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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.
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 23 (1987)
    Issue (Month): 2 (December)
    Pages: 183-208

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    Handle: RePEc:eee:jmvana:v:23:y:1987:i:2:p:183-208

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    Related research

    Keywords: change-point maximum likelihood estimator weak convergence Brownian motion;

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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. Jushan, Bai, 1995. "Estimation of multiple-regime regressions with least absolutes deviation," MPRA Paper 32916, University Library of Munich, Germany, revised Feb 1998.
    3. Bai, Jushan, 1993. "Least squares estimation of a shift in linear processes," MPRA Paper 32878, University Library of Munich, Germany.
    4. Boldea, O. & Hall, A.R. & Han, S., 2012. "Asymptotic distribution theory for break point estimators in models estimated via 2SLS," Open Access publications from Tilburg University urn:nbn:nl:ui:12-5241457, Tilburg University.
    5. 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.
    6. 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.
    7. 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.
    8. Alastair R. Hall & Sanggohn Han & Otilia Boldea, 2009. "Inference regarding multiple structural changes in linear models with endogenous regressors," Centre for Growth and Business Cycle Research Discussion Paper Series 125, Economics, The Univeristy of Manchester.

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