Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

6.2. Rao–Cram ́er Lower Bound and Efficiency 375

(a)Find the Fisher informationI(θ).

(b)Show that the mle ofθ, which was derived in Exercise 6.1.1, is an efficient

estimator ofθ.

(c)Using Theorem 6.2.2, obtain the asymptotic distribution of

√

n(̂θ−θ).

(d)For the data of Example 6.1.1, find the asymptotic 95% confidence interval

forθ.

6.2.8.LetXbeN(0,θ), 0 <θ<∞.

(a)Find the Fisher informationI(θ).

(b)IfX 1 ,X 2 ,...,Xnis a random sample from this distribution, show that the

mle ofθis an efficient estimator ofθ.

(c)What is the asymptotic distribution of

√

n(θ̂−θ)?

6.2.9.IfX 1 ,X 2 ,...,Xnis a random sample from a distribution with pdf

f(x;θ)=

{

3 θ^3

(x+θ)^40 <x<∞,^0 <θ<∞

0elsewhere,

show thatY=2Xis an unbiased estimator ofθand determine its efficiency.

6.2.10.LetX 1 ,X 2 ,...,Xnbe a random sample from aN(0,θ) distribution. We
want to estimate the standard deviation

√

θ. Find the constantcso thatY =

c

∑n

i=1|Xi|is an unbiased estimator of

√

θand determine its efficiency.

6.2.11.LetXbe the mean of a random sample of sizenfrom aN(θ, σ^2 ) distribu-

tion,−∞<θ<∞,σ^2 >0. Assume thatσ^2 is known. Show thatX

2

−σ

2

n is an

unbiased estimator ofθ^2 and find its efficiency.

6.2.12.Recall that̂θ=−n/

∑n
i=1logXiis the mle ofθfor a beta(θ,1) distribution.
Also,W=−

∑n

i=1logXihas the gamma distribution Γ(n,^1 /θ).

(a)Show that 2θWhas aχ^2 (2n) distribution.

(b)Using part (a), findc 1 andc 2 so that

P

(

c 1 <

2 θn

θ̂

<c 2

)

=1−α, (6.2.35)

for 0<α<1. Next, obtain a (1−α)100% confidence interval forθ.

(c)Forα=0.05 andn= 10, compare the length of this interval with the length

of the interval found in Example 6.2.6.

6.2.13.The data filebeta30.rdacontains 30 observations generated from a beta(θ,1)
distribution, whereθ= 4. The file can be downloaded at the site discussed in the
Preface.