Description

Department of Statistics
STAT 210A: Introduction to Mathematical Statistics
Problem Set 5
Fall 2014

If X1,…,Xn are i.i.d. from N(θ,θ), then two natural estimators of θ are the sample mean X¯ and the sample variance S2. Determine the asymptotic relative efficiency of S2 with respect to X¯.
Problem 5.2
Let X1,…,Xn be i.i.d. from an exponential distribution with unit failure rate.
1. Suppose we are interested in the limiting distribution for X(2), the second order statistic.
p
Naturally, X(2) → 0 as n → ∞. For an interesting limit theory we should scale X(2) by an appropriate power of n, but the correct power is not 1/2. Suppose x > 0. Find a value p so that P(npX(2) ≤ x) converges to a value between 0 and 1. (If p is too small, the probability will tend to 1, and if p is too large the probability will tend to 0.)
2. Determine the limiting distribution for X(n) − logn.
Problem 5.3
Consider the loss function
| if a ≤ θ if a > θ
where k1 > 0 and k2 > 0 are constants. In a Bayesian setting, suppose that the random variable (θ | X = x) has finite mean for each x. Show that under this loss function, Bayes estimators are pth quantiles of the posterior distribution, where p is a suitable function of k1 and k2.
Problem 5.4
Given a fixed known integer r > 1, let Xij, j = 1,…r and i = 1,…,n be i.i.d. samples from N(µi,σ2). Find the MLE of θ = (µ1,…,µn,σ2), and show that it is inconsistent for σ2 as n → +∞.
Problem 5.5
Let (X1,…,Xn) be an i.i.d. sample from the mixture distribution with density
fθ(x) = θf1(x) + (1 − θ)f2(x),
where fi,i = 1,2 are two different known densities, and θ ∈ (0,1) is unknown.
1
(a) Show that the conditions
1 and
are necessary and sufficient for the score equation (setting the derivative of the log likelihood to zero) to have a unique solution. Show that if there is a solution, then it is the MLE.
(b) Derive the MLE of θ when the score equation has no solution.
2