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Semiparametric likelihood ratio inference. (English) Zbl 0928.62036

Summary: 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.

MSC:

62G10 Nonparametric hypothesis testing
62G15 Nonparametric tolerance and confidence regions
62G20 Asymptotic properties of nonparametric inference
62F25 Parametric tolerance and confidence regions
62E20 Asymptotic distribution theory in statistics
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[1] Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer, New York. · Zbl 0769.62061
[2] Bickel, P., Klaassen, C., Ritov, Y. and Wellner, J. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press. · Zbl 0786.62001
[3] Birgé, L. and Massart, P. (1993). Rates of convergence for minimum contrast estimators. Probab. Theory Related Fields 97 113-150. · Zbl 0805.62037
[4] Chang, M. N. (1990). Weak convergence of a self-consistent estimator of the survival function with doubly censored data. Ann. Statist. 18 391-404. · Zbl 0706.62044
[5] 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. · Zbl 0629.62040
[6] Cox, D. R. and Hinkley, D. V. (1974). Theoretical Statistics. Chapman and Hall, London. · Zbl 0334.62003
[7] Gill, R. D. (1989). Nonand semi-parametric maximum likelihood estimators and the von-Mises method (part I). Scand. J. Statist. 16 97-128. · Zbl 0688.62026
[8] Gill, R. D., van der Laan, M. J. and Wijers, B. J. (1995). The line segment problem.
[9] Giné, E. and Zinn, J. (1986). Lectures on the central limit theorem for empirical processes. Lecture Notes in Math. 1221 50-11. Springer, Berlin. · Zbl 0605.60026
[10] Groeneboom, P. (1987). Asy mptotics for interval censored observations. Report 87-18, Dept. Mathematics, Univ. Amsterdam.
[11] Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Basel. · Zbl 0757.62017
[12] 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. · Zbl 0788.62029
[13] Hall, P. and La Scala, B. (1990). Methodology and algorithms of empirical likelihood. Internat. Statist. Rev. 58 109-127. · Zbl 0716.62003
[14] Huang, J. (1996). Efficient estimation for the Cox model with interval censoring. Ann. Statist. 24 540-568. · Zbl 0859.62032
[15] Huang, J. and Wellner, J. A. (1995). Efficient estimation for the Cox model with Case 2 interval censoring. · Zbl 0876.62027
[16] 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. · Zbl 0073.14701
[17] Klaassen, C. A. J. (1987). Consistent estimation of the influence function of locally efficient estimates. Ann. Statist. 15 617-627. · Zbl 0629.62041
[18] Li, G. (1995). On nonparametric likelihood ratio estimation of survival probabilities for censored data. Statist. Probab. Lett. 25 95-104. · Zbl 0851.62026
[19] 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. · Zbl 0827.62033
[20] 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. · Zbl 0747.62093
[21] Ossiander, M. (1987). A central limit theorem under metric entropy with L2 bracketing. Ann. Probab. 15 897-919. · Zbl 0665.60036
[22] Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237-249. JSTOR: · Zbl 0641.62032
[23] Pfanzagl, J. (1990). Estimation in Semiparametric Models. Lecture Notes in Statist. 63. Springer, New York. · Zbl 0704.62034
[24] Qin, J. (1993). Empirical likelihood in biased sample problems. Ann. Statist. 21 1182-1196. · Zbl 0791.62052
[25] Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations. Ann. Statist. 22 300-325. · Zbl 0799.62049
[26] Qin, J. and Wong, A. (1996). Empirical likelihood in a semi-parametric model. Scand. J. Statist. 23 209-220. · Zbl 0854.62032
[27] Rao, R. R. (1963). The law of large numbers for D 0 1 -valued random variables. Theory Probab. Appl. 8 7-74. · Zbl 0122.13303
[28] 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. JSTOR: · Zbl 0869.62081
[29] Rudin, W. (1973). Functional Analy sis. McGraw-Hill, New York. · Zbl 0253.46001
[30] Thomas, D. R. and Grunkemeier, G. L. (1975). Confidence interval estimation of survival probabilities for censored data. J. Amer. Statist. Assoc. 70 865-871. JSTOR: · Zbl 0331.62028
[31] Van der Laan, M. (1993). Efficient and inefficient estimation in semiparametric models. Ph.D. dissertation, Univ. Utrecht. · Zbl 0838.62003
[32] 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. · Zbl 0732.62035
[33] van der Vaart, A. W. (1996). Efficient estimation in semiparametric models. Ann. Statist. 24 862-878. · Zbl 0860.62029
[34] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002
[35] Wijers, B. J. (1995). Nonparametric estimation for a windowed line-segment process. Ph.D. dissertation, Univ. Utrecht. · Zbl 0876.62029
[36] Wong, W. H. and Shen, X. (1995). Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. Ann. Statist. 23 339-362. · Zbl 0829.62002
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