Page 295 - Fundamentals of Probability and Statistics for Engineers

278 Fundamentals of Probability and Statistics for Engineers

9.3.1.1 Method of Moments

The oldest systematic method of point estimation was proposed by Pearson
(1894) and was extensively used by him and his co-workers. It was neglected for
a number of years because of its general lack of optimum properties and
because of the popularity and universal appeal associated with the method of
maximum likelihood, to be discussed in Section 9.3.1.2. The moment method,
however, appears to be regaining its acceptance, primarily because of its
expediency in terms of computational labor and the fact that it can be improved
upon easily in certain cases.
The method of moments is simple in concept. Consider a selected probability
density function f x; 1 , 2 , ... , m ) for which parameters j , j ˆ 1, 2, ..., m, are
to be estimated based on sample X 1 , X 2 ,..., X n of X. The theoretical or popu-
lation moments of X are

Z 1
i
i ˆ x f x; 1 ; ... ; m †dx; i ˆ 1; 2; ... : 9:55†
1
They are, in general, functions of the unknown parameters; that is,
i ˆ i 1 ; 2 ; ... ; m †: 9:56†

However, sample moments of various orders can be found from the sample by
[see Equation (9.14)]
n
1 X i
M i ˆ X ; i ˆ 1; 2; .. . : 9:57†
j
n
jˆ1
^
The method of moments suggests that, in order to determine estimators 1 , ... ,
^
and m from the sample, we equate a sufficient number of sample moments to
the corresponding population moments. By establishing and solving as many
resulting moment equations as there are parameters to be estimated, estimators
for the parameter are obtained. Hence, the procedure for determining
^
^
^
1 , 2 ,... , and m consists of the following steps:
. Step 1: let
^
^
i 1 ; ... ; m †ˆ M i ; i ˆ 1; 2; ... ; m: 9:58†
^
These yield m moment equations in m unknowns j , j ˆ 1, ... , m.
^
. Step 2: solve for j , j ˆ 1, .. . , m, from this system of equations. These are
called the moment estimators for 1 ,... , and m .

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