ABSTRACT
The method of higher moments differs from the usual method of moments only in that it involves moments of an order that exceeds the number of parameters to be estimated by one for each tail that is removed. To estimate parameters of a four-parameter distribution from a doubly truncated sample, we equate the first six population moments to corresponding sample moments, whereas the usual method of moments would involve only the first four moments. A major advantage resulting from the use of higher-order moments is that estimating equations are linear and thus can be more easily solved. We let k = I, 2, 3, 4, 5 in (11.3.5), replace parameters with corresponding estimates, and equate mk = vb where vk is the kth sample moment about the left point of truncation. We thereby obtain the following five linear estimating equations:
(11.4.1)
Estimates h*, b5, bf, b"f, and J* are obtained as the simultaneous solution of these equations. The estimate H* can be obtained from (11.3.6) as
(11.4.2) Estimates of the noncentral moments of the complete distribution are calculated by substituting estimates h*, b5, bf, b~ into equations (11.2.8). Estimates of central moments are then calculated by substitution in (11.2.9), and estimates of standard moments follow from (11.2.10). The resulting estimates are summarized as
where 13 1 and 132 are Pearson's notation for the shape parameters. Note that estimates are distinguished from parameters throughout this presentation by starring (*) the estimates.
