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On the (im-)possibility of representing probability distributions as a difference of i.i.d. noise terms


Ewerhart, Christian; Serena, Marco (2024). On the (im-)possibility of representing probability distributions as a difference of i.i.d. noise terms. Mathematics of operations research:Epub ahead of print.

Abstract

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.

Abstract

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:03 Faculty of Economics > Department of Economics
Dewey Decimal Classification:330 Economics
Uncontrolled Keywords:Differences of random variables, density functions, characteristic function, uniform distribution
Scope:Discipline-based scholarship (basic research)
Language:English
Date:7 March 2024
Deposited On:07 May 2024 12:27
Last Modified:30 Jun 2024 01:41
Publisher:Institute for Operations Research and the Management Science
ISSN:0364-765X
OA Status:Green
Publisher DOI:https://doi.org/10.1287/moor.2023.0081
  • Content: Accepted Version
  • Language: English