Fixed point characterizations of continuous univariate probability distributions and their applications

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Fixed point characterizations of continuous univariate probability distributions and their applications Steffen Betsch1 · Bruno Ebner1 Received: 11 December 2018 / Revised: 17 September 2019 © The Institute of Statistical Mathematics, Tokyo 2019

Abstract By extrapolating the explicit formula of the zero-bias distribution occurring in the context of Stein’s method, we construct characterization identities for a large class of absolutely continuous univariate distributions. Instead of trying to derive characterizing distributional transformations that inherit certain structures for the use in further theoretic endeavors, we focus on explicit representations given through a formula for the density- or distribution function. The results we establish with this ambition feature immediate applications in the area of goodness-of-fit testing. We draw up a blueprint for the construction of tests of fit that include procedures for many distributions for which little (if any) practicable tests are known. To illustrate this last point, we construct a test for the Burr Type XII distribution for which, to our knowledge, not a single test is known aside from the classical universal procedures. Keywords Burr Type XII distribution · Density approach · Distributional characterizations · Goodness-of-fit tests · Non-normalized statistical models · Probability distributions · Stein’s method

1 Introduction Over the last decades, Stein’s method for distributional approximation has become a viable tool for proving limit theorems and establishing convergence rates. At its heart

Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10463019-00735-1) contains supplementary material, which is available to authorized users.

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Steffen Betsch [email protected] Bruno Ebner [email protected]

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Institute of Stochastics, Karlsruhe Institute of Technology (KIT), Englerstrasse 2, 76131 Karlsruhe, Germany

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S. Betsch, B. Ebner

lies the well-known Stein characterization which states that a real-valued random variable Z has a standard normal distribution if, and only if,   E f  (Z ) − Z f (Z ) = 0

(1)

holds for all functions f of a sufficiently large class of test functions. To exploit this characterization for testing the hypothesis   H0 : P X ∈ N(μ, σ 2 ) | (μ, σ 2 ) ∈ R × (0, ∞) (2) of normality, where P X is the distribution of a real-valued random variable X , against general alternatives, Betsch and Ebner (2019b) used that (1) can be untied from the class of test functions with the help of the so-called zero-bias transformation introduced by Goldstein and Reinert (1997). To be specific, a real-valued random variable X ∗ is said to have the X -zero-bias distribution if     E f  (X ∗ ) = E X f (X ) holds for any of the respective test functions f . If EX = 0 and Var(X ) = 1, the X -zero-bias distribution exists and is unique, and it has distribution function   T X (t) = E X (X − t)1{X ≤ t} , t ∈ R.

(3)

By (1), the standard Gaussian distribution is the unique fixed point of the