Page 526 - Probability and Statistical Inference

10. Bayesian Methods 503

known positive numbers. With fixed 0 < α < 1, derive the 100(1 – α)% HPD
credible interval for . {Hint: Use the form of the posterior from the Exercise
10.3.5.}
10.5.6 (Exercise 10.4.7 Continued) Let X , ..., X be iid Uniform(–θ, θ)
n
1
given that = θ where the parameter (> 0) is unknown. Suppose that has
the Pareto(τ, β) prior where τ, β are known positive numbers. With fixed 0 <
α < 1, derive the 100(1 – α)% HPD credible interval for . Under the squared
error loss function, derive the expression of the Bayes estimate for .
{Hint: Use the form of the posterior from the Exercise 10.3.6.}
10.5.7 Suppose that X has its pdf f(x; θ) = 2θ (θ – x)I(0 < x < θ) given
–2
that = θ where the parameter (> 0) is unknown. Suppose that has the
Gamma(3, 2) distribution.
(i) Show that the posterior pdf k(θ x) = ¼ (θ – x)e –(θ–x)/2 × I (0 < x < θ);
(ii) With fixed 0 < α < 1, derive the 100(1 – α)% HPD credible interval
for .
10.5.8 (Exercise 10.3.7 Continued) Let X , X , X be independent, X be
3
1
2
1
distributed as N (θ, 1), X be distributed as N (2θ, 3), and X be distributed as
3
2
N (θ, 3) given that = θ where (∈ ℜ) is the unknown parameter. Consider
the minimal sufficient statistic T for θ given that = θ. Let us suppose that
the prior distribution of on the space Θ = ℜ is N (2, 9). Suppose that the
following data has been observed:
Derive the 95% HPD credible interval for . Under the squared error loss
function, derive the expression of the Bayes estimate for . {Hint: First
find the posterior along the lines of the Exercise 10.3.7.}
10.6.1 (Example 10.6.1 Continued) Let X , ..., X be iid N (θ, 4.5) given
1
10
that = θ where (∈ ℜ) is the unknown parameter. Let us suppose that the
prior distribution of on the space Θ = ℜ is N (4, 2). Consider the statistic
which is minimal sufficient for θ given that = θ. Suppose that
the observed value of T is t = 48.5. Use the Bayes test from (10.6.2) to
choose between a null hypothesis H : < 3 against an alternative hypothesis
0
H : ≥ 5.
1
10.6.2 (Example 10.3.3 Continued) Let X , X , X be iid Bernoulli(θ) given
3
2
1
that = θ where is the unknown probability of success, 0 < v < 1. Let us
assume that the prior distribution of v on the space Θ = (0, 1) is described
as Beta(2, 4). Consider the statistic which is minimal suffi-
cient for θ given that = θ. Suppose that the observed value of T is t =
2. Use the Bayes test from (10.6.2) to choose between a null hypothesis