Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

7.6. Functions of a Parameter 441

NowE(Y)=nθandE(Y^2 )=nθ(1−θ)+n^2 θ^2. Hence

E

[

Y

n

(

1 −

Y

n

)]

=(n−1)

θ(1−θ)

n

.

If we multiply both members of this equation byn/(n−1), we find that the statistic
δˆ=(n/(n−1))(Y/n)(1−Y/n)=(n/(n−1)) ̃δis the unique MVUE ofδ. Hence
the MVUE ofδ/n, the variance ofY/n,isδ/nˆ.

It is interesting to compare the mle ̃δwithˆδ. Recall from Chapter 6 that the
mleδ ̃is a consistent estimate ofδand that

√

n(δ ̃−δ) is asymptotically normal.

Because

ˆδ−δ ̃=δ ̃^1

n− 1

P

→δ·0=0,

it follows thatˆδis also a consistent estimator ofδ.Further,

√

n(δˆ−δ)−

√

n(δ ̃−δ)=

√

n

n− 1

̃δ→P 0. (7.6.1)

Hence

√

n(δˆ−δ) has the same asymptotic distribution as

√

n(δ ̃−δ). Using the

Δ-method, Theorem 5.2.9, we can obtain the asymptotic distribution of

√

n(δ ̃−δ).

Letg(θ)=θ(1−θ). Theng′(θ)=1− 2 θ. Hence, by Theorem 5.2.9 and (7.6.1), the

asymptotic distribution of

√

n(δ ̃−δ)isgivenby

√

n(δˆ−δ)

D

→N(0,θ(1−θ)(1− 2 θ)^2 ),

providedθ =1/2; see Exercise 7.6.12 for the caseθ=1/2.

In the next example, we consider the uniform (0,θ) distribution and obtain the
MVUE for all differentiable functions ofθ. This example was sent to us by Professor
Bradford Crain of Portland State University.

Example 7.6.2.SupposeX 1 ,X 2 ,...,Xnare iid random variables with the com-
mon uniform (0,θ) distribution. LetYn=max{X 1 ,X 2 ,...,Xn}. In Example 7.4.2,
we showed thatYnis a complete and sufficient statistic ofθand the pdf ofYnis
given by (7.4.1). Letg(θ) be any differentiable function ofθ. Then the MVUE of
g(θ)isthestatisticu(Yn), which satisfies the equation

g(θ)=

∫θ

0

u(y)

nyn−^1

θn

dy, θ > 0 ,

or equivalently,

g(θ)θn=

∫θ

0

u(y)nyn−^1 dy, θ > 0.

Differentiating both sides of this equation with respect toθ,weobtain

nθn−^1 g(θ)+θng′(θ)=u(θ)nθn−^1.