Math 304: Homework 3
Due: February 23
1. WMS 9.3 & 9.15: Let Y1,...,Yndenote a random sample from the uniform distribution on
the interval (θ, θ + 1). Let
ˆ
θ1=¯
Y−1
2and ˆ
θ2=Y(n)−n
n+ 1
where Y(n)= max{Y1,...Yn}is the largest order statistic.
(a) Show that both ˆ
θ1and ˆ
θ2are unbiased.
(b) Find the efficiency of ˆ
θ1relative to ˆ
θ2.
(c) Show that both ˆ
θ1and ˆ
θ2are consistent estimators of θ.
2. HMC 6.2.7: Let Xhave a gamma distribution with α= 4 and β=θ > 0.
(a) Find the Fisher information I(θ).
(b) If X1,...,Xnis a random sample from this distribution, show that the mle of θis an
efficient estimator of θ.
(c) What is the asymptotic distribution of √n(ˆ
θ−θ)?
3. HMC 6.2.8: Let X∼N(0, θ),0< θ < ∞.
(a) Find the Fisher information I(θ).
(b) If X1,...,Xnis a random sample from this distribution, show that the mle of θis an
efficient estimator of θ.
(c) What is the asymptotic distribution of √n(ˆ
θ−θ)?
4. HMC 6.2.11: Let ¯
Xbe the mean of a random sample of size nfrom a N(θ, σ2) distribution,
−∞ < θ < ∞, σ2>0.Assume that σ2is known. Show that ¯
X2−σ2
nis an unbiased estimator
of θ2and find its efficiency.
5. HMC 6.2.14: Let S2be the sample variance of a random sample of size n > 1 from N(µ, θ ),0<
θ < ∞, where µis known. We know that E(S2) = θ. What is the efficiency of S2?
6. WMS 9.100: Suppose that Y1,...,Ynconstitue a random sample of size nfrom an exponential
distribution with mean θ. Find an approximate 100(1 −α)% confidence interval for θ2.
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