*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)