Description

Department of Statistics
STAT 210A: Introduction to Mathematical Statistics
Problem Set 7
Fall 2014

Problem 7.1
Method of moments estimation. Let X1,X2,… be i.i.d. observations from some family of distributions indexed by θ ∈ Ω ⊂R. Let X¯n denote the average of the first n observations, and let µ(θ) = EθXi and σ2(θ) = Varθ(Xi). Assume that µ is strictly monotonic and continuously differentiable. The method of moments estimator θˆn solves µ(θ) = X¯n. If
√
µ0(θ) 6= 0, find the limiting distribution for n(θˆn − θ).
Problem 7.2
Let X1,X2,… be i.i.d. from a uniform distribution on (0,1), and let Tn ∈ [0,1] be the unique solution of the equation
n n
X Xi X 2 t = Xi .
i=1 i=1
p
(a) Show that Tn → c as n →∞, identifying the constant c.
√
(b) Find the limiting distribution for n(Tn − c) as n →∞.
Problem 7.3
Suppose that X1,X2,…,Xn are i.i.d. samples from a normal location model N(θ,1), and that we are interested in estimating the quantity 1/θ. In order to do so, we use the estimator δ(X) = 1/X¯n where is the sample mean.
(a) Show that δ is asymptotically normal—viz.:
(b) Show that the expectation E[1/X¯n] fails to exist for all n. Why does this not contradict the result of part (a)?
Problem 7.4
A variance stabilizing approach. Let X1,X2,… be i.i.d. from a Poisson distribution with mean θ, and let θˆn = X¯n be the MLE of θ.
(a) Find a function g : [0,∞) →R such that
√ ˆ ) − g(θ)) →d N(0,1). Zn = n(g(θn
(b) Find a 1 − α asymptotic confidence interval for θ based on the approximate pivot Zn.
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Problem 7.5
Let X1,X2,… be i.i.d. from N(µ,σ2). Suppose we know that σ is a known function of µ: σ = g(µ). Let ˆµn denote the MLE for µ under this assumption, based on X1,X2,…,Xn.
(a) Give a 1−α asymptotic confidence interval for µ centered at ˆµn. Hint: If Z ∼ N(0,1), then Var(Z2) = 2 and Cov(Z,Z2) = 0.
(b) Compare the width of the asymptotic confidence interval in part (a) with the widthof the t-confidence interval that would be appropriate if µ and σ were not functionally related. Specificially, show that the ratio of the two widths converges in probability as n →∞, identifying the limiting value. (The limit should be a function of µ.)
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