Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-22T09:59:36.268Z Has data issue: false hasContentIssue false

Testing and estimating change-points in time series

Published online by Cambridge University Press:  01 July 2016

Dominique Picard*
Affiliation:
Université de Paris-Sud
*
Postal address: Université de Paris-Sud, XI, Bâtiment de Mathématique 425, E.R.A. CNRS 532, Statistique Appliquée, 91405 Orsay, France.

Abstract

The aim of this paper is to present a few techniques which may be useful in the analysis of time series when a failure is suspected. We present two categories of tests and investigate their asymptotic properties: one, of nonparametric type, is intended to detect a general failure in spectrum; the other investigates the properties of likelihood ratio tests in parametric models which have a non-standard behaviour in this situation. Finally, we obtain the asymptotic distribution of the likelihood estimators of the change parameters.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1985 

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.)

References

1. Anderson, T. W. (1977) The Statistical Analysis of Time Series. Wiley, New York.Google Scholar
2. Basseville, M. and Benveniste, A. (1983) Sequential detection of abrupt changes in the spectral characteristics of digital signals. IEEE Trans. Information Theory IT-29, 709723.CrossRefGoogle Scholar
3. Birnbaum, Z. W. and Marshall, A. W. (1961) Some multivariate Chebichev inequalities with extensions to continuous parameter processes. Ann. Math. Statist. 32, 687703.CrossRefGoogle Scholar
4. Blum, J. R., Kiefer, J. and Rosenblatt, M. (1961) Distribution-free tests of independence. Ann. Math. Statist. 32, 485497.CrossRefGoogle Scholar
5. Box, G. E. P. and Jenkins, G. M. (1970) Time Series Analysis, Forecasting and Control. Holden Day, San Francisco.Google Scholar
6. Campbell, M. J. and Walker, A. M. (1977) A survey of statistical work on the Mackenzie river series of annual Canadian lynx trapping for the years 1821-1934 and a new analysis (with discussion). J. R. Statist. Soc. A 140, 411431.Google Scholar
7. Coursol, J. and Dacunha-Castelle, D. (1982) Remarques sur l’approximation de la vraisemblance d’un processus gaussien stationnaire. Teor. Veroyatnost. i Primenen. 27, 155160.Google Scholar
8. Csensov, N. N. (1960) Limit theorems for some classes of random functions. Proc. Union Conf. Theory Prob. Math. Statist., Erevan 1955. Izd. Akad. Nauk Aroujan SSSR, Erevan , 280285.Google Scholar
9. Dellacherie, C. and Meyer, P. A. (1978) Probabilities and Potential. North-Holland, Amsterdam.Google Scholar
10. Dehayes, J. (1983) Rupture de modèles en statistique. Thèse d’Etat, Orsay.Google Scholar
11. Deshayes, J. and Picard, D. (1981) Testing for a change-point in a statistical model. Rapport technique, Université de Paris-Sud, Orsay.Google Scholar
12. Deshayes, J. and Picard, D. (1981) Convergence de processus à double indice; Applications aux tests de rupture dans un modèle. C.R. Acad. Sci. Paris 292, 449452.Google Scholar
13. Deshayes, D. and Picard, D. (1982) Tests de rupture de regression: comparaison asymptotique. Teor. Veroyastnost. i Primenen. 27, 95108.Google Scholar
14. Deshayes, D. and Picard, D. (1984) Principe d’invariance sur le processus de vraisemblance. Am. Inst. H. Poincaré 20, 120.Google Scholar
15. Gabr, M. M. and Subba Rao, T. (1981) The estimation and prediction of subset bilinear time series models with applications. J. Time Series Analysis 2, 155171.CrossRefGoogle Scholar
16. Hinkley, D. V. (1970) Inference about the change-point in a sequence of random variables. Biometrika 57, 117.Google Scholar
17. Hinkley, D. V. and Hinkley, E. A. (1970) Inference about the change-point in a sequence of binomial variables. Biometrika 57, 477488.Google Scholar
18. Ibragimov, I. A. (1962) On estimation of the spectral function of a stationary Gaussian process. Theory Prob. Appl. 8, 366400.Google Scholar
19. Kedem, B. and Slud, E. (1982) Time series discrimination by high-order crossings. Ann. Statist. 10, 786794.Google Scholar
20. Komlos, J., Major, P. and Tusnady, G. (1975), (1976) An approximation of partial sums of independent RV’s and the sample DF I, II. Z. Wahrscheinlichkeitsth. 32, 11131; 34, 33-58.Google Scholar
21. Mcleish, D. L. (1975) Invariance principles for dependent variables. Z. Wahrscheinlichkeitsth. 32, 165178.Google Scholar
22. Malevich, T. L. (1964) The asymptotic behaviour of an estimate for the spectral function of a stationary Gaussian process. Theory Prob. Appl. 9, 349353.Google Scholar
23. Nikiforov, I. V. (1982) Sequential methods for detecting changes in properties of industrial processes and plants. Soviet-Finnish Symposium on automation in Process Industries, Espoo, Finland, December 1982.Google Scholar
24. Picard, D. (1985) Détecter un changement dans un champ gaussien. Proc. 4th Franco-Belgian Meeting of Statisticians , Publications des Facultés Universitaires Saint Louis, Bruxelles, 93103.Google Scholar
25. Picard, D. (1983) Rupture de modèles en statistique. Thèse d’Etat, Orsay.Google Scholar
26. Priestley, M. B. (1981) Spectral Analysis and Time Series, Vol. 1. Probability and Mathematical Statistics. Academic Press, New York.Google Scholar
27. Shepp, L. A. (1979) The point density of the maximum and its location for a Wiener process with drift. J. Appl. Prob. 16, 423427.CrossRefGoogle Scholar
28. Soumbey, W. (1983) Rupture dans une serie chronologique. Application à la détection de rupture dans les champs gaussiens stationnaires. Thèse 3eme cycle, Orsay.Google Scholar
29. Tong, H. (1977) Some comments on the Canadian lynx data. J.R. Statist. Soc. A 140, 432436.Google Scholar
30. Wilsky, A. S. (1976) A survey of design methods for failure detection in dynamic systems. Automatics 12, 601611.Google Scholar