418 Sufficiency7.1.3.LetY 1 <Y 2 <Y 3 be the order statistics of a random sample of size 3 from
the uniform distribution having pdff(x;θ)=1/θ, 0 <x<θ, 0 <θ<∞, zero
elsewhere. Show that 4Y 1 , 2 Y 2 ,and^43 Y 3 are all unbiased estimators ofθ.Findthe
variance of each of these unbiased estimators.
7.1.4.LetY 1 andY 2 be two independent unbiased estimators ofθ. Assume that
the variance ofY 1 is twice the variance ofY 2. Find the constantsk 1 andk 2 so that
k 1 Y 1 +k 2 Y 2 is an unbiased estimator with the smallest possible variance for such a
linear combination.
7.1.5.In Example 7.1.2 of this section, takeL[θ, δ(y)] =|θ−δ(y)|. Show that
R(θ, δ 1 )=^15
√
2 /πandR(θ, δ 2 )=|θ|. Of these two decision functionsδ 1 andδ 2 ,
which yields the smaller maximum risk?
7.1.6.LetX 1 ,X 2 ,...,Xndenote a random sample from a Poisson distribution with
parameterθ, 0 <θ<∞.LetY =
∑n
1 Xiand letL[θ, δ(y)] = [θ−δ(y)](^2) .Ifwe
restrict our considerations to decision functions of the formδ(y)=b+y/n,whereb
does not depend ony, show thatR(θ, δ)=b^2 +θ/n. What decision function of this
form yields a uniformly smaller risk than every other decision function of this form?
With this solution, sayδ,and0<θ<∞, determine maxθR(θ, δ)ifitexists.
7.1.7. LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution that is
N(μ, θ), 0 <θ<∞,whereμis unknown. LetY =
∑n
1 (Xi−X)
(^2) /nand let
L[θ, δ(y)] = [θ−δ(y)]^2. If we consider decision functions of the formδ(y)=by,where
bdoes not depend upony, show thatR(θ, δ)=(θ^2 /n^2 )[(n^2 −1)b^2 − 2 n(n−1)b+n^2 ].
Show thatb=n/(n+ 1) yields a minimum risk decision function of this form. Note
thatnY /(n+ 1) is not an unbiased estimator ofθ.Withδ(y)=ny/(n+1) and
0 <θ<∞, determine maxθR(θ, δ)ifitexists.
7.1.8. LetX 1 ,X 2 ,...,Xndenote a random sample from a distribution that is
b(1,θ), 0 ≤θ≤1. LetY =
∑n
1 Xiand letL[θ, δ(y)] = [θ−δ(y)]
(^2) .Consider
decision functions of the formδ(y)=by,wherebdoes not depend upony.Prove
thatR(θ, δ)=b^2 nθ(1−θ)+(bn−1)^2 θ^2. Show that
max
θ
R(θ, δ)=
b^4 n^2
4[b^2 n−(bn−1)^2 ]
,
provided that the valuebis such thatb^2 n>(bn−1)^2 .Provethatb=1/ndoes not
minimize maxθR(θ, δ).
7.1.9. LetX 1 ,X 2 ,...,Xnbe a random sample from a Poisson distribution with
meanθ>0.
(a)StatisticianAobserves the sample to be the valuesx 1 ,x 2 ,...,xnwith sum
y=
∑
xi.Findthemleofθ.
(b)StatisticianBloses the sample valuesx 1 ,x 2 ,...,xnbut remembers the sum
y 1 and the fact that the sample arose from a Poisson distribution. Thus
Bdecides to create some fake observations, which he callsz 1 ,z 2 ,...,zn(as