On an intriguing distributional identity

Jones, M. C.; Marchand, E. and Strawderman, W. E. (2018). On an intriguing distributional identity. The American Statistician, 73(1) pp. 16–21.

DOI: https://doi.org/10.1080/00031305.2017.1375984

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.

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