Page 553 - Probability and Statistical Inference

530 11. Likehood Ratio and Other Tests

11.2.2 Suppose that X , ..., X are iid N(0, σ ) where σ(> 0) is the un-
2
1
n
known parameter. With preassigned α ∈ (0, 1), derive a level α LR test for the
null hypothesis H : σ = against an alternative hypothesis H :
2
0 1
in the implementable form. {Note: Recall from the Exercise 8.5.5 that no
UMP level α test exists for testing H versus H }.
1
0
+
2
11.2.3 Suppose that X , ..., X are iid N(µ, σ ) where σ (∈ ℜ ) is the
1
n
unknown parameter but µ(∈ ℜ) is assumed known. With preassigned α ∈ (0,
1), derive a level α LR test for a null hypothesis H : against an
0
alternative hypothesis H : in the implementable form. {Note: Recall
1
from the Exercise 8.5.5 that no UMP level a test exists for testing H versus
0
H }.
1
2
11.2.4 Suppose that X , X are iid N(µ, σ ) where µ(∈ ℜ), σ(∈ ℜ ) are
+
1 2
both assumed unknown parameters. With preassigned α ∈ (0, 1), reconsider
the level α LR test from (11.2.21) for choosing between a null hypothesis H 0
: against an alternative hypothesis H : Show that the
1
same test can be expressed as follows: Reject H if and only if |X – X | >
0 1 2
11.2.5 Suppose that X , ..., X are iid having the common exponential pdf
n
1
f(x; θ) = θ exp{–x/θ}I(x > 0) where θ(> 0) is assumed unknown. With
–1
preassigned α ∈ (0, 1), derive a level α LR test for a null hypothesis H : θ =
0
θ (> 0) against an alternative hypothesis H : θ ≠ θ in the implementable form.
0
1
0
{Note: Recall from the Exercise 8.5.4 that no UMP level a test exists for
testing H versus H }.
1
0
11.2.6 Suppose that X and X are independent random variables respec-
1
2
tively distributed as N(µ, σ ), N(3µ, 2σ ) where µ ∈ ℜ is the unknown param-
2
2
eter and σ ∈ ℜ is assumed known. With preassigned α ∈ (0, 1), derive a level
+
α LR test for H : µ = µ versus H : µ ≠ µ where µ (∈ ℜ) is a fixed number,
0
1
0
0
0
in the implementable form. {Hint: Write down the likelihood function along
the line of (8.3.31) and then proceed directly as in Section 11.2.1.}
11.2.7 Suppose that X , ..., X are iid having the Rayleigh distribution with
n
1
the common pdf f(x; θ) = 2θ xexp(–x /θ)I(x > 0) where θ(> 0) is the un-
2
–1
known parameter. With preassigned α ∈ (0, 1), derive a level α LR test for H 0
: θ = θ versus H : θ ≠ θ where θ (∈ ℜ ) is a fixed number, in the
+
0
1
0
0
implementable form.
11.2.8 Suppose that X , ..., X are iid having the Weibull distribution with
n
1
the common pdf f(x; a) = a bx exp(–x /a)I(x > 0) where a(> 0) is an
–1
b–1
b
unknown parameter but b(> 0) is assumed known. With preassigned a ∈ (0,
1), derive a level α LR test for H : a = a versus H : a ≠ a where a is a
0
0
0
1
0
positive number, in the implementable form.