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Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case

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  • Bhattacharya, P.K.

Abstract

In a sequence ofn independent random variables the pdf changes fromf(x, 0) tof(x, 0 + [delta]vn-1) after the firstn[lambda] variables. The problem is to estimate[lambda] [set membership, variant] (0, 1 ), where 0 and [delta] are unknownd-dim parameters andvn --> [infinity] slower thann1/2. Letn denote the maximum likelihood estimator (mle) of[lambda]. Analyzing the local behavior of the likelihood function near the true parameter values it is shown under regularity conditions that ifnn2(- [lambda]) is bounded in probability asn --> [infinity], then it converges in law to the timeT([delta]j[delta])1/2 at which a two-sided Brownian motion (B.M.) with drift1/2([delta]'J[delta])1/2[short parallel]t[short parallel]on(-[infinity], [infinity]) attains its a.s. unique minimum, whereJ denotes the Fisher-information matrix. This generalizes the result for small change in mean of univariate normal random variables obtained by Bhattacharya and Brockwell (1976,Z. Warsch. Verw. Gebiete37, 51-75) who also derived the distribution ofT[mu] for[mu] > 0. For the general case an alternative estimator is constructed by a three-step procedure which is shown to have the above asymptotic distribution. In the important case of multiparameter exponential families, the construction of this estimator is considerably simplified.

Suggested Citation

  • Bhattacharya, P.K., 1987. "Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case," Journal of Multivariate Analysis, Elsevier, vol. 23(2), pages 183-208, December.
  • Handle: RePEc:eee:jmvana:v:23:y:1987:i:2:p:183-208
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    Citations

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    Cited by:

    1. Elliott, Graham & Muller, Ulrich K., 2007. "Confidence sets for the date of a single break in linear time series regressions," Journal of Econometrics, Elsevier, vol. 141(2), pages 1196-1218, December.
    2. Bai, Jushan, 1993. "Least squares estimation of a shift in linear processes," MPRA Paper 32878, University Library of Munich, Germany.
    3. Cheng, Tsung-Lin, 2009. "An efficient algorithm for estimating a change-point," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 559-565, March.
    4. Hall, Alastair R. & Han, Sanggohn & Boldea, Otilia, 2008. "Inference regarding multiple structural changes in linear models estimated via two stage least squares," MPRA Paper 9251, University Library of Munich, Germany, revised 20 Jun 2008.
    5. Liang Jiang & Xiaohu Wang & Jun Yu, 2014. "On Bias in the Estimation of Structural Break Points," Working Papers 22-2014, Singapore Management University, School of Economics.
    6. Habibi Reza, 2011. "Exact Distribution of Argmax (Argmin)," Stochastics and Quality Control, De Gruyter, vol. 26(2), pages 155-162, January.
    7. Jin, Hao & Zhang, Jinsuo, 2010. "Subsampling tests for variance changes in the presence of autoregressive parameter shifts," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2255-2265, November.
    8. Hall, Alastair R. & Han, Sanggohn & Boldea, Otilia, 2012. "Inference regarding multiple structural changes in linear models with endogenous regressors," Journal of Econometrics, Elsevier, vol. 170(2), pages 281-302.
    9. Badi H. Baltagi & Chihwa Kao & Long Liu, 2017. "Estimation and identification of change points in panel models with nonstationary or stationary regressors and error term," Econometric Reviews, Taylor & Francis Journals, vol. 36(1-3), pages 85-102, March.
    10. Fiteni, Inmaculada, 1998. "Robust estimation of structural break points," DES - Working Papers. Statistics and Econometrics. WS 10685, Universidad Carlos III de Madrid. Departamento de Estadística.
    11. Jushan, Bai, 1995. "Estimation of multiple-regime regressions with least absolutes deviation," MPRA Paper 32916, University Library of Munich, Germany, revised Feb 1998.
    12. Fiteni, Inmaculada, 2004. "[tau]-estimators of regression models with structural change of unknown location," Journal of Econometrics, Elsevier, vol. 119(1), pages 19-44, March.
    13. Kim, Tae-Hwan & Leybourne, Stephen & Newbold, Paul, 2002. "Unit root tests with a break in innovation variance," Journal of Econometrics, Elsevier, vol. 109(2), pages 365-387, August.
    14. Chen, Gongmeng & Choi, Yoon K. & Zhou, Yong, 2005. "Nonparametric estimation of structural change points in volatility models for time series," Journal of Econometrics, Elsevier, vol. 126(1), pages 79-114, May.
    15. Hall, Alastair R. & Han, Sanggohn & Boldea, Otilia, 2008. "Asymptotic Distribution Theory for Break Point Estimators in Models Estimated via 2SLS," MPRA Paper 9472, University Library of Munich, Germany.
    16. Jandhyala, Venkata K. & Fotopoulos, Stergios B. & Hawkins, Douglas M., 2002. "Detection and estimation of abrupt changes in the variability of a process," Computational Statistics & Data Analysis, Elsevier, vol. 40(1), pages 1-19, July.

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