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

Author

Listed:
  • Christian Ewerhart

    (Department of Economics, University of Zurich, 8001 Zurich, Switzerland)

  • Marco Serena

    (School of Business, Economics and Law, CUNEF Universidad, 28040 Madrid, Spain)

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.

Suggested Citation

  • Christian Ewerhart & Marco Serena, 2025. "On the (Im-)Possibility of Representing Probability Distributions as a Difference of I.I.D. Noise Terms," Mathematics of Operations Research, INFORMS, vol. 50(1), pages 390-409, February.
    • Handle: RePEc:inm:ormoor:v:50:y:2025:i:1:p:390-409
    DOI: 10.1287/moor.2023.0081
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