Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

466 Sufficiency

7.9.4.LetXandY be random variables such thatE(Xk)andE(Yk) =0exist
fork=1, 2 , 3 ,....IftheratioX/Yand its denominatorYare independent, prove
thatE[(X/Y)k]=E(Xk)/E(Yk),k=1, 2 , 3 ,....
Hint: WriteE(Xk)=E[Yk(X/Y)k].
7.9.5.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample of sizen
from a distribution that has pdff(x;θ)=(1/θ)e−x/θ, 0 <x<∞, 0 <θ<∞, zero
elsewhere. Show that the ratioR=nY 1 /

∑n
1 Yiand its denominator (a complete
sufficient statistic forθ) are independent. Use the result of the preceding exercise
to determineE(Rk),k=1, 2 , 3 ,....
7.9.6.LetX 1 ,X 2 ,...,X 5 be iid with pdff(x)=e−x, 0 <x<∞, zero elsewhere.
Show that (X 1 +X 2 )/(X 1 +X 2 +···+X 5 ) and its denominator are independent.
Hint: The pdff(x)isamemberof{f(x;θ):0<θ<∞},wheref(x;θ)=
(1/θ)e−x/θ, 0 <x<∞, zero elsewhere.

7.9.7.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample from the
normal distributionN(θ 1 ,θ 2 ),−∞<θ 1 <∞, 0 <θ 2 <∞. Show that the joint
complete sufficient statisticsX=YandS^2 forθ 1 andθ 2 are independent of each
of (Yn−Y)/Sand (Yn−Y 1 )/S.

7.9.8.LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample from a
distribution with the pdf

f(x;θ 1 ,θ 2 )=

1

θ 2

exp

(

−

x−θ 1

θ 2

)

,

θ 1 <x<∞, zero elsewhere, where−∞<θ 1 <∞, 0 <θ 2 <∞. Show that the
joint complete sufficient statisticsY 1 andX=Y for the parametersθ 1 andθ 2 are
independent of (Y 2 −Y 1 )/

∑n
1 (Yi−Y^1 ).
7.9.9.LetX 1 ,X 2 ,...,X 5 be a random sample of sizen= 5 from the normal
distributionN(0,θ).

(a)Argue that the ratioR=(X 12 +X 22 )/(X 12 +···+X 52 ) and its denominator

(X^21 +···+X 52 ) are independent.

(b)Does 5R/2haveanF-distribution with 2 and 5 degrees of freedom? Explain

your answer.

(c)ComputeE(R) using Exercise 7.9.4.

7.9.10.Referring to Example 7.9.5 of this section, determinecso that

P(−c<T 1 −θ<c|T 2 =t 2 )=0. 95.

Use this result to find a 95% confidence interval forθ,givenT 2 =t 2 ; and note how
its length is smaller when the range oft 2 is larger.

7.9.11.Show thatY=|X|is a complete sufficient statistic forθ>0, whereXhas
the pdffX(x;θ)=1/(2θ), for−θ<x<θ, zero elsewhere. Show thatY=|X|and
Z=sgn(X) are independent.