Skip to main content
SpringerLink
Log in

Off-line statistical analysis of change-point models using non parametric and likelihood methods

  • Part II Likelihood And Related Methods, With An Emphasis On The Detection Of Changes In Spectral Characteristics
  • Conference paper
  • First Online:
Detection of Abrupt Changes in Signals and Dynamical Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 77))

Abstract

We investigate the problem of testing for a change-point in stationary statistical models and of estimating the change parameters in an off-line framework. This paper provides asymptotic results which emphasize the need for weighting the classical test statistics when the change time is completely unknown. The asymptotic expansions can also give new detectors which are simpler for using.

First, we present the different models under investigation (§I) and the corresponding tests (§II). The third paragraph will be devoted to the different ways for comparing these tests -both asymptotic and non asymptotic ones- and the partial results we can give. The fourth part is central: it proves the need for normalizing the test statistics near the edges of the observation window and contains the application on three well-known time-series. The last paragraph details the asymptotic tools used before: invariance theorems for the likelihood ratio process provide the asymptotic behaviour of the test statistics and the asymptotic distribution of the estimators of a change.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T.W. ANDERSON: "The statistical analysis of time series" Wiley (1977).

    Google Scholar 

  2. U. APPEL, A.V. BRANDT: "Adaptive sequential segmentation of piecewise stationary time series" Information Sciences vol. 29 (April 1983) pp. 27–56.

    Google Scholar 

  3. U. APPEL, A.V. BRANDT: "Performance comparison of two segmentation algorithms using growing reference windows" Proceedings of VIth International Conference on Analysis and Optimization of Systems (June 1984) part 1, pp. 156–170.

    Google Scholar 

  4. R.R. BAHADUR: "An optimal property of the likelihood ratio statistic" Vth Berkeley Symposium (1966) part I, pp. 12–26.

    Google Scholar 

  5. G.K. BHATTACHARYYA, R.A. JOHNSON: "Non parametric tests for shifts at unknown time point" A.M.S. vol. 39 (1968) pp. 1731–1743.

    Google Scholar 

  6. P. BILLINGSLEY: "Convergence of probability measures" Wiley (1968).

    Google Scholar 

  7. Z.W. BIRNBAUM, A.W. MARSHALL: "Some multivariate Chebychev inequalities with extensions to continuous parameter processes" A.M.S. vol. 32 (1961) pp. 687–703.

    Google Scholar 

  8. J.R. BLUM, J. KIEFER, J.I. ROSENBLATT: "Distribution-free tests of independance" A.M.S. vol. 32 (1961) pp. 485–497.

    Google Scholar 

  9. G.E.P. BOX, G.M. JENKINS: "Time series analysis" Holdenday (1976).

    Google Scholar 

  10. L.D. BROWN: "Non local asymptotic optimality of appropriate likelihood ratio tests" A.M.S. vol. 42 (1971) pp. 1206–1240.

    Google Scholar 

  11. R.L. BROWN, J. DURBIN, J.M. EVANS: "Techniques for testing the constancy of regression relationships over time" J.R.S.S.B vol. 37 (1975) pp. 149–192.

    Google Scholar 

  12. M.J. CAMBELL, R.A.N. WALKER: "A survey of statistical work on the Mackenzie River series of annual Canadian lynx trappings for years 1821–1934 and a new analysis" J.R.S.S.A vol. 140 pp. 411–431.

    Google Scholar 

  13. H. CHERNOFF, S. ZACKS: "Estimating the current mean of a normal distribution which is subjected to changes in time" A.M.S. vol. 35 (1964) pp. 998–1018.

    Google Scholar 

  14. J. COURSOL, D. DACUNHA-CASTELLE: "Remarques sur l'approximation de la vraisemblance d'un processus gaussien stationnaire" Theory of Prob. and Applic. vol. 27 (1982) pp. 155–160.

    Google Scholar 

  15. J. DESHAYES, D. PICARD: "Tests de rupture de régression: comparaison asymptotique" Theory of Prob. and Applic. vol. 27 (1982) pp. 95–108.

    Google Scholar 

  16. J. DESHAYES, D. PICARD: "Principe d'invariance sur le processus de vraisemblance" Annales de l'I.H.P. vol. 20 (1984) pp. 1–20.

    Google Scholar 

  17. J. DESHAYES, D. PICARD: "Lois asymptotiques des tests et estimateurs de rupture dans un modèle statistique classique" Annales de l'I.H.P. vol. 20 (1984) pp. 309–327.

    Google Scholar 

  18. J. DESHAYES, D. PICARD: "Ruptures de modèles en statistique" Thèses d'état, Université Paris Sud (1983).

    Google Scholar 

  19. I.A. IBRAGIMOV, R.Z. KHASMINSKII: "Asymptotic behaviour of statistical estimators in the smooth case-I-study of the likelihood ratio" Theory of Prob. and Applic. vol. 17 (1972) pp. 445–462.

    Google Scholar 

  20. I.A. IBRAGIMOV, R.Z. KHASMINSKII: "Statistical estimation-asymptotic theory" Springer Verlag-Applications of Mathematics t.16(1981).

    Google Scholar 

  21. KOMLOS, MAJOR, TUSNADY: "An approximation of partial sums of independent random variables and the sample distribution function" Zeit. Werw. Gebiete, vol. 32 (1975) pp. 111–131, and vol. 34 (1976) pp. 33–58.

    Google Scholar 

  22. L. LECAM: "On the assumption used to prove asymptotic normality of maximum likelihood estimates" A.M.S., vol. 41 (1970), pp. 802–828.

    Google Scholar 

  23. D.L. MAC LEISH: "Invariance principles for dependent variables" Zeit. Wahr. Verw. Gebiete, Vol. 32 (1975) pp. 165–178.

    Google Scholar 

  24. T.L. MALEVITCH: "The asymptotic behaviour of an estimate for the spectral function of a stationary Gaussian process" Theory of Prob. and Applic. vol. 9 (1964) pp. 349–353.

    Google Scholar 

  25. L. MOLINARI, Kinderspital Zürich, Personal communication.

    Google Scholar 

  26. I.V. NIKIFOROV: "Sequential detection of changes in stochastic systems" chapter VII in this book.

    Google Scholar 

  27. A.N. PETTITT: "A non parametric approach to the change-point problem" Appl. Stat. vol. 28 (1979) pp. 126–135.

    Google Scholar 

  28. A.N. PETTITT: "A simple cumulative sum type statistic for the change-point problem with zero-one observations" Biometrika, vol. 67 (1980) pp. 79–84.

    Google Scholar 

  29. D. PICARD: "Testing and estimating change-points in time series" to be published in Journal of Appl. Prob. (1985).

    Google Scholar 

  30. J. PRAAGMAN: "Bahadur efficiency of rank tests for the change-point problem" Annals of Stat. (to appear).

    Google Scholar 

  31. L.A. SHEPP: "The joint density of the maximum and its location for a Wiener process with drift". Journal of Appl. Prob. vol. 16 (1979) pp. 423–427.

    Google Scholar 

  32. A.F.M. SMITH: "A bayesian approach to inference about a change-point in a sequence of random variables" Biometrika, vol. 62 (1975) pp. 407–416.

    Google Scholar 

  33. W. SOUMBEY: "Rupture dans une série chronologique-Application à la détection de rupture dans les champs gaussiens stationnaires" Thèse de 3ème cycle. Université Paris Sud (1983).

    Google Scholar 

  34. M. STONE: "Large deviations of empirical probability measures" Annals of Stat. vol. 2 (1974) pp. 362–366.

    Google Scholar 

  35. A.S. WILLSKY: "Detection of abrupt changes in dynamic systems" chapter II in this book.

    Google Scholar 

  36. G.G. ROUSSAS: "Contuiguity of probability measures, some applications in statistics" Cambridge University Press (1972).

    Google Scholar 

  37. H. TONG: "Some comments on the Canadian lynx data" J.R.S.S.A. 140 (1977) pp. 432–436.

    Google Scholar 

  38. M.B. PRIESTLEY: "Spectral analysis and time series" vol. 1, Prob. and Mat.Stat. Academic Press (1981).

    Google Scholar 

  39. M.M. GABR, T. SUBBA RAO: "The estimation and prediction of subset bilinear time series model with applications" Journal of time series Analysis, vol. 2 no 3 (1981).

    Google Scholar 

  40. D.V. HINKLEY: "Inference about the change-point in a sequence of random variables" Biometrika, vol. 57 (1970) pp. 1–17.

    Google Scholar 

  41. D.V. HINKLEY, E.A. HINKLEY: "Inference about the change-point in a sequence of binomial variables" Biometrika, vol. 57 (1970) pp. 477–488.

    Google Scholar 

  42. I.A. IBRAGIMOV: "On estimation of the spectral function of a stationary Gaussian process" Theory of Prob. and Applic. vol. 8 (1962) pp. 366–400.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. C.N.R.S. : E.R.A. no 1044–532 and G.R.E.C.O. 69 SARTA, E.N.S.T. Département SYC, 75634, Paris Cedex 13

    Jean Deshayes

  2. C.N.R.S. : E.R.A. no 1044–532 and G.R.E.C.O. 69 SARTA, Université Paris-Sud Bât. de Maths, 91405, Orsay Cedex

    Dominique Picard

Authors
  1. Jean Deshayes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dominique Picard
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Michèle Basseville Albert Benveniste

Rights and permissions

Reprints and permissions

Copyright information

© 1985 Springer-Verlag

About this paper

Cite this paper

Deshayes, J., Picard, D. (1985). Off-line statistical analysis of change-point models using non parametric and likelihood methods. In: Basseville, M., Benveniste, A. (eds) Detection of Abrupt Changes in Signals and Dynamical Systems. Lecture Notes in Control and Information Sciences, vol 77. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0006392

Download citation

  • DOI: https://doi.org/10.1007/BFb0006392

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16043-4

  • Online ISBN: 978-3-540-39726-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation