Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

426 Sufficiency

7.2.8.What is the sufficient statistic forθif the sample arises from a beta distri-

bution in whichα=β=θ>0?

7.2.9.We consider a random sampleX 1 ,X 2 ,...,Xnfrom a distribution with pdf
f(x;θ)=(1/θ)exp(−x/θ), 0 <x<∞, zero elsewhere, where 0<θ. Possibly, in a
life-testing situation, however, we only observe the firstrorder statisticsY 1 <Y 2 <
···<Yr.

(a)Record the joint pdf of these order statistics and denote it byL(θ).

(b)Under these conditions, find the mle,θˆ, by maximizingL(θ).

(c)Find the mgf and pdf ofθˆ.

(d)With a slight extension of the definition of sufficiency, isˆθa sufficient statistic?

7.3 PropertiesofaSufficientStatistic....................

SupposeX 1 ,X 2 ,...,Xnis a random sample on a random variable with pdf or pmf
f(x;θ), whereθ∈Ω. In this section we discuss how sufficiency is used to determine
MVUEs. First note that a sufficient estimate is not unique in any sense. For if
Y 1 =u 1 (X 1 ,X 2 ,...,Xn) is a sufficient statistic andY 2 =g(Y 1 ) is a statistic, where
g(x) is a one-to-one function, then

f(x 1 ;θ)f(x 2 ;θ)···f(xn;θ)=k 1 [u 1 (y 1 );θ]k 2 (x 1 ,x 2 ,...,xn)

= k 1 [u 1 (g−^1 (y 2 ));θ]k 2 (x 1 ,x 2 ,...,xn);

hence, by the factorization theorem,Y 2 is also sufficient. However, as the theorem
below shows, sufficiency can lead to a best point estimate.
We first refer back to Theorem 2.3.1 of Section 2.3: IfX 1 andX 2 are random
variables such that the variance ofX 2 exists, then

E[X 2 ]=E[E(X 2 |X 1 )]

and

Var(X 2 )≥Var[E(X 2 |X 1 )].

For the adaptation in the context of sufficient statistics, we let the sufficient statistic
Y 1 beX 1 andY 2 ,anunbiasedstatisticofθ,beX 2 .Thus,withE(Y 2 |y 1 )=φ(y 1 ),
we have

θ=E(Y 2 )=E[φ(Y 1 )]

and

Var(Y 2 )≥Var[φ(Y 1 )].