Zhehao Zhang, Thomas S. Richardson
arXiv 14 May 2024 · Mathematics — Statistics Theory
arXiv:2405.08806 · PDF · DOI · OpenAlex · Extracted main text
We revisit the following problem, proposed by Kolmogorov: given prescribed marginal distributions $F$ and $G$ for random variables $X,Y$ respectively, characterize the set of compatible distribution functions for the sum $Z=X+Y$. Bounds on the distribution function for $Z$ were first given by Markarov (1982) and R\"uschendorf (1982) independently. Frank et al. (1987) provided a solution to the same problem using copula theory. However, though these authors obtain the same bounds, they make different assertions concerning their sharpness. In addition, their solutions leave 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 $Y_1 - Y_0$ given the marginal distributions of $Y_1$ and $Y_0$; the latter being identified from a randomized experiment. We use our new insights to sharpen and correct the results due to Fan and Park (2010) concerning individual treatment effects, and to fill some other logical gaps.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Frank, M. J., Nelsen, R. B., and Schweizer, B (1987) Best-possible bounds for the distribution of a sum—a problem of kolmogorov | 1.000 | 33 | 7 | 100% |
| 2 | Williamson, R. C. and Downs, T (1990) Probabilistic arithmetic. i. numerical methods for calculating convolutions and dependency bounds | 1.000 | 31 | 7 | 100% |
| 3 | Fan, Y. and Park, S. S (2010) Sharp bounds on the distribution of treatment effects and their statistical inference | 1.000 | 17 | 4 | 100% |
| 4 | Nelsen, R. B (2006) An introduction to copulas | 1.000 | 13 | 4 | 100% |
| 5 | Rüschendorf, L (1982) Random variables with maximum sums | 1.000 | 10 | 6 | 100% |
| 6 | Makarov, G (1982) Estimates for the distribution function of a sum of two random variables when the marginal distributions are fixed | 1.000 | 8 | 5 | 100% |
| 7 | Rüschendorf, L (1983) Solution of a statistical optimization problem by rearrangement methods | 0.811 | 4 | 2 | 100% |
| 8 | Firpo, S. and Ridder, G (2019) Partial identification of the treatment effect distribution and its functionals | 0.737 | 3 | 2 | 100% |
| 9 | Lu, J., Ding, P., and Dasgupta, T (2018) Treatment effects on ordinal outcomes: Causal estimands and sharp bounds | 0.737 | 3 | 2 | 100% |
| 10 | Sklar, M (1959) Fonctions de répartition à $N$ dimensions et leurs marges | 0.737 | 3 | 2 | 100% |
Showing the top 10 of 30 scored citations.
arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.
| Citing paper | Intensity | Mentions | Sections | |
|---|---|---|---|---|
| 1 | Individual Treatment Effect: Prediction Intervals and Sharp Bounds | 0.659 | 7 | 3 |