Description

Department of Statistics
STAT 210A: Introduction to Mathematical Statistics
Problem Set 1
Fall 2014

Problem 1.1
Suppose X1,…,Xn are independent random variables and that for i = 1,…,n, Xi has a
Poisson distribution with mean λi = exp(α + βti), where t1,…,tn are observed constants and α and β are unknown parameters. Show that the joint distributions for X1,…,Xn form a two-parameter exponential family and identify the statistics T1 and T2.
Problem 1.2
Let {pθ(x) : θ ∈ Ω} be an exponential family of densities with respect to some measure µ, where
.
In some situations, a potential observation X with density pθ(x) can only be observed if it happens to lie in some region S. For regularity, assume that Pθ(X ∈ S) > 0. In this case, the appropriate distribution for the observed variable Y is given by
Pθ(Y ∈ B) = Pθ(X ∈ B |X ∈ S).
This distribution for Y is called the truncation of the distribution for X to the set S.
1. Show that Y has a density with respect to µ, giving a formula for its density qθ.
2. Show that the densities {qθ(x)},θ ∈ Ω, form an exponential family.
Problem 1.3
Consider an i.i.d. sample {X1,…,Xn} from the uniform distribution on [0,θ], and the estimator Mn = max{X1,…,Xn}.
p
(a) Prove that Mn → θ as n → +∞.
(b) Compute the bias and variance of the estimator Mn as a function of θ.
(c) Compute the risk of Mn under quadratic loss.
Problem 1.4
Find the natural parameter space Ξ and densities pη for a canonical one-parameter exponential family with µ counting measure on {1,2,…}, h(x) = x2 and T(x) = −x. Also, determine the mean and variance for a random variable X with this density. Hint: Consider what Theorem 2.4 in Keener has to say about the derivatives of .
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Problem 1.5
A parameterization θ 7→ Pθ is identifiable if θ1 6= θ2 implies that Pθ1 6= Pθ2. Which of the following parameterizations are identifiable?
(a) Suppose that X1,…,Xp are independent with Xi ∼ N(αi + ν,σ2). Set
θ = (α1,…,αp,ν,σ2)
and consider the family of joint distributions Pθ over (X1,…,Xp).
(b) Suppose that X is a Bernoulli variable, and we parameterize its probability mass function with θ = (θ0,θ1) and p(x;θ) ∝ exp(θ0x + θ1(1 − x)).
(c) Autoregressive process: consider the collection of RVs√ (X1,…,Xp) given by X1 ∼

N(0,σ2), and Xi+1 = αXi + 1 − α2Wi where Wi ∼ N(0,σ2), independent of the Xi, and α ∈ [0,1]. Let θ = (α,σ2) and let Pθ index the joint distribution over (X1,…,Xp).
Problem 1.6
Suppose that (X1,…,Xn) are i.i.d. Poisson random variables with parameter θ. Show that is sufficient in two ways:
(a) first use direct methods: compute the conditional distribution given T = t.
(b) apply the factorization theorem.
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