Page 522 - Probability and Statistical Inference

10. Bayesian Methods 499

10.3.4 Suppose that X , ..., X are iid Uniform(0, θ) given that = θ where
n
1
the parameter (> 0) is unknown. We say that v has the Pareto prior, denoted
by Pareto(α, β), when the prior pdf is given by

where α, β are known positive numbers. Denote the sufficient statistic T =
X , the largest order statistic, given that = θ. Show that the posterior
n:n
distribution of v given T = t turns out to be Pareto(max(t, α), n + β).
10.3.5 (Exercise 10.3.4 Continued) Let X , ..., X be iid Uniform(0, aθ)
1 n
given that = θ where the parameter (> 0) is unknown, but a(> 0) is
assumed known. Suppose that has the Pareto(α, β) prior where α, β are
known positive numbers. Denote the sufficient statistic T = X , the largest
n:n
order statistic, given that = θ. Show that the posterior distribution of
given T = t is an appropriate Pareto distribution.
10.3.6 Let X , ..., X be iid Uniform(–θ, θ) given that = θ where the
1
n
parameter (> 0) is unknown. Suppose that has the Pareto(α, β) prior
where α, β are known positive numbers. Denote the minimal sufficient statis-
tic T = |X| , the largest order statistic among |X |, ..., |X |, given that = θ.
1
n:n
n
Show that the posterior distribution of given T = t is an appropriate Pareto
distribution. {Hint: Can this problem be reduced to the Exercise 10.3.4?}
10.3.7 Let X , X , X be independent, X be distributed as N(θ, 1), X be
2
3
1
1
2
distributed as N(2θ, 3), and X be distributed as N(θ, 3) given that = θ
3
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, τ ) where τ(> 0) is a known number. Derive the
2
posterior distribution of given that T = t, t ∈ ℜ. {Hint: Given = θ, is the
statistic T normally distributed?}
10.3.8 We denote and suppose that X , ..., X are
1 n
+
iid N(θ , θ ) given that = θ where the parameters 1 (∈ ℜ), 2 (∈ ℜ ) are
2
2
1
assumed both unknown. Consider the minimal sufficient statistic T =
for θ given that = θ. We are given the joint prior distribution of as
follows:
where h (θ ) stands for the pdf of the distribution and stands
½
1
for the pdf of the IGamma(2, 1) distribution. The form of the inverted gamma
pdf was given in the Exercise 10.3.3.