## Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 16-21 |

Number of pages | 6 |

Journal | American Statistician |

Volume | 73 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2 2019 |

## All Science Journal Classification (ASJC) codes

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

## Keywords

- Equidistribution
- Identity
- Uniform distribution