On An Intriguing Distributional Identity |
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| Authors: | M. C. Jones Éric Marchand William E. Strawderman |
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| Affiliation: | 1. School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, UK;2. Département de Mathématiques, Université de Sherbrooke, Sherbrooke, QC, Canada;3. Department of Statistics and Biostatistics, Rutgers University, Piscataway, NJ |
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| 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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| Keywords: | Equidistribution Identity Uniform distribution |
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