type I error
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probability for any other θ < θ is smaller than α. !
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Sample size determination is important in practice when we wish to
control both Type I and II error probabilities. See Exercise 8.4.25.

8.5 Simple Null Versus Two-Sided Alternative

8.5 Hypotheses

Consider testing a simple null hypothesis H : θ = θ versus a two-sided alter-
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native hypothesis H : θ ≠ θ where θ is a fixed value in the parameter space
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0
0
Θ. Will there exist a UMP level α test? The answer is “yes” in some situations
and “no” in some others. We refrain from going into general discussions of
what may or may not happen when the alternative hypothesis is two-sided.
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In the case of the N(µ, σ ) distribution with µ unknown but σ known, a
UMP level α test fails to exist for deciding between a simple null hypothesis
H : µ = µ and a two-sided alternative hypothesis H : µ ≠ µ . On the other
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0
0
0
hand, for the Uniform(0, θ) distribution with θ unknown, a UMP level θ test
exists for deciding between a simple null hypothesis H : θ = θ and a two-
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sided alternative hypothesis H : θ ≠ θ . In the next two subsections, we
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provide the details.
8.5.1 An Example Where UMP Test Does Not Exist
Suppose that X , ..., X are iid N(µ, σ ) with unknown µ ∈ ℜ but known σ ∈
2
n
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ℜ . Consider the statistic which is sufficient for µ. We wish
+
to test the simple null hypothesis H : µ = µ against the two-sided alternative
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0
H : µ ≠ µ with level α where µ is a fixed real number. We wish to show that
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there exists no UMP level α test in this situation.
Suppose that there is a UMP level α test and let its test function be denoted
by ψ* (X). Observe that ψ* (X) is then a UMP level α test for deciding
between H : µ = µ versus H : µ > µ . In the Example 8.4.1, however, the
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0
0
UMP level a for deciding between H versus H’ was written as
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where z is the upper 100α% point of the standard normal distribution. See,
α
for example, the Figure 8.3.1. The two test functions ψ* and ψ must coin-
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cide on the sets where ψ is zero or one. One can similarly show that the
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