Coordinate-Free Exponential Families on Contingency Tables
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We propose a class of coordinate-free multiplicative models on the set of positive distributions on contingency tables and on some sets of cells of a more general structure. The models are called relational and are generated by subsets of cells, some of which may not be induced by marginals of the table, and, under the model, every cell parameter is the product of effects associated with subsets the cell belongs to. Such models are useful in analyzing incomplete tables and generalized independence structures not pertaining to subsets of variables forming the table. We reveal when a relational model is regular or else a curved exponential family. We establish necessary and sufficient conditions for the existence and uniqueness of the MLE in the curved case. We also determine the conditions under which the properties of the MLE under relational models are comparable to those under hierarchical log-linear models and prove a generalization of Birch's theorem. We propose a generalization of the iterative proportional fitting procedure that can be used for maximum likelihood estimation under relational models and prove its convergence. Finally, we use the relational model framework to contribute to the ongoing debate whether British social mobility is declining and compare the patterns of occupational mobility in Great Britain in 1991 and 2005.
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