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LARGE SAMPLE PROPERTIES OF ESTIMATES OF

A DISCRETE GRADE OF MEMBERSHIP MODEL

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H. DENNIS TOLLEY^{1} AND KENNETH G. MANTON^{2}

^{1} *Department of Statistics, Brigham Young University, Provo, UT 84602, U.S.A.*

^{2} *Center for Demographic Studies, Duke University, Durham, NC 27706, U.S.A.*
(Received December 28, 1989; revised June 11, 1990)

**Abstract.**
Increasingly, fuzzy partitions are being used in
multivariate classification problems as an alternative to the crisp
classification procedures commonly used. One such fuzzy partition, the
grade of membership model, partitions individuals into fuzzy sets using
multivariate categorical data. Although the statistical methods used to
estimate fuzzy membership for this model are based on maximum likelihood
methods, large sample properties of the estimation procedure are problematic
for two reasons. First, the number of incidental parameters increases with
the size of the sample. Second, estimated parameters fall on the boundary
of the parameter space with non-zero probability. This paper examines the
consistency of the likelihood approach when estimating the components of a
particular probability model that gives rise to a fuzzy partition. The
results of the consistency proof are used to determine the large sample
distribution of the estimates. Common methods of classifying individuals
based on multivariate observations attempt to place each individual into
crisply defined sets. The fuzzy partition allows for individual to
individual heterogeneity, beyond simply errors in measurement, by defining a
set of pure type characteristics and determining each individual's distance
from these pure types. Both the profiles of the pure types and the
heterogeneity of the individuals must be estimated from data. These
estimates empirically define the fuzzy partition. In the current paper,
this data is assumed to be categorical data. Because of the large number of
parameters to be estimated and the limitations of categorical data, one may
be concerned about whether or not the fuzzy partition can be estimated
consistently. This paper shows that if heterogeneity is measured with
respect to a fixed number of moments of the grade of membership scores of
each individual, the estimated fuzzy partition is consistent.

*Key words and phrases*:
Consistency, fuzzy partition, grade of
membership.

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