Page 446 - Probability and Statistical Inference

8. Tests of Hypotheses 423

real number. In view of the Karlin-Rubin Theorem and Remark 8.4.2, the
UMP level α test will look like this:

or equivalently it can be written as

One will note that the Type I error probability at the boundary point µ = µ in
0
the null space is exactly α. One may also check directly that the same for any
other µ < µ is smaller than α as follows: Writing Z for a standard normal
0
variable, for µ < µ , we get
0

Since , we can now conclude that P{Z > z +
a
!
Example 8.4.9 Suppose that X , ..., X are iid N(µ, σ ) with known µ ∈ ℜ,
2
n
1
but assume that σ ∈ ℜ is unknown. Let and observe
+
that σ T(X) is distributed as the random variable. Thus, the pdf of T,
-2
which is a sufficient statistic for σ, has the MLR increasing property in T.
One may use (8.4.5) and the remark about the one-parameter exponential
family. We wish to test the null hypothesis H : σ ≤ σ versus H : σ > σ with
1
0
0
0
level α where σ is a fixed positive real number. In view of the Karlin-Rubin
0
Theorem and Remark 8.4.2, the UMP level α test will look like this:
or equivalently it can be written as

where recall that is the upper 100α % point of the distribution. See,
for example, the Figure 8.3.4. One will note that the Type I error probability at
the boundary point σ = σ in the null space is exactly α. One may also check
0
directly that the same for any other σ < σ is smaller than α as follows: For σ
0
< σ , we get
0