*7.7Evaluating a Point Estimator 269

to the conservation organization, then they should estimate the acidity of the sampled
water from the lake by

d=

∑n

i= 1

di/σi^2

∑n

i= 1

1/σi^2

The mean square error ofdis as follows:

r(d,θ)=Var(d) sincedis unbiased

=

(n

∑

i= 1

1/σi^2

)− (^2) n
∑
i= 1
(
1
σi^2
) 2
σi^2

1
∑n
i= 1
1/σi^2
■
A generalization of the result that the mean square error of an unbiased estimator is
equal to its variance is that the mean square error of any estimator is equal to its variance
plus the square of its bias. This follows since
r(d,θ)=E[(d(X)−θ)^2 ]
=E[(d−E[d]+E[d]−θ)^2 ]
=E[(d−E[d])^2 +(E[d]−θ)^2 +2(E[d]−θ)(d−E[d])]
=E[(d−E[d])^2 ]+E[(E[d]−θ)^2 ]

• 2 E[(E[d]−θ)(d−E[d])]
  =E[(d−E[d])^2 ]+(E[d]−θ)^2 +2(E[d]−θ)E[d−E[d]]
  sinceE[d]−θis constant
  =E[(d−E[d])^2 ]+(E[d]−θ)^2
  The last equality follows since
  E[d−E[d]] = 0
  Hence
  r(d,θ)=Var(d)+bθ^2 (d)