Description

Department of Statistics
STAT 210A: Introduction to Mathematical Statistics
Problem Set 2 Fall 2014

Problem 2.1
Suppose that (Xi,Yi), i = 1,…,n are sampled i.i.d. from the two-dimensional normal distribution
.
with θ ∈ Ω = (−1,1).
(a) Find a two-dimensional minimal sufficient statistic.
(b) Prove that the minimal sufficient statistic found in (a) is not complete.
(c) Prove that and are both ancillary, but that (Z1,Z2) is not ancillary.
Problem 2.2
Let X1,…,Xn be i.i.d. from a uniform distribution on (−θ,θ), where θ > 0 is an unknown parameter.
(a) Find a minimal sufficient statistic T.
(b) Let and define
,
where X(n) = maxi{Xi} and X(1) = mini{Xi}. Show that T and V are independent.
Problem 2.3
Consider the family of variates (X1,X2,…,Xn) with Xi ∼ Uni[
(a) Show that T = (X(1),X(n)) is sufficient.
(b) Suppose that we wish to estimate θ under the quadratic loss L(θ,a) = (θ − a)2. The sample mean X¯ might seem to be one reasonable estimate of θ, but turns out to be inadmissible. Indeed, show that the estimator

has a strictly better MSE than the MSE of X¯, and moreover that δ(X1,…,Xn) =
.
1
Problem 2.4
Suppose that X1,…,Xn are an i.i.d. sample from a location-scale family specified by the distribution function F ((x − a)/b). (Here F is a known cumulative distribution function; the real numbers a and b are the location and scale parameters respectively.) (a) If b is known, show that the differences (X1 − Xi)/b for i = 2,…,n are ancillary.
(b) If a is known, show that the ratios (X1 − a)/(Xi − a) for i = 2,…,n are ancillary.
(c) If neither a nor b are known, show that the ratios (X1 −Xi)/(X2 −Xi) for i = 3,…,n are ancillary.
Problem 2.5
Let X1,…,Xn be i.i.d. Poisson random variables with mean λ, and suppose that we wish to estimate g(λ) = exp(−λ) = Pλ[X = 0].
(a) Show that S1 = I[X1 = 0] and = 0] are both unbiased for estimating g(λ).
(b) Show that is sufficient.
(c) Compute the Rao-Blackwellized estimators Si∗ = E[Si | T] for i = 1,2. What does your answer have to do with completeness?
Problem 2.6
Let X be a single observation from a Poisson distribution with mean λ. Determine the UMVU estimator for
e−2λ = [Pλ(X = 0)]2 .
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