Hostname: page-component-6766d58669-kn6lq Total loading time: 0 Render date: 2026-05-24T04:14:49.336Z Has data issue: false hasContentIssue false
  • English
  • Français

Goodness-of-Fit Tests for Probability Distributions and Spectral Distributions

Published online by Cambridge University Press:  27 July 2009

T. W. Anderson
T. W. Anderson
Affiliation:
Department of Statistics, Stanford University, Stanford, California 94305
Get access Rights & Permissions [Opens in a new window]

Abstract

In the fall of 1948 in my course on Least Squares in the Department of Mathematical Statistics at Columbia University (and in the spring in Correlation and Chi-Square), I was particularly impressed by one of the students— Gerald J. Lieberman. I was disappointed that this promising student left Columbia after one year, but it was not long until our paths met again. It is a pleasure to dedicate this paper to my colleague and close friend!

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 9 , Issue 1 , January 1995 , pp. 27 - 37
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Anderson, T.W. (1993). Goodness of fit tests for spectral distributions. Annals of Statistics 21: 830–847.CrossRefGoogle Scholar
2.Anderson, T.W. (1994). Goodness of fit tests for autoregressive processes. Unpublished.Google Scholar
3.Anderson, T.W. & Darling, D.A. (1952). Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes. Annals of Mathematical Statistics 23: 193–212.CrossRefGoogle Scholar
4.Anderson, T.W. & Darling, D.A. (1954). A test of goodness of fit. Journal of the American Statistical Association 49: 765–769.CrossRefGoogle Scholar
5.Anderson, T.W. & Stephens, M.A. (1993). The modified Cramér-von Mises goodness-of-fit criterion for time series. Sankhyā 55: 357–369.Google Scholar
6.Anderson, T.W. & You, L. (1994). Adequacy of asymptotic theory for goodness-of-fit criteria for spectral distributions. Unpublished.Google Scholar
7.Bartlett, M.S. (1954). Problémes de l'analyse spectral des séries temporelles stationnaries. Publications de l'Institut de Statistique de l'Université de Paris III–3: 119–134.Google Scholar
8.Bartlett, M.S. (1966). An introduction to stochastic processes with special reference to methods and applications, 2nd ed.Cambridge: Cambridge University Press.Google Scholar
9.Cramér, H. (1928). On the composition of elementary errors. Skandinavisk Aktuarietidskrift 13–74, 141–180.Google Scholar
10.Darling, D.A. (1955). The Cramér-Smirnov test in the parametric case. Annals of Mathematical Statistics 26: 1–20.CrossRefGoogle Scholar
11.Doob, J.L. (1949). Heuristic approach to the Kolmogorov-Smirnov theorems. Annals of Mathematical Statistics 20:393–403.CrossRefGoogle Scholar
12.Durbin, J. (1973). Distribution theory for tests based on the sample distribution function. CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
13.Grenander, U. & Rosenblatt, M. (1952). On spectral analysis of stationary time-series. Proceedings of the National Academy of Sciences U.S.A. 38: 519–521.CrossRefGoogle ScholarPubMed
14.Grenander, U. & Rosenblatt, M. (1953). Statistical spectral analysis of time series arising from stationary processes. Annals of Mathematical Statistics 24: 537–558.CrossRefGoogle Scholar
15.Grenander, U. & Rosenblatt, M. (1957). Statistical analysis of stationary time series. Stockholm: Almqvist and Wiksell.CrossRefGoogle Scholar
16.Kac, M. (1949). On deviations between theoretical and empirical distributions. Proceedings of the National Academy Sciences U.S.A. 35: 252–257.CrossRefGoogle ScholarPubMed
17.Kiefer, J. (1959). K-sample analogues of the Kolmogorov-Smirnov and Cramér-von Mises tests. Annals of Mathematical Statistics 30: 420–447.CrossRefGoogle Scholar
18.Kolmogorov, A.N. (1933). Sulla determinazione empirica di une legge di distribution. Giornale dell'Istituto Italiano degli Atturai 4: 83–91.Google Scholar
19.Lewis, P.A.W. (1961). Distribution of the Anderson-Darling statistic. Annals of Mathematical Statistics 32: 1118–1124.CrossRefGoogle Scholar
20.von Mises, R. (1931). Wahrscheinlichkeitsrechnung und ihre Anwendung in Statistik und Theoretische Physik. Leipzig-Wien.Google Scholar
21.Scholz, F.W. & Stephens, M.A. (1987). K-sample Anderson-Darling tests. Journal of the American Statistical Association 82: 918–924.Google Scholar
22.Shorack, G.R. & Wellner, J.A. (1986). Empirical processes with applications to statistics. New York: John Wiley and Sons.Google Scholar
23.Smirnov, N.V. (1936). Sur la distribution de ω2. Compte Rendus Seance de l'Academie des Sciences, Paris 202: 449–452.Google Scholar
24.Smirnov, N.V. (1948). Table for estimating the goodness of fit of empirical distributions. Annals of Mathematical Statistics 19: 279–281.CrossRefGoogle Scholar
25.Stephens, M.A. (1970). Use of the Kolmogorov-Smirnov, Cramér-von Mises and related statistics without extensive tables. Journal of the Royal Statistical Society B 32: 115–122.Google Scholar
26.Stephens, M.A. (1974). EDF statistics for goodness-of-fit and some comparisons. Journal of the American Statistical Association 69: 730–737.CrossRefGoogle Scholar
27.Stephens, M.A. (1976). Asymptotic results for goodness-of-fit statistics with unknown parameters. Annals of Statistics 4: 357–369.CrossRefGoogle Scholar
28.Stephens, M.A. (1986). Tests based on EDF statistics. In D'Agostino, R.B. & Stephens, M.A. (eds.), Goodness-of-fit techniques. New York: Marcel Dekker, pp. 97–193.Google Scholar
29.Stephens, M.A. (1992). Introduction to Kolmogorov (1933). On the empirical determination of a distribution function. In Kotz, S. & Johnson, N.L. (eds.), Breakthroughs in statistics. Vol. II: Methodology and distribution. New York: Springer-Verlag, pp. 93–105.CrossRefGoogle Scholar
30.You, L. (1993). Tests and confidence regions for spectral distributions based on the Kolmogorov-Smirnov statistic. Doctoral dissertation, Stanford University, Stanford, CA.Google Scholar