Page 458 - Probability and Statistical Inference

8. Tests of Hypotheses 435

8.4.5 Let X , ..., X be iid having the common negative exponential pdf
1
n
f(x;µ) = σ exp{-(x - µ)/σ}I(x > µ) where µ(∈ ℜ) is an unknown parameter
-1
but σ (∈ ℜ + ) is assumed known. Show that the family of distributions has
the MLR increasing property in T = X , the sufficient statistic for µ.
n:1
8.4.6 Let X , ..., X be iid having the common Cauchy pdf f(x; θ) = 1/π{1
1
n
+ (x - θ) } I(x ∈ ℜ) where θ(∈ ℜ) is an unknown parameter. Show that this
2 -1
family of distributions does not enjoy the MLR property. {Hint: Note that for
any θ* > ?, the likelihood ratio L(x; θ*)/L(x; θ) θ → 1 as x ’s converge to ±∞.)
i
8.4.7 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
+
have fixed numbers α ∈ (0, 1) and σ (> 0). Derive the UMP level α test for H 0
0
: σ > σ versus H : σ = σ in the simplest implementable form.
0 1 0
(i) Use the Neyman-Pearson approach;
(ii) Use the MLR approach.
8.4.8 Suppose that X , ..., X are iid having the Uniform(0, θ) distribution
1
n
with unknown θ(> 0). With preassigned α ∈ (0, 1), derive the UMP level α
test for H : θ > θ versus H : θ ≤ θ is a positive number, in the simplest
0
0
0
1
implementable form.
(i) Use the Neyman-Pearson approach;
(ii) Use the MLR approach.
8.4.9 Let X , ..., X be iid with the common gamma pdf f(x; δ, b) =
n
1
-δ
+
+
b [Γ(δ)] x exp(-x/b)I(x > 0) with two unknown parameters (δ, b) ∈ ℜ × ℜ .
-1 d1
With preassigned α ∈ (0, 1), derive the UMP level α test, in the simplest
implementable form, for H : (b < b , δ = δ*) versus H : (b ≥ b , δ = δ*)
0
1
0
0
where b is a positive number and δ* is also positive number.
0
(i) Use the Neyman-Pearson approach;
(ii) Use the MLR approach.
2
8.4.10 Suppose that X , ..., X are iid random variables from a N(µ, σ )
n
1
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
0
suppose that we reject H if and only if > c where c is a fixed number. Is
0
there any α ∈ (0, 1) for which this test is UMP level α?
8.4.11 Suppose that X , ..., X are iid having the Uniform(0, θ) distribution
1
n
with unknown θ(> 0). In order to choose between the two hypotheses H : θ
0
> θ versus H : θ ≤ θ where θ is a positive number, suppose that we reject
0
0
0
1
H if and only if X < c where c is a fixed positive number. Is there any α ∈
n:n
0
(0, 1) for which this test is UMP level α?