Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.5. The Exponential Class of Distributions 439

7.5.3.LetX 1 ,X 2 ,...,Xndenote a random sample of sizenfrom a distribution

with pdff(x;θ)=θxθ−^1 , 0 <x<1, zero elsewhere, andθ>0.

(a)Show that thegeometric mean(X 1 X 2 ···Xn)^1 /nof the sample is a complete

sufficient statistic forθ.

(b)Find the maximum likelihood estimator ofθ, and observe that it is a function

of this geometric mean.

7.5.4.LetXdenote the mean of the random sampleX 1 ,X 2 ,...,Xnfrom a gamma-
type distribution with parametersα>0andβ=θ>0. ComputeE[X 1 |x].
Hint: Can you find directly a functionψ(X)ofXsuch thatE[ψ(X)] =θ?Is
E(X 1 |x)=ψ(x)? Why?

7.5.5.LetXbe a random variable with the pdf of a regular case of the exponential
class, given byf(x;θ)=exp[θK(x)+H(x)+q(θ)], a<x<b, γ<θ<δ. Show
thatE[K(X)] =−q′(θ)/p′(θ), provided these derivatives exist, by differentiating
both members of the equality

∫b

a

exp[p(θ)K(x)+H(x)+q(θ)]dx=1

with respect toθ. By a second differentiation, find the variance ofK(X).

7.5.6. Given thatf(x;θ)=exp[θK(x)+H(x)+q(θ)], a<x<b, γ<θ<δ,
represents a regular case of the exponential class, show that the moment-generating
functionM(t)ofY=K(X)isM(t)=exp[q(θ)−q(θ+t)],γ<θ+t<δ.

7.5.7.In the preceding exercise, given thatE(Y)=E[K(X)] =θ,provethatY is
N(θ,1).
Hint: ConsiderM′(0) =θand solve the resulting differential equation.

7.5.8.IfX 1 ,X 2 ,...,Xnis a random sample from a distribution that has a pdf which
is a regular case of the exponential class, show that the pdf ofY 1 =

∑n
1 K(Xi)is
of the formfY 1 (y 1 ;θ)=R(y 1 )exp[p(θ)y 1 +nq(θ)].
Hint: LetY 2 =X 2 ,...,Yn=Xnben−1 auxiliary random variables. Find the
joint pdf ofY 1 ,Y 2 ,...,Ynand then the marginal pdf ofY 1.

7.5.9.LetY denote the median and letXdenote the mean of a random sample of
sizen=2k+ 1 from a distribution that isN(μ, σ^2 ). ComputeE(Y|X=x).
Hint: See Exercise 7.5.4.

7.5.10. LetX 1 ,X 2 ,...,Xn be a random sample from a distribution with pdf
f(x;θ)=θ^2 xe−θx, 0 <x<∞,whereθ>0.

(a)Argue thatY=

∑n

1 Xiis a complete sufficient statistic forθ.

(b)ComputeE(1/Y) and find the function ofYthat is the unique MVUE ofθ.

7.5.11.LetX 1 ,X 2 ,...,Xn,n>2, be a random sample from the binomial distri-
butionb(1,θ).