Page 460 - Probability and Statistical Inference

8. Tests of Hypotheses 437

α ∈ (0, 1), derive the UMP level a test for H : b ≤ b versus H : b > b where
1
0
0
0
b is a positive real number, in the simplest implementable form.
0
8.4.20 (Exercise 8.3.19 Continued) Suppose that X , ..., X are iid positive
n
1
random variables having the common lognormal pdf
with x > 0, –∞ < µ < ∞, 0 < σ < ∞. Here, µ is the only unknown parameter.
Given the value α ∈ (0, 1), derive the UMP level α test for deciding between
the null hypothesis H : µ = µ and the alternative hypothesis H : µ > µ 0
0
1
0
where µ is a fixed real number. Describe the test in its simplest implementable
0
form.
8.4.21 (Exercise 8.4.20 Continued) Let us denote the lognormal pdf

with w > 0, –∞ < µ < ∞, 0 < σ < ∞. Suppose that X , ..., X are iid positive
1
m
random variables having the common pdf f(x; µ, 2), Y , ..., Y are iid positive
n
1
random variables having the common pdf f(y; 2µ, 3), and also that the X’s
and Y’s are independent. Here, µ is the unknown parameter and m ≠ n. De-
scribe the tests in their simplest implementable forms.
(i) Find the minimal sufficient statistic for µ;
(ii) Given α ∈ (0, 1), find the MP level a test to choose between the
null hypothesis H : µ = µ and the alternative hypothesis H : µ =
0
0
1
µ (> µ ) where µ , µ are fixed real numbers;
1 0 0 1
(iii) Given α ∈ (0, 1), find the UMP level a test to choose between the
null hypothesis H : µ = 1 and the alternative hypothesis H : µ > 1.
0 a
{Hints: In part (i), use the Lehmann-Scheffé Theorems from Chapter 6 to
claim that is the minimal sufficient statis-
tic. In parts (ii)-(iii), use the likelihood functions along the lines of the Ex-
amples 8.3.11-8.3.12 and show that one will reject H if and only if
0

8.4.22 Denote the gamma pdf f(w; µ) = [µ Γ(a)] exp{-w/µ} w with w >
a
a-1
-1
0, 0 < µ < ∞, 0 < a < ∞. Suppose that X , ..., X are iid positive random
1 m
variables having the common pdf f(x; µ, 1), Y , ..., Y are iid positive random
1
n
variables having the common pdf f(y; µ, 3), and also that the X’s and Y’s are
independent. Here, µ is the only unknown parameter and m ≠ n.