Skip to main content
Log in

Maximum likelihood estimation in the multi-path change-point problem

  • Estimation
  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

Maximum likelihood estimators of the parameters of the distributions before and after the change and the distribution of the time to change in the multi-path change-point problem are derived and shown to be consistent. The maximization of the likelihood can be carried out by using either the EM algorithm or results from mixture distributions. In fact, these two approaches give equivalent algorithms. Simulations to evaluate the performance of the maximum likelihood estimators under practical conditions, and two examples using data on highway fatalities in the United States, and on the health effects of urea formaldehyde foam insulation, are also provided.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Billingsley, P. (1986).Probability and Measure, 2nd ed., Wiley, New York.

    Google Scholar 

  • Carlstein, E. (1988). Nonparametric change-point estimation,Ann. Statist.,16, 188–197.

    Google Scholar 

  • Cobb, G. W. (1978). The problem of the Nile: conditional solution to a change point problem,Biometrika,62, 243–251.

    Google Scholar 

  • Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm,J. Roy. Statist. Soc. Ser. B,39, 1–38.

    Google Scholar 

  • Hinkley, D. V. (1970). Inference about the change-point in a sequence of random variables,Biometrika,57, 1–16.

    Google Scholar 

  • Joseph, L. (1989). The multi-path change-point, Ph.D. Thesis, Department of Mathematics and Statistics, McGill University, Montreal.

    Google Scholar 

  • Joseph, L. and Wolfson, D. B. (1992). Estimation in multi-path change-point problems,Comm. Statist. Theory and Methods,21, 897–913.

    Google Scholar 

  • Kiefer, J. and Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters,Ann. Math. Statist.,27, 887–906.

    Google Scholar 

  • L'Abbé, K. (1984). Health effects of urea-formaldehyde foam insulation, Master's Thesis, Department of Epidemiology and Biostatistics, McGill University, Montreal.

    Google Scholar 

  • Neyman, J. and Scott, E. L. (1948). Consistent estimates based on partially consistent observations,Econometrica,16, 1–32.

    Google Scholar 

  • Peters, B. C. and Walker, H. F. (1978). An iterative procedure for obtaining maximum likelihood estimates of the parameters for a mixture of normal distributions,SIAM J. Appl. Math.,35, 362–378.

    Google Scholar 

  • Picard, D. (1985). Testing and estimating change-points in time series,Adv. in Appl. Probab.,17, 841–867.

    Google Scholar 

  • Redner, R. (1981). Note on the consistency of the maximum likelihood estimate for nonidentifiable distributions,Ann. Statist.,9, 225–228.

    Google Scholar 

  • Redner, R. and Walker, H. F. (1984). Mixture densities, maximum likelihood and the EM algorithm,SIAM Rev.,26, 195–239.

    Google Scholar 

  • Shaban, S. A. (1980). Change-point problem and two-phase regression: an annotated bibliography,Internat. Statist. Rev.,48, 83–93.

    Google Scholar 

  • Wald, A. (1949). Note on the consistency of the maximum likelihood estimate,Ann. Math. Statist.,20, 595–601.

    Google Scholar 

  • Worsely, K. J. (1986). Confidence regions and test for a change-point in a sequence of exponential family random variables,Biometrika,73, 91–104.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

This work was supported in part by the Natural Science and Engineering Council of Canada, and the Fonds pour la Formation de chercheurs et l'aide à la Recherche Gouvernment du Québec.

Lawrence Joseph is also a member of the Department of Epidemiology and Biostatistics of McGill University.

About this article

Cite this article

Joseph, L., Wolfson, D.B. Maximum likelihood estimation in the multi-path change-point problem. Ann Inst Stat Math 45, 511–530 (1993). https://doi.org/10.1007/BF00773352

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00773352

Key words and phrases

Navigation