CONVEX MODELS OF HIGH DIMENSIONAL DISCRETE DATA

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

1Duke University, Center for Demographic Studies, 2117 Campus Drive, Box 90408, Durham, NC 27708, U.S.A. 2Brigham 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.