Page 462 - Probability and Statistical Inference

8. Tests of Hypotheses 439

8.5.2 (Exercise 8.4.21 Continued) Denote the lognormal pdf f(w; µ) =
with w > 0, –∞ < µ < ∞, 0 < σ < ∞.
Suppose that X , ..., X are iid positive random variables having the common
m
1
pdf f(x; µ, 2), Y , ..., Y are iid positive random variables having the common
1
n
pdf f(y; 2µ, 3), and also that the X’s and Y’s are independent. Here, µ is the
only unknown parameter and m ≠ n. Argue whether there does or does not
exist a UMP level a test for deciding between the null hypothesis H : µ = 1
0
and the alternative hypothesis H : µ ≠ 1. {Hint: Try to exploit arguments
a
similar to those used in the Section 8.5.1.}
8.5.3 Let X , X , X , X be iid Uniform(0, θ) where θ > 0. In order to test
1
2
4
3
H : θ = 1 against H : θ ≠ 1, suppose that we propose to use the critical region
1
0
R = {X ∈ ℜ : X < ½ or X > 1}. Evaluate the level α and the power
+4
4:4
4:4
function.
8.5.4 Let X , ..., X be iid having the common exponential pdf f(x; θ) =
n
1
θ exp{-x/θ}I(x > 0) where θ(> 0) is assumed unknown. With preassigned α
-1
∈ (0, 1), show that no UMP level α test for H : θ = θ versus H : θ ≠ θ exists
1
0
0
0
where ? is a fixed positive number. {Hint: Try to exploit arguments similar to
0
those used in the Section 8.5.1.}
+
2
8.5.5 Let X , ..., X be iid N(0, σ ) with unknown σ ∈ ℜ . With preas-
n
1
signed α ∈ (0, 1), we wish to test H : σ = σ versus H : σ ≠ σ at the level
1
0
0
0
α, where σ is a fixed positive number. Show that no UMP level α test exists
0
for testing H versus H . {Hint: Try to exploit arguments similar to those
1
0
used in the Section 8.5.1.}