Page 296 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 279

Let us remark that it is not necessary to consider m consecutive moment
equations as indicated by Equations (9.58); any convenient set of m equations that
^
lead to the solution for j , j ˆ 1, .. . , m, is sufficient. Lower-order moment equa-
tions are preferred, however, since they require less manipulation of observed data.
An attractive feature of the method of moments is that the moment equations
are straightforward to establish, and there is seldom any difficulty in solving
them. However, a shortcoming is that such desirable properties as unbiasedness
or efficiency are not generally guaranteed for estimators so obtained.
However, consistency of moment estimators can be established under general
conditions. In order to show this, let us consider a single parameter whose
^
moment estimator satisfies the moment equation
^
i †ˆ M i ; 9:59†
for some i. The solution of Equation (9.59) for ^ can be represented by
^
^
= (M i ), for which the Taylor’s expansion about i ) gives
^ 2†
^
^ 1†
2
ˆ ‡ ‰ i †Š‰M i i †Š ‡ ‰ i †Š ‰M i i †Š ‡ ; 9:60†
2!
where superscript (k) denotes the kth derivative with respect to M i . Upon
performing successive differentiations of Equation (9.59) with respect to M i ,
Equation (9.60) becomes

2

d i † 1 1 2 d i † d i † 3
^
ˆ‰M i i †Š ‰M i i †Š ‡ :
d 2 d 2 d
9:61†
^
The bias and variance of can be found by taking the expectation of
Equation (9.61) and the expectation of the square of Equation (9.61), respect-
ively. Up to the order of 1/n, we find
9
2 3
1
^
2
Ef g ˆ 2i † d i d i >
>
i
>
2n d 2 d ; >
>
=
9:62†
2

1

>
^
2
varf gˆ 2i † d i : >
>
>
>
i
n d ;
Assuming that all the indicated moments and their derivatives exist, Equations
(9.62) show that
^
lim Ef gˆ ;
n!1
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