434 Sufficiency(b)N(0,θ), where 0<θ<∞.7.4.3.LetX 1 ,X 2 ,...,Xnrepresent a random sample from the discrete distribution
having the pmf
f(x;θ)={
θx(1−θ)^1 −x x=0, 1 , 0 <θ< 1
0elsewhere.Show thatY 1 =∑n
1 Xiis a complete sufficient statistic forθ. Find the unique
function ofY 1 that is the MVUE ofθ.
Hint: DisplayE[u(Y 1 )] = 0, show that the constant termu(0) is equal to zero,
divide both members of the equation byθ = 0, and repeat the argument.
7.4.4.Consider the family of probability density functions{h(z;θ):θ∈Ω},where
h(z;θ)=1/θ, 0 <z<θ, zero elsewhere.(a)Show that the family is complete provided that Ω ={θ:0<θ<∞}.
Hint: For convenience, assume thatu(z) is continuous and note that the
derivative ofE[u(Z)] with respect toθis equal to zero also.(b)Show that this family is not complete if Ω ={θ:1<θ<∞}.
Hint: Concentrate on the interval 0<z<1 and find a nonzero function
u(z) on that interval such thatE[u(Z)] = 0 for allθ>1.7.4.5. Show that the first order statisticY 1 of a random sample of sizenfrom
the distribution having pdff(x;θ)=e−(x−θ),θ<x<∞,−∞<θ<∞, zero
elsewhere, is a complete sufficient statistic forθ. Find the unique function of this
statistic which is the MVUE ofθ.
7.4.6.Let a random sample of sizenbe taken from a distribution of the discrete
type with pmff(x;θ)=1/θ, x=1, 2 ,...,θ, zero elsewhere, whereθis an unknown
positive integer.
(a)Show that the largest observation, sayY, of the sample is a complete sufficient
statistic forθ.(b)Prove that
[Yn+1−(Y−1)n+1]/[Yn−(Y−1)n]
is the unique MVUE ofθ.7.4.7.LetXhave the pdffX(x;θ)=1/(2θ), for−θ<x<θ, zero elsewhere, where
θ>0.
(a)Is the statisticY=|X|a sufficient statistic forθ?Why?(b)LetfY(y;θ)bethepdfofY. Is the family{fY(y;θ):θ> 0 }complete? Why?7.4.8.LetXhave the pmfp(x;θ)=^12(n
|x|)
θ|x|(1−θ)n−|x|,forx=± 1 ,± 2 ,...,±n,
p(0,θ)=(1−θ)n, and zero elsewhere, where 0<θ<1.(a)Show that this family{p(x;θ):0<θ< 1 }is not complete.