Page 447 - Probability and Statistical Inference

424 8. Tests of Hypotheses

Since we claim that
Example 8.4.10 Suppose that X , ..., X are iid with the common pdf
1 n
f(x; λ) = λe I (x > 0) with unknown λ ∈ ℜ . Let us consider
-?x
+
which is sufficient for λ and its pdf has the MLR decreasing property in T. We
wish to test the null hypothesis H : λ ≤ λ versus H : λ > λ with level α
0
1
0
0
where λ is a fixed positive real number. In view of the Karlin-Rubin Theorem
0
and Remark 8.4.2, the UMP level a test will look like this:
or equivalently it can be written as

where recall that is the lower 100α% point of the distribution.
See, for example, the Figure 8.4.1. One will note that the Type I error prob-
ability at the boundary point λ = λ in the null space is exactly α. One may also
0
check directly, as in the two previous examples, that the same for any other
λ < λ is smaller than α. Note that for small values of we are rejecting
0
the null hypothesis H : λ ≤ λ in favor of the alternative hypothesis H :
0 0 1
λ > λ . This is due to the fact that we have the MLR decreasing property in T
0
instead of the increasing property. Also, observe that E (X) = λ and thus
-1
λ
“small” values of would be associated with “large” values of ? which
should lead to the rejection of H . Recall Remark 8.4.2 in this context. !
0
Example 8.4.11 Suppose that X , ..., X are iid Uniform(0, θ) with un-
1
n
known θ ∈ ℜ . Let us consider T(X) = X which is sufficient for θ and its
+
n:n
pdf has the MLR increasing property in T. We wish to test the null hypothesis
H : θ ≤ θ versus H : θ > θ with level α where θ is a fixed positive real
1
0
0
0
0
number. In view of the Karlin-Rubin Theorem and Remark 8.4.2, the UMP
level a test will look like this:
Let us now choose k so that the Type I error probability at the boundary point
n-1
-n
θ = θ in the null space is exactly α. The pdf of T is given by nt θ I (0 < t
0
< θ) when θ is the true value. So, we proceed as follows.