The Annals of Statistics

Semiparametric likelihood ratio inference

S. A. Murphy and A. W. van der Vaart

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Likelihood ratio tests and related confidence intervals for a real parameter in the presence of an infinite dimensional nuisance parameter are considered. In all cases, the estimator of the real parameter has an asymptotic normal distribution. However, the estimator of the nuisance parameter may not be asymptotically Gaussian or may converge to the true parameter value at a slower rate than the square root of the sample size. Nevertheless the likelihood ratio statistic is shown to possess an asymptotic chi-squared distribution. The examples considered are tests concerning survival probabilities based on doubly censored data, a test for presence of heterogeneity in the gamma frailty model, a test for significance of the regression coefficient in Cox's regression model for current status data and a test for a ratio of hazards rates in an exponential mixture model. In both of the last examples the rate of convergence of the estimator of the nuisance parameter is less than the square root of the sample size.

Article information

Ann. Statist., Volume 25, Number 4 (1997), 1471-1509.

First available in Project Euclid: 9 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G15: Tolerance and confidence regions 62G20: Asymptotic properties 62F25: Tolerance and confidence regions

Least favorable submodel profile likelihood confidence interval


Murphy, S. A.; van der Vaart, A. W. Semiparametric likelihood ratio inference. Ann. Statist. 25 (1997), no. 4, 1471--1509. doi:10.1214/aos/1031594729.

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  • Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York.
  • Bickel, P., Klaassen, C., Ritov, Y. and Wellner, J. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press.
  • Birg´e, L. and Massart, P. (1993). Rates of convergence for minimum contrast estimators. Probab. Theory Related Fields 97 113-150.
  • Chang, M. N. (1990). Weak convergence of a self-consistent estimator of the survival function with doubly censored data. Ann. Statist. 18 391-404.
  • Chang, M. N. and Yang, G. L. (1987). Strong consistency of a nonparametric estimator of the survival function with doubly censored data. Ann. Statist. 15 1536-1547.
  • Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics. Chapman and Hall, London.
  • Gill, R. D. (1989). Nonand semi-parametric maximum likelihood estimators and the von-Mises method (part I). Scand. J. Statist. 16 97-128.
  • Gill, R. D., van der Laan, M. J. and Wijers, B. J. (1995). The line segment problem. Preprint.
  • Gin´e, E. and Zinn, J. (1986). Lectures on the central limit theorem for empirical processes. Lecture Notes in Math. 1221 50-11. Springer, Berlin.
  • Groeneboom, P. (1987). Asy mptotics for interval censored observations. Report 87-18, Dept. Mathematics, Univ. Amsterdam.
  • Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkh¨auser, Basel.
  • Gu, M. G. and Zhang, C. H. (1993). Asy mptotic properties of self-consistent estimators based on doubly censored data. Ann. Statist. 21 611-624.
  • Hall, P. and La Scala, B. (1990). Methodology and algorithms of empirical likelihood. Internat. Statist. Rev. 58 109-127.
  • Huang, J. (1996). Efficient estimation for the Cox model with interval censoring. Ann. Statist. 24 540-568.
  • Huang, J. and Wellner, J. A. (1995). Efficient estimation for the Cox model with Case 2 interval censoring. Preprint.
  • Kiefer, J. and Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many nuisance parameters. Ann. Math. Statist. 27 887-906.
  • Klaassen, C. A. J. (1987). Consistent estimation of the influence function of locally efficient estimates. Ann. Statist. 15 617-627.
  • Li, G. (1995). On nonparametric likelihood ratio estimation of survival probabilities for censored data. Statist. Probab. Lett. 25 95-104.
  • Murphy, S. A. (1994). Consistency in a proportional hazards model incorporating a random effect. Ann. Statist. 22 712-731. Murphy, S. A. (1995a). Asy mptotic theory for the frailty model Ann. Statist. 23 182-198.
  • Nielsen, G. G., Gill, R. D., Andersen, P. K. and Sorensen, T. I. (1992). A counting process approach to maximum likelihood estimation in frailty models. Scand. J. Statist. 19 25-44.
  • Ossiander, M. (1987). A central limit theorem under metric entropy with L2 bracketing. Ann. Probab. 15 897-919.
  • Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237-249.
  • Pfanzagl, J. (1990). Estimation in Semiparametric Models. Lecture Notes in Statist. 63. Springer, New York.
  • Qin, J. (1993). Empirical likelihood in biased sample problems. Ann. Statist. 21 1182-1196.
  • Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations. Ann. Statist. 22 300-325.
  • Qin, J. and Wong, A. (1996). Empirical likelihood in a semi-parametric model. Scand. J. Statist. 23 209-220.
  • Rao, R. R. (1963). The law of large numbers for D 0 1 -valued random variables. Theory Probab. Appl. 8 7-74.
  • Roeder, K., Carroll, R. J. and Lindsay, B. G. (1996). A semiparametric mixture approach to case-control studies with errors in covariables. J. Amer. Statist. Assoc. 91 722-732.
  • Rudin, W. (1973). Functional Analy sis. McGraw-Hill, New York.
  • Thomas, D. R. and Grunkemeier, G. L. (1975). Confidence interval estimation of survival probabilities for censored data. J. Amer. Statist. Assoc. 70 865-871.
  • Van der Laan, M. (1993). Efficient and inefficient estimation in semiparametric models. Ph.D. dissertation, Univ. Utrecht.
  • van der Vaart, A. W. (1991). On differentiable functionals. Ann. Statist. 19 178-204. van der Vaart, A. W. (1994a). Infinite dimensional M-estimators In Proceedings of the 6th International Vilnius Conference (B. Grigelionis, J. Kubilius, H. Pragarauskas and V. Statulevicius, eds.) 715-734. VSP International Science Publishers, Zeist. van der Vaart, A. W. (1994b). Bracketing smooth functions. Stochastic Process. Appl. 52 93-105. van der Vaart, A. W. (1994c). On a model of Hasminskii and Ibragimov. In Proceedings of the Kolmogorov Semester at the Euler International Mathematical Institute, St. Petersburg (A. A. Zaitsev, ed.). North-Holland, Amsterdam. To appear.
  • van der Vaart, A. W. (1996). Efficient estimation in semiparametric models. Ann. Statist. 24 862-878.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • Wijers, B. J. (1995). Nonparametric estimation for a windowed line-segment process. Ph.D. dissertation, Univ. Utrecht.
  • Wong, W. H. and Shen, X. (1995). Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. Ann. Statist. 23 339-362.