On An Intriguing Distributional Identity

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.