Page 456 - Probability and Statistical Inference

8. Tests of Hypotheses 433

will reject H if and only if | X | < k. Determine k as a function of α. Perform
0
the power calculations. {Hint: Follow the Example 8.3.10.}
8.3.11 Suppose that X , ..., X are iid having Uniform (–θ, θ) distribution
1
n
with the unknown parameter θ (> 0). In order to choose between the two
hypotheses H : θ = θ versus H : θ = θ (> θ ) with two positive numbers θ ,
0 0 1 1 0 0
θ , suppose that we reject H if and only if | X n:n | > c where c is a fixed
1
0
positive number. Is there any α ∈ (0, 1) for which this test is MP level α?
{Hint: Write the likelihood function and look at the minimal sufficient statistic
for θ.}
8.3.12 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 MP
level α test for H : p = p versus H : p = p (> p ) where p , p are two
1
0
0
0
0
1
1
numbers from (0, 1). Explicitly find k and γ numerically when n = 4, 5, 6, α
= .05 and p = .1, .5, .7.
0
8.3.13 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
2
-1
parameter. With preassigned α ∈ (0, 1), derive the MP level α test for H : θ
0
= θ versus H : θ = θ (< θ ) where θ , θ are two positive numbers, in the
1
0
1
1
0
0
simplest implementable form.
8.3.14 Let X , ..., X be iid having the Weibull distribution with the com-
n
1
b
b-1
-1
mon pdf f(x;a) = a bx exp(-x /a)I(x > 0) where a(> 0) is an unknown pa-
rameter but b(> 0) is assumed known. With preassigned α ∈ (0, 1), derive the
MP level a test for H : a = a versus H : a = a (> a ) where a , a are two
1
0
0
1
0
1
0
positive numbers, in the simplest implementable form.
8.3.15 (Example 8.3.11 Continued) Evaluate the power of the MP level α
test described by (8.3.32).
8.3.16 (Example 8.3.11 Continued) Suppose that X and X are indepen-
2
1
dent random variables respectively distributed as N(µ, σ ), N(3µ, 2σ ) where
2
2
µ ∈ ℜ is the unknown parameter and σ ∈ ℜ is assumed known. Derive the
+
MP level a test for H : µ = µ versus H : µ = µ (< µ ). Evaluate the power of
0
1
1
0
0
the test. {Hint: Write the likelihood function along the line of (8.3.31) and
proceed accordingly.}
8.3.17 (Example 8.3.12 Continued) Evaluate the power of the MP level a
test described by (8.3.34).
8.3.18 (Example 8.3.12 Continued) Let us denote X’ = (X , X ) where X is
1 2
assumed to have the bivariate normal distribution, Here, we
consider µ(∈ ℜ) as the unknown parameter. Derive the MP level a test for H 0
: µ = µ versus H : µ = µ (< µ ). Evaluate the power of the test. {Hint: Write
0
0
1
1
the likelihood function along the line of (8.3.33) and proceed accordingly.}