Page 525 - Probability and Statistical Inference

502 10. Bayesian Methods

true average gas mileage per gallon, . Give the Bayes estimate of the true
average gas mileage per gallon under the squared error loss.
10.4.10 (Fubini’s Theorem) Suppose that a function of two real variables
g(x , x ) is either non-negative or integrable on the space χ = χ × χ (⊆ ℜ ×
1
2
2
1
ℜ). That is, for all (x , x ) ∈ χ, the function g(x , x ) is either non-negative or
2
1
1
2
integrable. Then, show that the order of the (two-dimensional) integrals can
be interchanged, that is, one can write
{Note: This is a hard result to verify. It is stated here for reference purposes
and completeness. Fubini’s Theorem was used in the proof of the Theorem
10.4.1 for changing the order of two integrals.}
10.5.1 (Exercise 10.4.2 Continued) Let X , X be independent, X be dis-
1
1
2
tributed as N (θ, 1) and X be distributed as N (2θ, 3) given that = θ where
2
(∈ ℜ) is the unknown parameter. Let us suppose that the prior distribution
of on the space Θ = ℜ is N (5, τ ) where τ(> 0) is a known number. With
2
fixed 0 < α < 1, derive the 100(1 – α)% HPD credible interval for . {Hint:
Use the form of the posterior from the Exercise 10.3.1.}
10.5.2 (Exercise 10.4.3 Continued) Let X , ..., X be iid Exponential(θ)
1
n
given that = θ where the parameter (> 0) is unknown. Suppose that has
the IGamma(1, 1) prior pdf h(θ) = θ exp {–1/θ}I(θ > 0). With fixed 0 < α <
–2
1, derive the 100(1 – α)% HPD credible interval for . {Hint: Use the form of
the posterior from the Exercise 10.3.2.}
10.5.3 (Exercise 10.4.4 Continued) Let X , ..., X be iid N (0, θ ) given that
2
1
n
= θ where the parameter v(> 0) is unknown. Suppose that has the
IGamma(1, 1). With fixed 0 < α < 1, derive the 100(1 – α)% HPD credible
interval for . {Hint: Use the form of the posterior from the Exercise 10.3.3.}
2
10.5.4 (Exercise 10.4.5 Continued) Let X , ..., X be iid Uniform(0, θ)
n
1
given that = θ where the parameter (> 0) is unknown. Suppose that v has
the Pareto(τ, β) prior, that is the prior pdf is given by h(θ) = βτ θ I(τ < θ
β –(β+1)
< ∞) where τ, β are known positive numbers. With fixed 0 < α < 1, derive the
100(1 - a)% HPD credible interval for . {Hint: Use the form of the posterior
from the Exercise 10.3.4.}
10.5.5 (Exercise 10.4.6 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(τ, β) prior where τ, β are