On An Intriguing Distributional Identity

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

Research output: Contribution to journalArticlepeer-review


For a continuous random variable X with support equal to (a, b), with c.d.f. F, and g: Ω 1 → Ω 2 a continuous, strictly increasing function, such that Ω 1 ∩Ω 2 ⊇(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) = d ψ − 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.

Original languageEnglish (US)
Pages (from-to)16-21
Number of pages6
JournalAmerican Statistician
Issue number1
StatePublished - Jan 2 2019

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty


  • Equidistribution
  • Identity
  • Uniform distribution


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