Description

Department of Statistics
STAT 210A: Introduction to Mathematical Statistics
Problem Set 8
Fall 2014

Problem 8.1
Consider estimating success probabilities θ1,…,θp for p independent binomial variables, X1,…,Xp, each based on m trials, under compound squared error loss, di)2.
(a) Following a Bayesian approach, model the unknown parameters as random variablesΘ1,…,Θp that are i.i.d. from a beta distribution with parameters α and β. Determine the Bayes estimators of Θ1,…,Θp.
(b) In the Bayesian model, X1,…,Xp are i.i.d. Determine the first two moments for their common marginal distribution, EXi and EXi2. Using these, suggest simple method of moments estimators for α and β.
(c) Give empirical Bayes estimators for θi combining the simple “empirical” estimates for α and β in (b) with the Bayes estimate for θi when α and β are known in (a).
Problem 8.2
Let X1,…,Xn be an i.i.d. sample from the uniform distribution on [0,θ].
(a) Consider the problem of testing H0 : θ ≤ θ0 versus H1 : θ > θ0. Show that any test δ for which Eθ0[δ(X)] = α, Eθ[δ(X)] ≤ α for all θ ≤ θ0 and δ(x) = 1 when x(n) = max{x1,…,xn} > θ0 is UMP at level α.
(b) Now consider the problem of testing H0 : θ = θ0 against H1 : θ 6= θ0. Show that a unique UMP test exists, and is given by
(1 if x(n) 1/n
> θ0 or x(n) < θ0(α)
δ(x) =
0 otherwise.
Problem 8.3
Suppose that X1,…,Xn are independent exponential random variables with E[Xi] = βti where t1,…,tn are known constants, and β > 0 is an unknown parameter.
(a) Show that the MLE of β is given by .
(b) Prove that
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(c) Suppose that we want to test H0 : β = 1 versus H1 : β 6= 1. In order to do so, we consider the statistic
,
where ) is the likelihood. Show that .
(d) Show that when H0 is true, we have 2 .
Problem 8.4
Consider simple versus simple testing from a Bayesian perspective. Let Θ have a Bernoulli distribution with P(Θ = 1) = p and P(Θ = 0) = 1 − p. Given Θ = 0, X will have density p0 and given Θ = 1, X will have density p1.
(a) Show that the chance of accepting the wrong hypothesis in the Bayesian model usinga test function ϕ is
R(ϕ) = E[I(Θ = 0)ϕ(X) + I(Θ = 1)(1 − ϕ(X))].
(b) Use the tower property to relate R(ϕ) to E0(ϕ) = E(ϕ(X)γΘ = 0) and E1(ϕ) = E(ϕ(X)γΘ = 1).
(b) Find the test function ϕ∗ minimizing R(ϕ). Show that ϕ∗ is a likelihood ratio test, identifying the critical value k.
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