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Cited by 3,542Statistical Distribution Estimation and ApplicationsProbability and Statistical ResearchStatistical Methods and Inference

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

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Anderson, T. W., Darling, D. A.·1952·The Annals of Mathematical Statistics
DOI 10.1214/aoms/1177729437
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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 $\psi(t)$ 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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Research works printed but not linked: 13

  1. [3][3] G. Doetscí, "Les équations aux dérivées partielles du type parabolique," Enseignement Math., Vol. 35 (1936), pp. 43–87. -
  2. [2][2] HARALD CRAmÉR, 'On the composition of elementary errors," Skandinavisk Aktuarietidskrift, Vol. 11 (1928), pp. 13–74, 141–180. -
  3. [5][5] J. L. Doob, "The brownian motion and stochastic equations,"Annals of Mathematics, Vol. 2 (1942), pp. 351–369. -
  4. [7][7] R. FortET, '"Les fonctions aléatoires du type de Markoff associées à certaines équations linéaires aux dérivées partielles du type parabolique," Jour. Math. Pures Appl., Vol. 22 (1943), pp. 177–243. -
  5. [8][8] E. GouRsAT, Cours d'Analyse Mathématique, Vol. 3, 2nd ed., Gauthier-Villars, Paris, 1915. -
  6. [11][11] A. N. KoLmogorov, 'Sulla determinazione empirica delle leggi di probabilita,"Giorn. Ist. Ital. Attuari, Vol. 4 (1933), pp. 1–11. -
  7. [13][13] G. M. MANIYA, "Generalization of the criterion of A. N. Kolmogorov," Doklady Akad. Nauk SSSR (NS), Vol. 69 (1949), pp. 495–497. -
  8. [14][14] G. PóLYA, "Remarks on computing the probability integral," Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1949, pp. 63–78. -
  9. [15][15] N. V. SmIrNov, 'Sur la distribution de ω2," C. R. Acad. Sci. Paris, Vol. 202 (1936), p. 449. -
  10. [16][16] N. V. SmIrNov, "On the distribution of the ω2 criterion,"Rec. Math. (Mat. Sbornik) (NS), Vol. 2 (1937), pp. 973–993. -
  11. [17][17] N. V. SmIrNov, 'On the deviation of the empirical distribution function," Rec. Math. (Mat. Sbornik) (NS), Vol. 6 (1939), pp. 3–26. -
  12. [20][20] R. voN MisEs, "Differentiable statistical functions," Annals of Math. Stat., Vol. 18 (1947), pp. 309–348. -
  13. [23][23] E. T. WHITTAKER AND G. N. WATsON, A Course of Modern Analysis, Cambridge University Press, 1927.

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Metadata

Authors
Anderson, T. W., & Darling, D. A.
Venue
The Annals of Mathematical Statistics
Publisher
Institute of Mathematical Statistics
Published
1952-06
Volume
23
Issue
2
Pages
193-212
ISSN
0003-4851 (ISSN-L)
DOI
10.1214/aoms/1177729437
OpenAlex
W1973429805
Cited by
3,542
References
10
Citations by year
2026: 172 · 2025: 231 · 2024: 194 · 2023: 168 · 2022: 190 · 2021: 190 · 2020: 199 · 2019: 164 · 2018: 143 · 2017: 164 · 2016: 157 · 2015: 141 · 2014: 141 · 2013: 133 · 2012: 133
FWCI
10.11
Type
journal article
Language
en
Open access
Locations
  • The Annals of Mathematical Statistics (publishedVersion)
  • Project Euclid (Cornell University) (submittedVersion)
Topics
  • Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
  • Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
  • Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
Keywords
goodness-of-fit tests, empirical distribution function, asymptotic distribution, eigenvalue problem, boundary value problems, differential equations, continuous distribution function, independent identically distributed random variables

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