Page 527 - Probability and Statistical Inference

504 10. Bayesian Methods

H : ≤ .3 against an alternative hypothesis H : ≥ .5. {Hint: Use integration
0
1
by parts when evaluating a , a from (10.6.1). Review (1.6.28) as needed.}
1
0
10.6.3 (Example 10.3.4 Continued) Let X , ..., X be iid Poisson(θ) given
4
1
that v = θ where (> 0) is the unknown population mean. Let us suppose that
the prior distribution of on the space Θ = (0, ∞) is Gamma(2, 3). Consider
the statistic which is minimal sufficient for θ given that = θ.
Suppose that the observed value of T is t = 1. Use the Bayes test from (10.6.2)
to choose between a null hypothesis H : ≤ 1.5 against an alternative hy-
0
pothesis H : ≥ 2.5. {Hint: Use integration by parts when evaluating α , α 1
0
1
from (10.6.1). Review (1.6.28) as needed.}
10.7.1 (Example 10.7.1 Continued) Using the definition of a pdf, show
that the function k(θ; x) given in (10.7.3) is indeed a probability density func-
tion of .
10.7.2 Let X , ..., X be iid N (θ 1) given that = θ where (∈ ℜ ) is the
+
n
1
unknown parameter. Let us suppose that the prior pdf of on the space Θ =
–αθ
+
ℜ is given by h(θ) = αe I(θ > 0) where α(> 0) is a known number. Con-
sider the sample mean, which is minimal sufficient for θ given that
= θ.
(i) Derive the marginal pdf m(t) of T for t ∈ ℜ;
(ii) Derive the posterior pdf k(θ t) for v given that T = t, for θ > 0 and
–∞ < t < ∞;
(iii) Under the squared error loss function, derive the Bayes estimate
and show, as in the Example 10.7.3, that can take only positive
values.
{Hint: Proceed along the lines of the Examples 10.7.1-10.7.2.}
10.7.3 (Exercise 10.7.2 Continued) Let X , ..., X be iid N (θ, 1) given that
1
10
= θ where (∈ ℜ ) is the unknwon parameter. Let us suppose that the
+
+
prior pdf of on the space Θ = ℜ is given by h(θ) = αe I (θ > 0) where
-αθ
α(> 0) is a known number. Suppose that the observed value of the sample
mean was 5.92. Plot the posterior pdf k(θ; t) assigning the values α = ½, ¼
and 1/8. Comment on the empirical behaviors of the corresponding posterior
pdf and the Bayes estimates of .
10.7.4 (Example 10.7.4 Continued) Verify the expressions of m(x) and k(θ
x) given by (10.7.8) and (10.7.9) respectively. Under the squared error loss
function, derive the form of the Bayes estimate .
10.7.5 Let X , ..., X be iid N (θ, 1) given that = θ where (∈ ℜ) is the
1 n
unknown parameter. Let us suppose that the prior pdf of on the space Θ =
ℜ is given by h(θ) = ½αe -α|θ| I (θ ∈ ℜ) where α(> 0) is a known number.
Consider the sample mean, which is minimal sufficient for