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

The DOI matches its Crossref record; 10 of the 10 references it cites were found in a registry or an index.

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

Open access
Anderson, T. W., Darling, D. A.·1952·The Annals of Mathematical Statistics
DOI 10.1214/aoms/1177729437
Open this paper in CoChat — free accountOpen PDF

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.

Overview

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.1214/aoms/1177729437

Cites

10 of 10 references verified0 citation errors · 0 not checkable

Statistics

Verification

Claims the paper makes

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Citations

Cites

Show all 10 references

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.

Cited by

Sorted by citations

Items 801–900 of 1,004

Related

See all

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

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review