Theory of Statistics - George Mason University

Exercises 381
4.14. Consider again the binomial(n, π) family of distributions in Example 4.6.
Let Pα,β be the beta(α, β) distribution.
a) Determine the gamma-minimax estimator of π under squared-error
loss within the class of priors Γ = {Pα,β : 0 < α, β}.
b) Determine the gamma-minimax estimator of π under squared-error
loss within the class of priors Γ = {Pα,β : 0 < α, β ≤ 1}.
4.15. Consider a generalization of the absolute-error loss function, |θ − d|:
c(d − θ) for d ≥ θ
L(θ, d) =
(1 − c)(θ − d) for d < θ
for 0 < c < 1 (equation (3.87)). Given a random sample X1, . . ., Xn on
the random variable X, determine the Bayes estimator of θ = E(X|θ).
(Assume whatever distributions are relevant.)
4.16. Let X ∼ U(0, θ) and the prior density of Θ be θ −2 I[1,∞)(θ). The posterior
is therefore
fΘ|x(θ|x) = 2c2
I[c,∞)(θ),
θ3 where c = max(1, x).
a) For squared-error loss, show that the Bayes estimator is the posterior
mean. What is the posterior mean?
b) Consider a reparametrization: ˜ θ = θ 2 , and let ˜ δ be the Bayes estimator
of ˜ θ. The prior density now is
1
2 ˜ θ 3/2 I[1,∞)( ˜ θ).
In order to preserve the connection, take the loss function to be
L( ˜ θ, ˜
δ) = ( ˜δ − ˜θ) 2 . What is the posterior mean? What is the
Bayes estimator of ˜ θ?
c) Compare the two estimators. Comment on the relevance of the loss
functions and of the prior for the relationship between the two estimators.
4.17. Let X1 depend on θ1 and X2 be independent of X1 and depend on θ2. Let
θ1 and θ2 have independent prior distributions. Assume a squared-error
loss. Let δ1 and δ2 be the Bayes estimators of θ1 and θ2 repectively.
a) Show that δ1 −δ2 is the Bayes estimator of θ1 −θ2 given X = (X1, X2)
and the setup described.
b) Now assume that θ2 > 0 (with probability 1), and let ˜ δ2 be the Bayes
estimator of 1/θ2 under the setup above. Show that δ1 ˜ δ2 is the Bayes
estimator of θ1/θ2 given X = (X1, X2).
4.18. In the problem of estimating π given X from a binomial(10, π) with
beta(α, β) prior and squared-error loss, as in Example 4.6, sketch the
risk functions, as in Figure 3.1 on page 273, for the unbiased estimator,
the minimax estimator, and the estimator resulting from Jeffreys’s noninformative
prior.
Theory of Statistics c○2000–2013 James E. Gentle