Description

Department of Statistics
STAT 210A: Introduction to Mathematical Statistics
Problem Set 4 Fall 2014

Problem 4.1
Let X have a geometric distribution with success probability ✓, so P✓(X = x) = ✓(1 ✓)x,x = 0,1,…. What is the smallest possible variance for an unbiased estimate of ✓? Compare this variance with the lower bound in Theorem 4.9 in Keener.
Problem 4.2
Let ⇥ = (0,+1) and A = [0,1), and suppose that X ⇠ Poi(✓). Consider the loss function L(✓,a) = (✓ a)2/✓.
(a) Find Bayes estimators with respect to the family of Gamma(a,b) priors.
(b) Show that the estimator (X) = X can be obtained by a suitable limit from (a).
Problem 4.3
Let (X1,…,Xn) be an i.i.d. sample from the uniform distribution on (0,✓), where ✓ > 0 is unknown. Suppose that the prior distribution of ✓ is log-normal with parameters (µ0, 02) where µ0 2R and 02 > 0 are known constants.
(a) Find the posterior density of log✓.
(b) Suppose that we are interested in estimating ✓ under the loss function
0 if = ✓
L( ,✓) = (
1 otherwise.
Find the Bayes estimator of ✓ under this loss function. (Hint: Part (a) is related.)
Problem 4.4
Consider the Bayesian model in which ⇥~ has distribution ⇤, and conditioned on ⇥~ = ✓, the random variable X has distribution P✓. Suppose that we are interested in estimating g(✓) under quadratic loss. Prove that no unbiased estimator (X) of g(✓) can be a Bayesian estimator unless the Bayesian risk r(⇤, ) = 0. This shows that Bayes estimators and unbiased estimators agree only in pathological cases.
Problem 4.5
Suppose that X has density in exponential family form
.
where each Ti is di↵erentiable.
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(a) Prove the following lemma. If X has support on the real line and g is any di↵erentiable function such that E|g0(X)| < +1, then
.
(b) Specializing to the N(µ, 2) case, use part (a) to show that cov{g(X),X} = 2E[g0(X)].
(c) Use part (b) to compute the third and fourth moments of the N(µ, 2) distribution.
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