When an estimator is unbiased, the bias is zero; that is,
EXAMPLE 7-1 Suppose that Xis a random variable with mean and variance. Let be a
random sample of size nfrom the population represented by X. Show that the sample mean
and sample variance are unbiased estimators of and , respectively.
First consider the sample mean. In Equation 5.40a in Chapter 5, we showed that
Therefore, the sample mean is an unbiased estimator of the population mean .
Now consider the sample variance. We have
The last equality follows from Equation 5-37 in Chapter 5. However, since
and we have
Therefore, the sample variance is an unbiased estimator of the population variance
Although is unbiased for ^2 , Sis a biased estimator of . For large samples, the bias is very
small. However, there are good reasons for using Sas 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.
S^2
S^2 ^2.
^2
1
n 1
1 n^2 n^2 n^2 ^22
E 1 S^22
1
n 1
ca
n
i 1
1 ^2 ^22 n 1 ^2 ^2 n2d
E 1 X^22 ^2 ^2 n,
E 1 X^2 i 2 ^2 ^2
1
n 1
ca
n
i 1
E 1 X^2 i 2 nE 1 X^2 2d
1
n 1
E (^) a
n
i 1
1 X^2 iX^2 2 XXi 2
1
n 1
E aa
n
i 1
X^2 inX^2 b
E 1 S^22 E £
a
n
i 1
1 XiX 22
n 1
§
1
n 1
E (^) a
n
i 1
1 XiX 22
X
E 1 X 2 .
S^2 ^2
X
^2 X 1 , X 2 ,p, Xn
E 1 ˆ 2 0.
The point estimator is an unbiased estimatorfor the parameter if
(7-1)
If the estimator is not unbiased, then the difference
(7-2)
is called the biasof the estimator .ˆ
E 1 ˆ 2
E 1 ˆ 2
ˆ
Definition
7-2 GENERAL CONCEPTS OF POINT ESTIMATION 223
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