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On curve estimation under order restrictions

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  • Pettersson, Kjell

    (Statistical Research Unit, Department of Economics, School of Business, Economics and Law, Göteborg University)

Abstract

Robust regression is of interest in many problems where assumptions of a parametric function may be inadequate. In this thesis, we study regression problems where the assumptions concern only whether the curve is increasing or decreasing. Examples in economics and public health are given. In a forthcoming paper, the estimation methods presented here will be the basis for likelihood based surveillance systems for detecting changes in monotonicity. Maximum likelihood estimators are thus derived. Distributions belonging to the regular exponential family, for example the normal and Poisson distributions, are considered. The approach is semiparametric, since the regression function is nonparametric and the family of distributions is parametric. In Paper I, “Unimodal Regression in the Two-parameter Exponential Family with Constant or Known Dispersion Parameter”, we suggest and study methods based on the restriction that the curve has a peak (or, equivalently, a trough). This is of interest for example in turning point detection. Properties of the method are described and examples are given. The starting point for Paper II, “Semiparametric Estimation of Outbreak Regression”, was the situation at the outbreak of a disease. A regression may be constant before the outbreak. At the onset, there is an increase. We construct a maximum likelihood estimator for a regression which is constant at first but then starts to increase at an unknown time. The consistency of the estimator is proved. The method is applied to Swedish influenza data and some of its properties are demonstrated by a simulation study.

Suggested Citation

  • Pettersson, Kjell, 2008. "On curve estimation under order restrictions," Research Reports 2007:15, University of Gothenburg, Statistical Research Unit, School of Business, Economics and Law.
    • Handle: RePEc:hhs:gunsru:2007_015
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    File URL: http://hdl.handle.net/2077/9514
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    References listed on IDEAS

    as
    1. E. Andersson, 2002. "Monitoring cyclical processes. A non-parametric approach," Journal of Applied Statistics, Taylor & Francis Journals, vol. 29(7), pages 973-990.
    2. Efromovich S., 2001. "Density Estimation Under Random Censorship and Order Restrictions: From Asymptotic to Small Samples," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 667-684, June.
    3. Marianne Frisén, 2003. "Statistical Surveillance. Optimality and Methods," International Statistical Review, International Statistical Institute, vol. 71(2), pages 403-434, August.
    4. Christian Sonesson & David Bock, 2003. "A review and discussion of prospective statistical surveillance in public health," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 166(1), pages 5-21, February.
    5. Andersson, Eva & Bock, David & Frisén, Marianne, 2007. "Modeling influenza incidence for the purpose of on-line monitoring," Research Reports 2007:5, University of Gothenburg, Statistical Research Unit, School of Business, Economics and Law.
    6. Bock, David & Andersson, Eva & Frisén, Marianne, 2007. "Statistical Surveillance of Epidemics: Peak Detection of Influenza in Sweden," Research Reports 2007:6, University of Gothenburg, Statistical Research Unit, School of Business, Economics and Law.
    7. David Bock & Eva Andersson & Marianne Frisén, 2005. "Statistical surveillance of cyclical processes with application to turns in business cycles," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 24(7), pages 465-490.
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    Cited by:

    1. Schiöler, Linus & Frisén, Marianne, 2008. "On statistical surveillance of the performance of fund managers," Research Reports 2008:4, University of Gothenburg, Statistical Research Unit, School of Business, Economics and Law.
    2. Andersson, Eva, 2008. "Hotelling´s T2 Method in Multivariate On-line Surveillance. On the Delay of an Alarm," Research Reports 2008:3, University of Gothenburg, Statistical Research Unit, School of Business, Economics and Law.
    3. Jonsson, Robert, 2008. "When does Heckman’s two-step procedure for censored data work and when does it not?," Research Reports 2008:2, University of Gothenburg, Statistical Research Unit, School of Business, Economics and Law.

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    More about this item

    Keywords

    Non-parametric; Order restrictions; Two-parameter exponential family; Known dispersion parameter; Poisson distribution; Constant Base-line; Monotonic change; Exponential family;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General

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