This research involves the development of new estimators, a projection pursuit estimator and a wavelet estimator, for cell probabilities of high-dimensional ordered contingency tables. A high-dimensional contingency table has two unique features: it is usually very sparse, and it has a large number of boundary cells. The projection pursuit method is used to study the features of high-dimensional data by looking at its low-dimensional projections. The projected data is no longer sparse and the projection pursuit method does not suffer from boundary effect since the method is not based on local averages. When a smoothing method is introduced, one usually assumes that the contingency table has an underlying density function which has a certain degree of smoothness. Both the order of a kernel function and the optimal choice of a smoothing parameter depend on the smoothness of the underlying density function. This assumption seems unnatural. One may not have a prior knowledge about the smoothness of an unknown density. The wavelet method can adapt to the smoothness of an unknown density, and it avoids the problem of how much to smooth. Therefore, one does not need to assume the existence of an underlying density function. Contingency or cross tabulation tables occur very frequently in biomedical science, social science, and educational research. For example, to study the public's opinions on five different policy issues (health care, tax, . . .) on a five point scale (strongly agree, agree, neutral, disagree, strongly disagree), a five-dimensional cross tabulation table with 3,125 cells would be needed. Frequently when the number of cells is large, the table is usually sparse, i.e., has many empty cells. Large and sparse tables provide many challenges for statistical analysis. The research develops new tools for analyzing this type of categorical data.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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James E. Gentle
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Michigan Technological University
United States
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