• Bernoulli
  • Volume 18, Number 2 (2012), 586-605.

Reparametrization of the least favorable submodel in semi-parametric multisample models

Yuichi Hirose and Alan Lee

Full-text: Open access


The method of estimation in Scott and Wild (Biometrika 84 (1997) 57–71 and J. Statist. Plann. Inference 96 (2001) 3–27) uses a reparametrization of the profile likelihood that often reduces the computation times dramatically. Showing the efficiency of estimators for this method has been a challenging problem. In this paper, we try to solve the problem by investigating conditions under which the efficient score function and the efficient information matrix can be expressed in terms of the parameters in the reparametrized model.

Article information

Bernoulli, Volume 18, Number 2 (2012), 586-605.

First available in Project Euclid: 16 April 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

efficiency efficient information bound efficient score multisample profile likelihood semi-parametric model


Hirose, Yuichi; Lee, Alan. Reparametrization of the least favorable submodel in semi-parametric multisample models. Bernoulli 18 (2012), no. 2, 586--605. doi:10.3150/10-BEJ342.

Export citation


  • [1] Bickel, P.J., Klaassen, C.A.J., Ritov, Y. and Wellner, J.A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Baltimore, MD: Johns Hopkins Univ. Press.
  • [2] Breslow, N., McNeney, B. and Wellner, J.A. (2003). Large sample theory for semiparametric regression models with two-phase, outcome dependent sampling. Ann. Statist. 31 1110–1139.
  • [3] Breslow, N.E., Robins, J.M. and Wellner, J.A. (2000). On the semi-parametric efficiency of logistic regression under case-control sampling. Bernoulli 6 447–455.
  • [4] Gilbert, P.B. (2000). Large sample theory of maximum likelihood estimates in semiparametric biased sampling models. Ann. Statist. 28 151–194.
  • [5] Gilbert, P.B., Lele, S.R. and Vardi, Y. (1999). Maximum likelihood estimation in semiparametric selection bias models with application to AIDS vaccine trials. Biometrika 86 27–43.
  • [6] Gill, R.D., Vardi, Y. and Wellner, J.A. (1988). Large sample theory of empirical distributions in biased sampling models. Ann. Statist. 16 1069–1112.
  • [7] Lawless, J.F., Kalbfleisch, J.D. and Wild, C.J. (1999). Semiparametric methods for response-selective and missing data problems in regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 413–438.
  • [8] Lee, A.J. and Hirose, Y. (2008). Semi-parametric efficiency bounds for regression models under case-control sampling: The profile likelihood approach. Ann. Inst. Statist. Math. 62 1023–1052.
  • [9] Newey, W.K. (1994). The asymptotic variance of semiparametric estimators. Econometrica 62 1349–1382.
  • [10] Robins, J.M., Hsieh, F.S. and Newey, W. (1995). Semiparametric efficient estimation of a conditional density with missing or mismeasured covariates. J. Roy. Statist. Soc. Ser. B 57 409–424.
  • [11] Robins, J.M., Rotnitzky, A. and Zhao, L.P. (1994). Estimation of regression coefficients when some regressors are not always observed. J. Amer. Statist. Assoc. 89 846–866.
  • [12] Scott, A.J. and Wild, C.J. (1997). Fitting regression models to case-control data by maximum likelihood. Biometrika 84 57–71.
  • [13] Scott, A.J. and Wild, C.J. (2001). Maximum likelihood for generalised case-control studies. J. Statist. Plann. Inference 96 3–27.
  • [14] Vardi, Y. (1985). Empirical distributions in selection bias models (with discussion). Ann. Statist. 13 178–205.