Skip to main content
SpringerLink
Log in

Penalized least absolute deviations estimation for nonlinear model with change-points

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

This paper studies the asymptotic properties of a smoothed least absolute deviations estimator in a nonlinear parametric model with multiple change-points occurring at the unknown times with independent and identically distributed errors. The model is nonlinear in the sense that between two successive change-points the regression function is nonlinear into respect to parameters. It is shown via Monte Carlo simulations that its performance is competitive with that of least absolute deviations estimator and it is more efficient than the least squares estimator, particularly in the presence of the outlier points. If the number of change-points is unknown, an estimation criterion for this number is proposed. Interest of this method is that the objective function is approximated by a differentiable function and if the model contains outliers, it detects correctly the location of the change-points.

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

Access this article

Log in via an institution

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

  • Babu GJ (1989) Strong representations for LAD estimators in linear models. Probab Theory Relat Fields 83: 547–558

    Article  MathSciNet  MATH  Google Scholar 

  • Bai J (1995) Least absolute deviation estimation of a shift. Econom Theory 11: 403–436

    Article  Google Scholar 

  • Bai J (1998) Estimation of multiple-regime regressions with least absolute deviation. J Stat Plan Inference 74: 103–134

    Article  MATH  Google Scholar 

  • Bai J, Perron P (1998) Estimating and testing linear models with multiple structural changes. Econometrica 66(1): 47–78

    Article  MathSciNet  MATH  Google Scholar 

  • Bassett G, Koenker R (1978) Asymptotic theory of least absolute error regression. J Am Stat Assoc 73: 618–622

    Article  MathSciNet  MATH  Google Scholar 

  • Bhattacharya PK (1994) Some aspects of change-point analysis. IMS Lecture Notes-Monograph Series, vol 23. Hayward, CA, pp 28–56

  • Ciuperca G (2009a) The M-estimation in a multi-phase random nonlinear model. Stat Probab Lett 75(5): 573–580

    Article  MathSciNet  Google Scholar 

  • Ciuperca G (2009b) Estimating nonlinear regression with and without change-points by the LAD method. Ann Inst Stat Math (in press). see http://www.ism.ac.jp/editsec/aism/forthcoming_all.html

  • Ciuperca G, Dapzol N (2008) Maximum likelihood estimator in a multi-phase random regression model. Statistics 42(4): 363–381

    Article  MathSciNet  MATH  Google Scholar 

  • DasGupta M, Mishra S (2007) Least absolute deviation estimation of linear econometric models: a literature review. MPRA paper no. 1781

  • Dielman TE (2005) Least absolute value regression: recent contributions. J Stat Comput Simul 75(4): 263–286

    Article  MathSciNet  MATH  Google Scholar 

  • Feder PI (1975a) On asymptotic distribution theory in segmented regression problems-identified case. Ann Stat 3: 49–83

    Article  MathSciNet  MATH  Google Scholar 

  • Feder PI (1975b) The log likelihood ratio in segmented regression. Ann Stat 3: 84–97

    Article  MathSciNet  MATH  Google Scholar 

  • Hitomi K, Kagihara M (2001) Calculation method for nonlinear dynamic least-absolute deviations estimator. J Jpn Stat Soc 31(1): 39–51

    MathSciNet  MATH  Google Scholar 

  • Kim HK, Choi SH (1995) Asymptotic properties of non-linear least absolute deviation estimators. J Korean Stat Soc 24: 127–139

    Article  MathSciNet  Google Scholar 

  • Koenker R, Bassett G (1982) Test of linear hypotheses and l 1 estimation. Econometrica 50: 1577–1583

    Article  MathSciNet  MATH  Google Scholar 

  • Koul HL, Qian L (2002) Asymptotics of maximum likelihood estimator in a two-phase linear regression model. J Stat Plan Inference 108: 99–119

    Article  MathSciNet  MATH  Google Scholar 

  • Koul HL, Qian L, Surgailis D (2003) Asymptotics of M-estimators in two-phase linear regression models. Stoch Process Appl 103: 123–154

    Article  MathSciNet  MATH  Google Scholar 

  • Lavielle M, Moulines E (1997) Dtection de ruptures multiples dans la moyenne d’un processus alatoire. CRAS 324: 239–243

    MathSciNet  MATH  Google Scholar 

  • Lavielle M, Moulines E (2000) Least-squares estimation of an unknown number of shifts in a time series. J Time Ser Anal 21(1): 33–59

    Article  MathSciNet  MATH  Google Scholar 

  • Liu J, Wu S, Zidek JV (1997) On segmented multivariate regressions. Stat Sin 7: 497–525

    MathSciNet  MATH  Google Scholar 

  • Oberhofer W (1982) The consistency of nonlinear regression minimizing the L 1-norm. Ann Stat 10(1): 316–319

    Article  MathSciNet  MATH  Google Scholar 

  • Pollard D (1991) Asymptotics for least absolute deviation regression estimators. Econom Theory 7: 186–199

    Article  MathSciNet  Google Scholar 

  • Rukhin AL, Vajda I (1997) Change-point estimation as a nonlinear regression problem. Statistics 30(3): 181–200

    Article  MathSciNet  MATH  Google Scholar 

  • Wang L, Wang J (2004) The limiting behavior of the least deviation estimators for threshold autoregressive models. J Multivar Anal 89: 243–260

    Article  MATH  Google Scholar 

  • Weiss AA (1991) Estimating nonlinear dynamic models using least absolute error estimation. Econom Theory 7: 46–68

    Article  Google Scholar 

  • Yao YC (1988) Estimating the number of change-points via Schwarz’s criterion. Stat Probab Lett 6: 181–189

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Camille Jordan, Université de Lyon, Université Lyon 1, CNRS, UMR 5208, Bat. Braconnier, 43, blvd du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Gabriela Ciuperca

Authors
  1. Gabriela Ciuperca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriela Ciuperca.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ciuperca, G. Penalized least absolute deviations estimation for nonlinear model with change-points. Stat Papers 52, 371–390 (2011). https://doi.org/10.1007/s00362-009-0236-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-009-0236-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation