Introduction to Probability and Statistics for Engineers and Scientists

*7.7Evaluating a Point Estimator 271

Therefore,

E[d 2 ]=

∫θ

0

x

nxn−^1

θn

dx=

n

n+ 1

θ (7.7.1)

Also

E[d 22 ]=

∫θ

0

x^2

nxn−^1

θn

dx=

n

n+ 2

θ^2

and so

Var(d 2 )=

n

n+ 2

θ^2 −

(

n

n+ 1

θ

) 2

(7.7.2)

=nθ^2

[

1

n+ 2

−

n

(n+1)^2

]

=

nθ^2

(n+2)(n+1)^2

Hence

r(d 2 ,θ)=(E(d 2 )−θ)^2 +Var(d 2 ) (7.7.3)

=

θ^2

(n+1)^2

+

nθ^2

(n+2)(n+1)^2

=

θ^2

(n+1)^2

[

1 +

n

n+ 2

]

=

2 θ^2

(n+1)(n+2)

Since
2 θ^2
(n+1)(n+2)

≤

θ^2

3 n

n=1, 2,...

it follows thatd 2 is a more superior estimator ofθthan isd 1.
Equation 7.7.1 suggests the use of even another estimator — namely, the unbiased
estimator (1+1/n)d 2 (X)=(1+1/n) maxiXi. However, rather than considering this
estimator directly, let us consider all estimators of the form

dc(X)=cmax

i

Xi=cd 2 (X)

wherecis a given constant. The mean square error of this estimator is

r(dc(X),θ)= Var(dc(X))+(E[dc(X)]−θ)^2

=c^2 Var(d 2 (X))+(cE[d 2 (X)]−θ)^2

=

c^2 nθ^2

(n+2)(n+1)^2

+θ^2

(

cn

n+ 1

− 1

) 2

by Equations 7.7.2 and 7.7.1 (7.7.4)