446 Sufficiency(a)Find the MVUE ofP(X≤1) = (1 +θ)e−θ.
Hint: Letu(x 1 )=1,x 1 ≤1, zero elsewhere, and findE[u(X 1 )|Y =y],
whereY=∑n
1 Xi.
(b)Express the MVUE as a function of the mle ofθ.(c)Determine the asymptotic distribution of the mle ofθ.(d)Obtain the mle of P(X ≤1). Then use Theorem 5.2.9 to determine its
asymptotic distribution.7.6.7.LetX 1 ,X 2 ,...,Xndenote a random sample from a Poisson distribution
with parameterθ>0. From Remark 7.6.1, we know thatE[(−1)X^1 ]=e−^2 θ.
(a)Show thatE[(−1)X^1 |Y 1 =y 1 ]=(1− 2 /n)y^1 ,whereY 1 =X 1 +X 2 +···+Xn.
Hint: First show that the conditional pdf ofX 1 ,X 2 ,...,Xn− 1 ,givenY 1 =y 1 ,
is multinomial, and hence that ofX 1 ,givenY 1 =y 1 ,isb(y 1 , 1 /n).(b)Show that the mle ofe−^2 θise−^2 X.(c)Sincey 1 =nx, show that (1− 2 /n)y^1 is approximately equal toe−^2 xwhenn
is large.7.6.8.As in Example 7.6.3, letX 1 ,X 2 ,...,Xnbe a random sample of sizen> 1
from a distribution that isN(θ,1). Show that the joint distribution ofX 1 andX
is bivariate normal with mean vector (θ, θ), variancesσ 12 =1andσ 22 =1/n,and
correlation coefficientρ=1/
√
n.7.6.9.Let a random sample of sizenbe taken from a distribution that has the pdf
f(x;θ)=(1/θ)exp(−x/θ)I(0,∞)(x). Find the mle and MVUE ofP(X≤2).
7.6.10. LetX 1 ,X 2 ,...,Xnbe a random sample with the common pdff(x)=
θ−^1 e−x/θ,forx>0, zero elsewhere; that is,f(x)isaΓ(1,θ)pdf.
(a)Show that the statisticX=n−^1∑n
i=1Xiis a complete and sufficient statistic
forθ.(b)Determine the MVUE ofθ.(c)Determinethemleofθ.(d)Often, though, this pdf is written asf(x)=τe−τx,forx>0, zero elsewhere.
Thusτ=1/θ. Use Theorem 6.1.2 to determine the mle ofτ.(e)Show that the statisticX=n−^1∑n
i=1Xiis a complete and sufficient statistic
forτ. Show that (n−1)/(nX) is the MVUE ofτ=1/θ. Hence, as usual,
the reciprocal of the mle ofθis the mle of 1/θ, but, in this situation, the
reciprocal of the MVUE ofθis not the MVUE of 1/θ.(f)Compute the variances of each of the unbiased estimators in parts (b) and
(e).