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7-2 GENERAL CONCEPTS OF POINT ESTIMATION 223

Definition
ˆ
The point estimator is an unbiased estimator for the parameter if
ˆ
E1 2 (7-1)
If the estimator is not unbiased, then the difference

ˆ
E1 2 (7-2)
ˆ
is called the bias of the estimator .

ˆ
When an estimator is unbiased, the bias is zero; that is, E1 2 0.
EXAMPLE 7-1 Suppose that X is a random variable with mean and variance 2 . Let X , X , p , X n be a
2
1
random sample of size n from the population represented by X. Show that the sample mean X
and sample variance S 2 are unbiased estimators of and 2 , respectively.
First consider the sample mean. In Equation 5.40a in Chapter 5, we showed that E1X 2 .
Therefore, the sample mean X is an unbiased estimator of the population mean .
Now consider the sample variance. We have

n
a 1X X 2 2 n
i
2
E1S 2 E £ i 1 § 1 E a 1X X 2 2
i
n 1 n 1
i 1
1 n 1 n
2
2
2
2
E a 1X i X 2X X 2 E a a X i nX b
i
n 1 i 1 n 1 i 1
1 n
2
2
c a E1X i 2 nE1X 2d
n 1
i 1
2
2
The last equality follows from Equation 5-37 in Chapter 5. However, since E1X i 2 2
2
2
2
and E1X 2 n, we have
1 n
2
2
2
2
2
E1S 2 c a 1 2 n1 n2d
n 1
i 1
1
2
2
2
2
1n n n 2
n 1
2
2
Therefore, the sample variance S 2 is an unbiased estimator of the population variance .
2
Although S 2 is unbiased for , S is a biased estimator of . For large samples, the bias is very
small. However, there are good reasons for using S as an estimator of in samples from nor-
mal distributions, as we will see in the next three chapters when are discuss confidence
intervals and hypothesis testing.