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7-2 GENERAL CONCEPTS OF POINT ESTIMATION 227

^
Θ
Distribution of 1
Figure 7-2 A biased
ˆ
estimator 1 that has ^
Θ
smaller variance than Distribution of 2
the unbiased estimator
ˆ
^
. 2 θ E( )
Θ 1
ˆ

That is, the mean square error of is equal to the variance of the estimator plus the squared bias.
ˆ
ˆ
ˆ
If is an unbiased estimator of , the mean square error of is equal to the variance of .

ˆ
The mean square error is an important criterion for comparing two estimators. Let 1
ˆ ˆ ˆ
and 2 be two estimators of the parameter , and let MSE ( 1 ) and MSE ( 2 ) be the mean
ˆ
ˆ ˆ ˆ to
square errors of 1 and 2 . Then the relative efficiency of 2 1 is defined as
ˆ
MSE1 2
1
ˆ
MSE1 2 (7-4)
2
ˆ
If this relative efficiency is less than 1, we would conclude that 1 is a more efficient estima-
ˆ
tor of than 2 , in the sense that it has a smaller mean square error.
Sometimes we find that biased estimators are preferable to unbiased estimators because they
have smaller mean square error. That is, we may be able to reduce the variance of the estimator
considerably by introducing a relatively small amount of bias. As long as the reduction in variance
is greater than the squared bias, an improved estimator from a mean square error viewpoint will
ˆ
result. For example, Fig. 7-2 shows the probability distribution of a biased estimator 1 that has
ˆ
ˆ
a smaller variance than the unbiased estimator 2 . An estimate based on 1 would more likely
ˆ
be close to the true value of than would an estimate based on 2 . Linear regression analysis
(Chapters 11 and 12) is an area in which biased estimators are occasionally used.
ˆ
An estimator that has a mean square error that is less than or equal to the mean square
error of any other estimator, for all values of the parameter , is called an optimal estimator
of . Optimal estimators rarely exist.
EXERCISES FOR SECTION 7-2
7-1. Suppose we have a random sample of size 2n from a (a) Is either estimator unbiased?
population denoted by X, and E1X 2 and V1X 2 2 . Let (b) Which estimator is best? In what sense is it best?
ˆ
ˆ
7-3. Suppose that 1 and 2 are unbiased estimators of the
ˆ
1 2n 1 n parameter . We know that V1 1 2 10 ˆ and V1 2 2 4 .
X 1 a X i and X 2 n a X i

2n i 1 i 1 Which estimator is best and in what sense is it best?
7-4. Calculate the relative efficiency of the two estimators
be two estimators of . Which is the better estimator of ? in Exercise 7-2.
Explain your choice. 7-5. Calculate the relative efficiency of the two estimators
7-2. Let X 1 , X 2 , p , X 7 denote a random sample from a in Exercise 7-3.
ˆ
ˆ
population having mean and variance 2 . Consider the 7-6. Suppose that 1 and 2 are estimators of the parame-
ˆ
ˆ
ˆ
following estimators of : ter . We know that E1 1 2 , E1 2 2 2, V 1 1 2 10,
ˆ
V 1 2 2 4 . Which estimator is best? In what sense is it best?
X 1 X 2 p X 7 ˆ ˆ ˆ
ˆ
1 7-7. Suppose that 1 , 2 , and 3 are estimators of . We
ˆ
ˆ
ˆ
7 know that E1 1 2 E1 2 2 , E 1 3 2 , V1 1 2 12,
ˆ
ˆ
2
ˆ
V 1 2 2 10 , and E1 3 2 6 . Compare these three esti-
ˆ
2 2X 1 X 6 X 4 mators. Which do you prefer? Why?
2