An experimental response model for bivariate association

A class of models for bivariate association between random variables X and Y is studied in which the conditional model of Y given X is an exponential response model for an unknown exponential family. Although such models have been found in the literature to a limited extent, properties of maximum-likelihood estimates have not been considered in the nontrivial case in which neither variable is polytomous. In such a case, maximum-likelihood estimation involves solution of a system of equations in which the number of unknown parameters increases as the sample size increases. Consequently conventional approaches to large-sample properties of maximum-likelihood estimates are inapplicable. Nonetheless, with the use of Frechet differentiability and the implicit function theorem, the maximum-likelihood estimates are shown to exist and to be expressible in terms of Frechet differentiable functions. In conjunction with the known stochastic properties of the Kolmogorov-Smirnov distance n12∥Fn-F∥ infinity=Op1 between the empirical and the population distribution function, the maximum-likelihood estimates are shown to be consistent and asymptotically normal when variables X and Y have bounded supports. Confidence intervals and hypothesis tests of the parameters can be performed based on the asymptotic normality in the usual manner. The functionals related to the estimated conditional distribution of Y given X are also shown to be consistent and asymptotically normal.