Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes

1952/06/01 by T. W. Anderson, D. A. Darling · 3,562 citations
Decision Sciences · Environmental Science · Mathematics · #Analysis of environmental and stochastic processes #Applied mathematics #Asymptotic distribution #Calculus (dental) #Combinatorics #Discrete mathematics #Distribution function #Empirical distribution function #Independent and identically distributed random variables #Mathematical Approximation and Integration #Mathematical analysis #Mathematics #Probability and Risk Models #Random variable #Statistics

paper · pdf · doi:10.1214/aoms/1177729437

published in The Annals of Mathematical Statistics 23(2), 193-212 (Institute of Mathematical Statistics)

openalex publication_date 1952/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The statistical problem treated is that of testing the hypothesis that n independent, identically distributed random variables have a specified continuous distribution function F(x). If F_n(x) is the empirical cumulative distribution function and ψ(t) is some nonnegative weight function (0 \leqq t \leqq 1), we consider n^(1)/(2) sup_-∞<x<∞ \| F(x) - F_n(x) | ψ^(1)/(2)\lbrack F(x) \rbrack\ and n∫^∞_-∞\lbrack F(x) - F_n(x) \rbrack² ψ\lbrack F(x)\rbrack dF(x). A general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations. For certain weight functions including ψ = 1 and ψ = 1/\lbrack t(1 - t) \rbrack we give explicit limiting distributions. A table of the asymptotic distribution of the von Mises ω² criterion is given.

Citations

• Table for Estimating the Goodness of Fit of Empirical Distributions
• Confidence Limits for Continuous Distribution Functions

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