Primary Submission Category: Heterogeneous Treatment Effects
Revisiting a problem of Kolmogorov with application to individual treatment effects
Authors: Zhehao Zhang, Thomas Richardson,
Presenting Author: Zhehao Zhang*
We revisit the following problem, proposed by Kolmogorov: given prescribed marginal distributions F and G for random variables X and Y respectively, characterize the set of compatible distribution functions for the sum Z=X+Y. Bounds on the distribution function for Z were given by Makarov (1982), and Frank, Nelson and Schweizer (1987), the latter using copula theory. However, though they obtain the same bounds, they make different assertions concerning their sharpness. In addition, their solutions left some open problems in the case when the given marginal distribution functions are discontinuous. These issues have led to some confusion and erroneous statements in subsequent literature, which we correct.
Kolmogorov’s problem is closely related to inferring possible distributions for individual treatment effects Y(x=1) – Y(x=0) given the marginal distributions Y(x=0) and Y(x=1); the latter being identified from a randomized experiment. We use our new insights to sharpen results due to Fan and Park (2010) concerning individual treatment effects, and to fill some other logical gaps.