Page 301 - Fundamentals of Probability and Statistics for Engineers

284 Fundamentals of Probability and Statistics for Engineers

X
X 2

r X 1

Figure 9.3 Measurement X, for Example 9.13

moments of X can be directly generated from Equation (9.77). We remark that,
although an estimator for 2 is not required, it is nevertheless an unknown

parameter and must be considered together with . In the applied literature, an
unknown parameter for which the value is of no interest is sometimes referred
to as a nuisance parameter.
Two moment equations are needed in this case. However, we see from
Equation (9.77) that the odd-order moments of X are quite complicated. For
simplicity, the second-order and fourth-order moment equations will be used.
We easily obtain from Equation (9.77)
2
2 ˆ ‡ 2 ; )
9:78†
4
2
2
4 ˆ ‡ 8 ‡ 8 :
The two moment equations are
^
‡ 2 ˆ M 2 ; )
c 2
9:79†
^ 2
2
‡ 8 ‡ 8 2 2 ˆ M 4 :
c
^ c
^
Solving for ,we have
^
2
ˆ 2M M 4 † 1=2 : 9:80†
2
Incidentally, a moment estimator 2 for 2 , if needed, is obtained from Equa-
c
tions (9.79) to be
1
^
ˆ M 2 †: 9:81†
c 2
2
Combined Moment Estimators. Let us take another look at Example 9.11 for
the purpose of motivating the following development. In this example, an
estimator for has been obtained by using the first-order moment equation.
Based on the same sample, one can obtain additional moment estimators for
by using higher-order moment equations. For example, since 2 ˆ 2/ 2 ,the
second-order moment equation,
2 ˆ M 2 ;

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