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 extent we can characterize (either partially or completely) a bivariate distribution on the basis of complete specification of one family of conditional distributions and partial or complete specification of the mode function of the other family of conditional distributions. This problem is related to an open question mentioned in the paper of Arnold, Castillo and Sarabia (2008)  [3].