Fixed point characterizations of continuous univariate probability distributions and their applications

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 endeavours, 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.

40 pages

References in corpus (12)

• Stein's method and the zero bias transformation with application to simple random sampling
• A Kernel Test of Goodness of Fit
• Nonnormal approximation by Stein's method of exchangeable pairs with application to the Curie--Weiss model
• New rates for exponential approximation and the theorems of Rényi and Yaglom
• Tests based on characterizations, and their efficiencies: a survey
• A new characterization of the Gamma distribution and associated goodness of fit tests
• Discretized normal approximation by Stein's method
• Bounds for the normal approximation of the maximum likelihood estimator
• Testing normality via a distributional fixed point property in the Stein characterization
• Stein's method for steady-state diffusion approximations of $M/Ph/n+M$ systems
• New $L^2$-type exponentiality tests
• A unified approach to Stein characterizations

Cited by in corpus (5)

• Characterizations of non-normalized discrete probability distributions and their application in statistics
• A Kernel Stein Test for Comparing Latent Variable Models
• New weighted $L^2$-type tests for the inverse Gaussian distribution
• Minimum $L^q$-distance estimators for non-normalized parametric models
• A new goodness of fit test for normal distribution based on Stein's characterization