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224 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

Sometimes there are several unbiased estimators of the sample population parameter. For
example, suppose we take a random sample of size n 10 from a normal population and
obtain the data x 12.8, x 9.4, x 8.7, x 11.6, x 13.1, x 9.8, x 14.1,
2
1
5
6
7
4
3
x 8 8.5, x 9 12.1, x 10 10.3. Now the sample mean is
12.8 9.4 8.7 11.6 13.1 9.8 14.1 8.5 12.1 10.3
x
10
11.04
the sample median is
10.3 11.6
~ 10.95
x
2
and a 10% trimmed mean (obtained by discarding the smallest and largest 10% of the sample
before averaging) is

8.7 9.4 9.8 10.3 11.6 12.1 12.8 13.1
x tr1102
8
10.98

We can show that all of these are unbiased estimates of . Since there is not a unique unbiased
estimator, we cannot rely on the property of unbiasedness alone to select our estimator. We
need a method to select among unbiased estimators. We suggest a method in Section 7-2.3.

7-2.2 Proof That S is a Biased Estimator of (CD Only)

7-2.3 Variance of a Point Estimator

ˆ
ˆ
Suppose that 1 and 2 are unbiased estimators of . This indicates that the distribution of
each estimator is centered at the true value of . However, the variance of these distributions
ˆ
ˆ
may be different. Figure 7-1 illustrates the situation. Since 1 has a smaller variance than ,
2
ˆ
the estimator 1 is more likely to produce an estimate close to the true value . A logical prin-
ciple of estimation, when selecting among several estimators, is to choose the estimator that
has minimum variance.
Definition
If we consider all unbiased estimators of , the one with the smallest variance is
called the minimum variance unbiased estimator (MVUE).

^
Θ
Distribution of
1
Figure 7-1 The ^
Distribution of Θ
sampling distributions 2
of two unbiased estima-
ˆ ˆ θ
tors 1 and 2 .