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Efficiency of profile likelihood in semi-parametric models

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Abstract

Profile likelihood is a popular method of estimation in the presence of an infinite-dimensional nuisance parameter, as the method reduces the infinite-dimensional estimation problem to a finite-dimensional one. In this paper we investigate the efficiency of a semi-parametric maximum likelihood estimator based on the profile likelihood. By introducing a new parametrization, we improve on the seminal work of Murphy and van der Vaart (J Am Stat Assoc, 95: 449–485, 2000): our improvement establishes the efficiency of the estimator through the direct quadratic expansion of the profile likelihood, which requires fewer assumptions. To illustrate the method an application to two-phase outcome-dependent sampling design is given.

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References

  • Andersen P.K., Borgan O., Gill R.D., Keiding N. (1993) Statistical models based on counting processes. Springer, New York

    MATH  Google Scholar 

  • Begun J.M., Hall W.J., Huang W.M., Wellner J.A. (1983) Information and asymptotic efficiency in parametric–nonparametric models. The Annals of Statistics 11: 432–452

    Article  MathSciNet  MATH  Google Scholar 

  • Bickel P.J., Klaassen C.A.J., Ritov Y., Wellner J.A. (1993) Efficient and adaptive estimation for semiparametric models. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  • Breslow N.E., Chatterjee N. (1999) Design and analysis of two-phase studies with binary outcome applied to Wilms tumor prognosis. Journal of the Royal Statistical Society: Series C 48: 457–468

    Article  MATH  Google Scholar 

  • Breslow N.E., Holubkov R. (1997) Maximum likelihood estimation of logistic regression parameters under two-phase outcome-dependent sampling. Journal of the Royal Statistical Society: Series B 59: 447–461

    Article  MathSciNet  MATH  Google Scholar 

  • Breslow, N. E., McNeney, B., Wellner, J. A. (2000a). Large sample theory for semiparametric regression models with two-phase outcome dependent sampling. Technical Report 381, Department of Statistics, University of Washington.

  • Breslow N.E., Robins J.M., Wellner J.A. (2000b) On the semi-parametric efficiency of logistic regression under case–control sampling. Bernoulli 6: 447–455

    Article  MathSciNet  MATH  Google Scholar 

  • Breslow N.E., McNeney B., Wellner J.A. (2003) Large sample theory for semiparametric regression models with two-phase outcome dependent sampling. The Annals of Statistics 31: 1110–1139

    Article  MathSciNet  MATH  Google Scholar 

  • Gilbert P.G. (2000) Large sample theory of maximum likelihood estimation in semiparametric selection biased sampling models. The Annals of Statistics 28: 151–194

    Article  MathSciNet  MATH  Google Scholar 

  • Gilbert P.G., Lele S.R., Vardi Y. (1999) Maximum likelihood estimation in semiparametric selection bias models with application to AIDS vaccine trials. Biometrika 86: 27–43

    Article  MathSciNet  MATH  Google Scholar 

  • Gill R.D. (1989) Non- and semi-parametric maximum likelihood estimators and the von Mises method (part 1). Scandinavian Journal of Statistics 16: 97–128

    MathSciNet  MATH  Google Scholar 

  • Gill R.D., Vardi Y., Wellner J.A. (1988) Large sample theory of empirical distribution in biased sampling models. The Annals of Statistics 3: 1069–1112

    Article  MathSciNet  Google Scholar 

  • Godambe V.P. (1991) Orthogonality of estimating functions and nuisance parameters. Biometrika 78: 143–151

    Article  MathSciNet  Google Scholar 

  • Hall W.S., Newell M.L. (1979) The mean value theorem for vector valued functions: A simple proof. Mathematics Magazine 52: 157–158

    Article  MathSciNet  Google Scholar 

  • Kiefer J., Wolfowitz J. (1956) Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Annals of Mathematical Statistics 27: 887–906

    Article  MathSciNet  MATH  Google Scholar 

  • Kolmogorov A.N., Fomin S.V. (1975) Introductory real analysis. Dover, New York

    Google Scholar 

  • Lawless J.L., Kalbfleish J.D., Wild C.J. (1999) Semiparametric methods for response-selective and missing data problems in regression. Journal of the Royal Statistical Society: Series B 61: 413–438

    Article  MATH  Google Scholar 

  • Lee, A. J., Hirose, Y. (2008). Semi-parametric efficiency bounds for regression models under case–control sampling: The profile likelihood approach. Annals of the Institute of Statistical Mathematics. doi:10.1007/s10463-008-0205-1.

  • Murphy S.A., van der Vaart A.W. (1999) Observed information in semi-parametric models. Bernoulli 5: 381–412

    Article  MathSciNet  MATH  Google Scholar 

  • Murphy S.A., van der Vaart A.W. (2000) On profile likelihood (with discussion). Journal of the American Statistical Association 95: 449–485

    Article  MathSciNet  MATH  Google Scholar 

  • Newey W.K. (1990) Semi-parametric efficiency bounds. Journal of Applied Economics 5: 99–135

    Article  MATH  Google Scholar 

  • Newey W.K. (1994) The asymptotic variance of semi-parametric estimators. Econometrica 62: 1349–1382

    Article  MathSciNet  MATH  Google Scholar 

  • Prentice R.L., Pyke R. (1979) Logistic disease incidence models and case–control studies. Biometrika 66: 403–411

    Article  MathSciNet  MATH  Google Scholar 

  • Robins J.M., Rotnitzky A., Zhao L.P. (1994) Estimation of regression coefficients when some regressors are not always observed. Journal of the American Statistical Association 89: 846–866

    Article  MathSciNet  MATH  Google Scholar 

  • Robins J.M., Hsieh F., Newey W.K. (1995) Semiparametric efficient estimation of a conditional density with missing or mismeasured covariates. Journal of the Royal Statistical Society: Series B 57: 409–424

    MathSciNet  MATH  Google Scholar 

  • Scott A.J., Wild C.J. (1997) Fitting regression models to case–control data by maximum likelihood. Biometrika 84: 57–71

    Article  MathSciNet  MATH  Google Scholar 

  • Scott A.J., Wild C.J. (2001) Maximum likelihood for generalised case–control studies. Journal of Statistical Planning and Inference 96: 3–27

    Article  MathSciNet  MATH  Google Scholar 

  • Seber G.A.F., Lee A.J. (2003) Linear regression analysis (2nd ed.). Wiley, New York

    MATH  Google Scholar 

  • Shapiro A. (1990) On concepts of directional differentiability. Journal of Optimization Theory and Applications 66: 477–487

    Article  MathSciNet  MATH  Google Scholar 

  • Tsiatis A.B. (2006) Semiparametric theory and missing data. Springer, New York

    MATH  Google Scholar 

  • van de Geer S.A. (2000) Empirical processes in M-estimation. Cambridge University of Press, Cambridge

    Google Scholar 

  • van der Vaart A.W. (1991) On differentiable functionals. The Annals of Statistics 19: 178–204

    Article  MathSciNet  MATH  Google Scholar 

  • van der Vaart A.W. (1998) Asymptotic statistics. Cambridge University of Press, Cambridge

    MATH  Google Scholar 

  • van der Vaart A.W., Wellner J.A. (1996) Weak convergence and empirical processes. Springer, New York

    MATH  Google Scholar 

  • van der Vaart, A. W., Wellner, J. A. (2000). Preservation theorems for Glivenko–Cantelli and uniform Glivenko–Cantelli class. In E. Gine, D. M. Mason, J. A. Wellner (Eds.), High dimensional probability (Vol. II, pp. 115–134). Boston: Birkhäuser.

  • van der Vaart A.W., Wellner J.A. (2001) Consistency of semiparametric maximum likelihood estimators for two-phase sampling. Canadian Journal of Statistics 29: 269–288

    Article  MATH  Google Scholar 

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  1. School of Mathematics, Statistics and Operations Research, Victoria University of Wellington, Wellington, New Zealand

    Yuichi Hirose

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  1. Yuichi Hirose
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Correspondence to Yuichi Hirose.

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Hirose, Y. Efficiency of profile likelihood in semi-parametric models. Ann Inst Stat Math 63, 1247–1275 (2011). https://doi.org/10.1007/s10463-010-0280-y

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  • DOI: https://doi.org/10.1007/s10463-010-0280-y

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