On the (Im-)Possibility of Representing Probability Distributions as a Difference of IID Noise Terms

• 成果类型: Article
• 署名作者: Ewerhart, Christian; Serena, Marco
• 署名单位: University of Zurich; CUNEF Universidad
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2023.0081
• 发表日期: 2025
• 关键词: positive-definite functions; tournaments; contests; models; DECOMPOSITION; THEOREM; roots; ratio; form; pay

摘要

A random variable is difference -form decomposable (DFD) if it may be written as the difference of two i.i.d. random terms. We show that densities of such variables exhibit a remarkable degree of structure. Specifically, a DFD density can be neither approximately uniform, nor quasiconvex, nor strictly concave. On the other hand, a DFD density need, in general, be neither unimodal nor logconcave. Regarding smoothness, we show that a compactly supported DFD density cannot be analytic and will often exhibit a kink even if its components are smooth. The analysis highlights the risks for model consistency resulting from the strategy widely adopted in the economics literature of imposing assumptions directly on a difference of noise terms rather than on its components.