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June, 1952 Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes
T. W. Anderson, D. A. Darling
Ann. Math. Statist. 23(2): 193-212 (June, 1952). DOI: 10.1214/aoms/1177729437


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 $\psi(t)$ is some nonnegative weight function $(0 \leqq t \leqq 1)$, we consider $n^{\frac{1}{2}} \sup_{-\infty<x<\infty} \{| F(x) - F_n(x) | \psi^\frac{1}{2}\lbrack F(x) \rbrack\}$ and $n\int^\infty_{-\infty}\lbrack F(x) - F_n(x) \rbrack^2 \psi\lbrack F(x)\rbrack dF(x).$ 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 $\psi = 1$ and $\psi = 1/\lbrack t(1 - t) \rbrack$ we give explicit limiting distributions. A table of the asymptotic distribution of the von Mises $\omega^2$ criterion is given.


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T. W. Anderson. D. A. Darling. "Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes." Ann. Math. Statist. 23 (2) 193 - 212, June, 1952.


Published: June, 1952
First available in Project Euclid: 28 April 2007

zbMATH: 0048.11301
MathSciNet: MR50238
Digital Object Identifier: 10.1214/aoms/1177729437

Rights: Copyright © 1952 Institute of Mathematical Statistics

Vol.23 • No. 2 • June, 1952
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