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

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Listed:
  • Christian Ewerhart
  • Marco Serena

Abstract

A random variable is difference-form decomposable (DFD) if it may be written as the di¤erence 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, 2023. "On the (Im-)Possibility of Representing Probability Distributions as a Difference of I.I.D. Noise Terms," Working Papers tax-mpg-rps-2023-04, Max Planck Institute for Tax Law and Public Finance.
    • Handle: RePEc:mpi:wpaper:tax-mpg-rps-2023-04
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    More about this item

    Keywords

    Differences of random variables; Density functions; Characteristic function; Uniform distribution;
    All these keywords.

    JEL classification:

    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling

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