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

1952/06/01 by T. W. Anderson, D. A. Darling · 79 citations
Environmental Science · Mathematics · Decision Sciences · #Analysis of environmental and stochastic processes #Mathematical Approximation and Integration #Probability and Risk Models

paper · pdf · doi:10.1214/aoms/1177729437

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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