Applied Statistics and Probability for Engineers

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 from a normal population and

obtain the data x 1 12.8, x 2 9.4, x 3 8.7, x 4 11.6, x 5 13.1, x 6 9.8, x 7 14.1,

x 8 8.5, x 9 12.1, x 10 10.3. Now the sample mean is

the sample median is

and a 10% trimmed mean (obtained by discarding the smallest and largest 10% of the sample

before averaging) is

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 Sis a Biased Estimator of  (CD Only)

7-2.3 Variance of a Point Estimator

Suppose that and 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

may be different. Figure 7-1 illustrates the situation. Since has a smaller variance than

the estimator 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.

ˆ 1

ˆ 1 ˆ 2 ,

ˆ 1 ˆ 2

10.98

xtr 1102 

8.79.49.810.311.612.112.813.1

8

x~

10.311.6

2

10.95

11.04

x

12.89.48.711.613.19.814.18.512.110.3

10

n 10

If we consider all unbiased estimators of , the one with the smallest variance is

called the minimum variance unbiased estimator(MVUE).

Definition

θ

Distribution of Θ^ 1

Figure 7-1 The Distribution of Θ^ 2

sampling distributions

of two unbiased estima-

tors and. ˆ 1 ˆ 2

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