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7-3 METHODS OF POINT ESTIMATION 229

7-3 METHODS OF POINT ESTIMATION

The definitions of unbiasness and other properties of estimators do not provide any guidance
about how good estimators can be obtained. In this section, we discuss two methods for ob-
taining point estimators: the method of moments and the method of maximum likelihood.
Maximum likelihood estimates are generally preferable to moment estimators because they
have better efficiency properties. However, moment estimators are sometimes easier to com-
pute. Both methods can produce unbiased point estimators.

7-3.1 Method of Moments

The general idea behind the method of moments is to equate population moments, which are
defined in terms of expected values, to the corresponding sample moments. The population
moments will be functions of the unknown parameters. Then these equations are solved to
yield estimators of the unknown parameters.

Definition
Let X , X , p , X n be a random sample from the probability distribution f(x), where
1
2
f(x) can be a discrete probability mass function or a continuous probability density
k
function. The kth population moment (or distribution moment) is E(X ), k
n k
1, 2, p . The corresponding kth sample moment is 11 n2 g i 1 X i , k 1, 2, p .

To illustrate, the first population moment is E(X) , and the first sample moment is
n
X X
11 n2 g i 1 i . Thus by equating the population and sample moments, we find that
X. That is, the sample mean is the moment estimator of the population mean. In the
ˆ
general case, the population moments will be functions of the unknown parameters of the dis-
, , p , .
tribution, say, 1 2 m
Definition
, X , p , X be a random sample from either a probability mass function
Let X 1 2 n
or probability density function with m unknown parameters , , p , . The
m
2
1
ˆ
ˆ
ˆ , , p ,
moment estimators 1 2 m are found by equating the first m population
moments to the first m sample moments and solving the resulting equations for the
unknown parameters.
EXAMPLE 7-3 Suppose that X , X , p , X n is a random sample from an exponential distribution with param-
1
2

eter . Now there is only one parameter to estimate, so we must equate E(X) to . For the
X
ˆ
exponential, E1X 2 1 . Therefore E1X 2 X results in 1 X, so 1 X is the

moment estimator of .
As an example, suppose that the time to failure of an electronic module used in an automobile
engine controller is tested at an elevated temperature to accelerate the failure mechanism.