7.5. The Exponential Class of Distributions 435(b)LetY=|X|. Show thatYis a complete and sufficient statistic forθ.7.4.9. LetX 1 ,...,Xnbe iid with pdff(x;θ)=1/(3θ),−θ<x< 2 θ, zero else-
where, whereθ>0.
(a)Find the mlêθofθ.(b)Isθ̂a sufficient statistic forθ?Why?(c)Is (n+1)θ/n̂ the unique MVUE ofθ?Why?7.4.10.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample of sizen
from a distribution with pdff(x;θ)=1/θ, 0 <x<θ, zero elsewhere. By Example
7.4.2, the statisticYnis a complete sufficient statistic forθand it has pdf
g(yn;θ)=nynn−^1
θn, 0 <yn<θ,and zero elsewhere.(a)Find the distribution functionHn(z;θ)ofZ=n(θ−Yn).(b)Find the limn→∞Hn(z;θ) and thus the limiting distribution ofZ.7.5 TheExponentialClassofDistributions.................
In this section we discuss an important class of distributions, called theexponential
class. As we show, this class possesses complete and sufficient statistics which are
readily determined from the distribution.
Consider a family{f(x;θ):θ∈Ω}of probability density or mass functions,
where Ω is the interval set Ω ={θ:γ<θ<δ},whereγandδare known constants
(they may be±∞), and where
f(x;θ)={
exp[p(θ)K(x)+H(x)+q(θ)] x∈S
0elsewhere,
(7.5.1)whereSis the support ofX. In this section we are concerned with a particular
class of the family called the regular exponential class.
Definition 7.5.1(Regular Exponential Class). A pdf of the form (7.5.1) is said
to be a member of theregular exponential classof probability density or mass
functions if
1.S, the support ofX, does not depend uponθ2.p(θ)is a nontrivial continuous function ofθ∈Ω- Finally,
(a) ifXis a continuous random variable, then each ofK′(x)
≡ 0 andH(x)
is a continuous function ofx∈S,