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CONVEX MODELS OF HIGH DIMENSIONAL DISCRETE DATA

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MAX A. WOODBURY^{ 1},
KENNETH G. MANTON^{ 1} AND
H. DENNIS TOLLEY^{ 2}

^{1} *Duke University, Center for
Demographic Studies, 2117 Campus Drive,*

Box 90408, Durham, NC 27708, U.S.A.

^{2} *Brigham Young University,
Department of Statistics,*

Room 226, TMCB, Provo, UT 84602, U.S.A.
(Received September 8, 1994; revised May 20, 1996)

**Abstract.**
Categorical data of high (but finite)
dimensionality generate
sparsely populated *J*-way
contingency tables because of finite sample sizes. A model
representing such data by a ``smooth'' low dimensional
parametric structure using a ``natural'' metric would be
useful. We discuss a model using a metric determined by
convex sets to represent moments of a discrete
distribution to order *J*. The model is shown, from
theorems on convex polytopes, to depend only on the linear
space spanned by the convex set--it is otherwise measure
invariant. We provide an empirical example to illustrate
the maximum likelihood estimation of parameters of a
particular statistical application (Grade of Membership
analysis) of such a model.

**Key words and phrases:**
Probability mixtures,
convex sets, polytopes, convex duality, Grades of
Membership.

**Source**
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