Page 454 - Probability and Statistical Inference

8. Tests of Hypotheses 431

8.2.6 Suppose that X , ..., X are iid N(θ, 4) where θ(∈ ℜ) is the unknown
1
10
parameter. In order to test H : θ = –1 against H : θ = 1, we propose the
1
0
critical region Find the level α and evaluate
the power at θ = 1.
8.3.1 Suppose that X , ..., X are iid random variables from the N(µ, σ )
2
1
n
+
population where µ is assumed known but σ is unknown, µ ∈ ℜ, σ ∈ ℜ . We
fix a number α ∈ (0, 1) and two positive numbers σ , σ .
0 1
(i) Derive the MP level a test for H : σ = σ versus H : σ = σ (> σ )
1
0
0
0
1
in the simplest implementable form;
(ii) Derive the MP level α test for H : σ = σ versus H : σ = σ (< σ )
0
0
1
0
1
in the simplest implementable form;
In each part, draw the power function.
8.3.2 (Example 8.3.4 Continued) Suppose that X , ..., X are iid having the
1
n
Uniform(0, θ) distribution with unknown θ(> 0). With preassigned α ∈ (0, 1)
and two positive numbers θ < θ , derive the MP level a test for H : θ = θ 0
1
0
0
versus H : θ = θ in the simplest implementable form. Perform the power
1
1
calculations.
8.3.3 (Example 8.3.5 Continued) Suppose that X , ..., X are iid with the
n
1
common pdf b [Γ(δ)] x exp(-x/b), with two unknown parameters (δ, b) ∈
-δ
-1 δ-1
ℜ . With preassigned α ∈ (0, 1), derive the MP level α test, in the simplest
+2
implementable form, for H : (b = b , δ = δ*) versus H : (b = b , δ = δ*)
1
1
0
0
where b < b are two positive numbers and δ* is a positive number.
1
0
8.3.4 Suppose that X , ..., X are iid random variables from the N(µ, σ )
2
1
n
population where µ is unknown but σ is assumed known, µ ∈ ℜ, σ ∈ ℜ . In
+
order to choose between the two hypotheses H : µ = µ versus H : µ = µ (>
0
0
1
1
µ ), suppose that we reject H if and only if where c is a fixed number.
0
0
Is there any α ∈ (0, 1) for which this particular test is MP level α?
8.3.5 (Exercise 8.3.2 Continued) Suppose that X , ..., X are iid having
n
1
the Uniform(0, θ) distribution with unknown θ (> 0). In order to choose
between the two hypotheses H : θ = θ versus H : θ = θ where θ < θ are
0
1
1
1
0
0
two positive numbers, suppose that we reject H if and only if X < c where
0
n:n
c is a fixed positive number. Is there any α ∈ (0, 1) for which this test is MP
level α ?
8.3.6 (Example 8.3.6 Continued) Suppose that X , ..., X are iid ran-
1
n
dom variables having the Bernoulli(p) distribution where p ∈ (0, 1) is the