Page 459 - Probability and Statistical Inference

436 8. Tests of Hypotheses

8.4.12 Suppose that X , ..., X are iid random variables having the
1
n
Bernoulli(p) distribution where p ∈ (0, 1) is the unknown parameter. With
preassigned α ∈ (0, 1), derive the randomized UMP level α test for H : p = p 0
0
versus H : p > p where p is a number between 0 and 1.
1 0 0
8.4.13 Suppose that X , ..., X are iid random variables having the Poisson(λ)
n
1
distribution where λ ∈ ℜ is the unknown parameter. With preassigned α ∈
+
(0, 1), derive the randomized UMP level α test for H : λ = λ versus H : λ <
0
0
1
λ where λ is a positive number.
0
0
8.4.14 Suppose that X , ..., X are iid having the Uniform(–θ, θ) distribu-
1
n
tion with unknown θ(> 0). In order to choose between the two hypotheses H 0
: θ ≤ θ versus H : θ > θ where θ is a positive number, suppose that we
0
1
0
0
reject H if and only if | X n:n | > c where c is a fixed known positive number. Is
0
there any α ∈ (0, 1) for which this test is UMP level α?
8.4.15 Suppose that X , ..., X are iid Geometric(p) where p ∈ (0, 1) is the
1
n
unknown parameter. With preassigned α ∈ (0, 1), derive the randomized UMP
level α test for H : p ≥ p versus H : p < p where p is a number between 0
0
0
1
0
0
and 1.
8.4.16 Let X , ..., X be iid having the Rayleigh distribution with the com-
1
n
mon pdf f(x; θ) = 2θ xexp(–x /θ)I(x > 0) where θ(> 0) is the unknown pa-
–1
2
rameter. With preassigned α ∈ (0, 1), derive the UMP level α test for H : θ ≤
0
θ versus H : θ > θ where θ is a positive number, in the simplest implementable
1
0
0
0
form.
(i) Use the Neyman-Pearson approach;
(ii) Use the MLR approach.
8.4.17 Let X , ..., X be iid having the Weibull distribution with the com-
n
1
mon pdf f(x; a) = a bx exp(-x /a)I(x > 0) where a(> 0) is an unknown
b
-1
b-1
parameter but b(> 0) is assumed known. With preassigned a ∈ (0, 1), derive
the UMP level a test for H : a ≤ a versus H : a > a where a is a positive
0
0
1
0
0
number, in the simplest implementable form.
(i) Use the Neyman-Pearson approach;
(ii) Use the MLR approach.
8.4.18 (Exercise 8.4.5 Continued) Let X , ..., X be iid having the common
n
1
–1
negative exponential pdf f(x; µ) = σ exp{-(x - µ)/σ}I(x > µ) where µ(∈ ℜ) is
an unknown parameter but σ(∈ ℜ ) is assumed known. With preassigned α ∈
+
(0, 1), derive the UMP level α test for H : µ ≤ µ versus H : µ > µ where µ 0
1
0
0
0
is a real number, in the simplest implementable form.
8.4.19 (Exercise 8.4.1 Continued) Let X , ..., X be iid having the common
1
n
Laplace pdf f(x; b) = ½b exp(– | x – a | /b)I(x ∈ ℜ) where b(> 0) is an
-1
unknown parameter but a(∈ ℜ) is assumed known. With preassigned