On An Intriguing Distributional Identity

M. C. Jones, Éric Marchand and William E. Strawderman

The American Statistician, 2019, vol. 73, issue 1, 16-21

Abstract: For a continuous random variable X with support equal to (a, b), with c.d.f. F, and g: Ω1 → Ω2 a continuous, strictly increasing function, such that Ω1∩Ω2⊇(a, b), but otherwise arbitrary, we establish that the random variables F(X) − F(g(X)) and F(g− 1(X)) − F(X) have the same distribution. Further developments, accompanied by illustrations and observations, address as well the equidistribution identity U − ψ(U) = dψ− 1(U) − U for U ∼ U(0, 1), where ψ is a continuous, strictly increasing and onto function, but otherwise arbitrary. Finally, we expand on applications with connections to variance reduction techniques, the discrepancy between distributions, and a risk identity in predictive density estimation.

Date: 2019
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DOI: 10.1080/00031305.2017.1375984

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