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226 CHAPTER 7 POINT ESTIMATION OF PARAMETERS

estimate. Since many point estimators are normally distributed (or approximately so) for large
n, this is a very useful result. Even in cases in which the point estimator is not normally
distributed, we can state that so long as the estimator is unbiased, the estimate of the parameter
will deviate from the true value by as much as four standard errors at most 6 percent of the time.
Thus a very conservative statement is that the true value of the parameter differs from the point
estimate by at most four standard errors. See Chebyshev’s inequality in the CD only material.

EXAMPLE 7-2 An article in the Journal of Heat Transfer (Trans. ASME, Sec. C, 96, 1974, p. 59) described
a new method of measuring the thermal conductivity of Armco iron. Using a temperature of
100 F and a power input of 550 watts, the following 10 measurements of thermal conductiv-
ity (in Btu/hr-ft- F) were obtained:

41.60, 41.48, 42.34, 41.95, 41.86,
42.18, 41.72, 42.26, 41.81, 42.04

A point estimate of the mean thermal conductivity at 100 F and 550 watts is the sample mean or

x 41.924 Btu/hr-ft- F

The standard error of the sample mean is 1n , and since is unknown, we may replace
X
it by the sample standard deviation s 0.284 to obtain the estimated standard error of X as
s 0.284
ˆ 0.0898
X
1n 110
Notice that the standard error is about 0.2 percent of the sample mean, implying that we have ob-
tained a relatively precise point estimate of thermal conductivity. If we can assume that thermal
conductivity is normally distributed, 2 times the standard error is 2 ˆ 210.08982 0.1796,
X
and we are highly confident that the true mean thermal conductivity is with the interval
41.924
0.1756 , or between 41.744 and 42.104.

7-2.5 Bootstrap Estimate of the Standard Error (CD Only)

7-2.6 Mean Square Error of an Estimator

Sometimes it is necessary to use a biased estimator. In such cases, the mean square error of the
ˆ
estimator can be important. The mean square error of an estimator is the expected squared
ˆ
difference between and .
Definition
ˆ
The mean square error of an estimator of the parameter is defined as
ˆ
ˆ
MSE1 2 E1 2 2 (7-3)

The mean square error can be rewritten as follows:
ˆ
ˆ
ˆ
ˆ
2
MSE1 2 E3 E1 24 3 E1 24 2
ˆ
V1 2 1bias2 2