Probability estimation via smoothing in sparse contingency tables with ordered categories

Probability estimation in sparse two-dimensional contingency tables with ordered categories is examined. Several smoothing procedures are compared to analysis of the unsmoothed table. It is shown that probability estimates obtained via maximum penalized likelihood smoothing are consistent under a sparse asymptotic framework if the underlying probability matrix is smooth, and are more accurate than kernel-based and other smoothing techniques. In fact, computer simulations indicate that smoothing based on a product kernel is less effective than no smoothing at all. An example is given to illustrate the smoothing technique. Possible extensions to model building and higher dimensional tables are discussed.