**A** rather intriguing question on X validated, namely a simulation approach to sampling a bivariate distribution fully specified by one conditional p(x|y) and the symmetric conditional expectation IE[Y|X=x]. The book Conditional Specification of Statistical Models, by Arnold, Castillo and Sarabia, as referenced by and in the question, contains (§7.7) illustrations of such cases. As for instance with some power series distribution on ℕ but also for some exponential families (think Laplace transform). An example is when

which means X conditional on Y=y is exponential E(λy). The expectation IE[Y|X=x] is then *sufficient* to identify the joint. As I figured out before checking the book, this result is rather immediate to establish by solving a linear system, but it does not help in finding a way to simulating the joint. (I am afraid it cannot be connected to the method of simulated moments!)

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