Partial or complete characterization of a bivariate distribution based on one conditional distribution and partial specification of the mode function of the other conditional distribution

There are various ways to characterize a bivariate distribution based on given distributional information. For example, information on both families of conditional densities, i.e., of X given Y and of Y given X, is sufficient to characterize the bivariate distribution. On the other hand, knowledge of both regression functions, i.e., E(X|Y=y) and E(Y|X=x), will be inadequate to determine the joint distribution. In this paper, we discuss to what …