## Weighted empirical likelihood in some two-sample semiparametric models with various types of censored data.(English)Zbl 1132.62083

Summary: The weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator $$(\widetilde \theta _n, \widetilde F _n)$$ for the underlying parameter $$\theta _{0}$$ and distribution $$F_{0}$$ is derived, and the strong consistency of $$(\widetilde \theta _n, \widetilde F_n)$$ and the asymptotic normality of $$\widetilde \theta _n$$ are established. Under biased sampling models, the weighted empirical log-likelihood ratio is shown to have an asymptotic scaled chi-squared distribution for censored data aforementioned. For right censored data, doubly censored data and partly interval-censored data, it is shown that $$\sqrt n (\widetilde F _n - F_0)$$ weakly converges to a centered Gaussian process, which leads to a consistent goodness-of-fit test for the case-control logistic regression models.

### MSC:

 62N01 Censored data models 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 62N02 Estimation in survival analysis and censored data 62N03 Testing in survival analysis and censored data 62G05 Nonparametric estimation
Full Text:

### References:

 [1] Bazaraa, M. S., Sherali, H. D. and Shetty, C. M. (1993). Nonlinear Programming, Theory and Algorithms , 2nd ed. Wiley, New York. · Zbl 1140.90040 [2] Bickel, P. J. and Ren, J. (1996). The m out of n bootstrap and goodness of fit tests with doubly censored data. Lecture Notes in Statist. 109 35-47. Springer, Berlin. · Zbl 0839.62054 [3] Bickel, P. J. and Ren, J. (2001). The Bootstrap in hypothesis testing. State of the Art in Statistics and Probability Theory. Festschrift for Willem R. van Zwet (M. de Gunst, C. Klaassen and A. van der Vaart, eds.) 91-112. IMS, Beachwood, OH. · Zbl 1380.62183 [4] 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 [5] Enevoldsen, A. K., Borch-Johnson, K., Kreiner, S., Nerup, J. and Deckert, T. (1987). Declining incidence of persistent proteinuria in type I (insulin-dependent) diabetic patient in Denmark. Diabetes 36 205-209. [6] Geskus, R. and Groeneboom, P. (1999). Asymptotically optimal estimation of smooth functionals for interval censoring, case 2. Ann. Statist. 27 627-674. · Zbl 0954.62034 [7] Gill, R. D. (1983). Large sample behavior of the product-limit estimator on the whole line. Ann. Statist. 11 49-58. · Zbl 0518.62039 [8] Gill, R. D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method (Part 1). Scand. J. Statist. 16 97-128. · Zbl 0688.62026 [9] 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. · Zbl 0668.62024 [10] Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation . Birkhäuser, Berlin. · Zbl 0757.62017 [11] Gu, M. G. and Zhang, C. H. (1993). Asymptotic properties of self-consistent estimators based on doubly censored data. Ann. Statist. 21 611-624. · Zbl 0788.62029 [12] Huang, J. (1999). Asymptotic properties of nonparametric estimation based on partly interval-censored data. Statist. Sinica 9 501-519. · Zbl 0933.62038 [13] Huang, J. and Wellner, J. A. (1995). Asymptotic normality of the NPMLE of the linear functionals for interval censored data, case 1. Statist. Neerlandica 49 153-163. · Zbl 0832.62029 [14] Iranpour, R. and Chacon, P. (1988). Basic Stochastic Processes . MacMillan, New York. · Zbl 0681.60035 [15] Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc . 53 457-481. JSTOR: · Zbl 0089.14801 [16] Kim, M. Y., De Gruttola, V. G. and Lagakos, S. W. (1993). Analyzing doubly censored data with covariates, with application to AIDS. Biometrics 49 13-22. · Zbl 0776.62083 [17] Mykland, P. A. and Ren, J. (1996). Self-consistent and maximum likelihood estimation for doubly censored data. Ann. Statist. 24 1740-1764. · Zbl 0867.62019 [18] Odell, P. M., Anderson, K. M. and D’Agostino, R. B. (1992). Maximum likelihood estimation for interval-censored data using a Weibull-based accelerated failure time model. Biometrics 48 951-959. [19] Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237-249. JSTOR: · Zbl 0641.62032 [20] Patil, G. P. and Rao, C. R. (1977). The weighted distributions: A survey of their applications. In Applications of Statistics (P. R. Krishnaiah, ed.) 383-405. North-Holland, Amsterdam. · Zbl 0371.62034 [21] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes in FORTRAN. The Art of Scientific Computing . Cambridge Univ. Press. · Zbl 0778.65002 [22] Prentice, R. L. and Pyke, P. (1979). Logistic disease incidence models and case-control studies. Biometrika 66 403-411. JSTOR: · Zbl 0428.62078 [23] Qin, J. (1993). Empirical likelihood in biased sample problems. Ann. Statist. 21 1182-1196. · Zbl 0791.62052 [24] Qin, J. and Zhang, B. (1997). A goodness-of-fit test for logistic regression models based on case-control data. Biometrika 84 609-618. JSTOR: · Zbl 0888.62045 [25] Ren, J. (2001). Weight empirical likelihood ratio confidence intervals for the mean with censored data. Ann. Inst. Statist. Math. 53 498-516. · Zbl 0985.62027 [26] Ren, J. (2003). Goodness of fit tests with interval censored data. Scand. J. Statist. 30 211-226. · Zbl 1034.62039 [27] Ren, J. and Gu, M. G. (1997). Regression M-estimators with doubly censored data. Ann. Statist. 25 2638-2664. · Zbl 0907.62045 [28] Ren, J. and Peer, P. G. (2000). A study on effectiveness of screening mammograms. Internat. J. Epidemiology 29 803-806. [29] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York. · Zbl 0538.62002 [30] Stute, W. and Wang, J. L. (1993). The strong law under random censorship. Ann. Statist. 21 1591-1607. · Zbl 0785.60020 [31] Turnbull, B. W. (1974). Nonparametric estimation of a survivorship function with doubly censored data. J. Amer. Statist. Assoc. 69 169-173. JSTOR: · Zbl 0281.62044 [32] Vardi, Y. (1982). Nonparametric estimation in the presence of length bias. Ann. Statist. 10 616-620. · Zbl 0491.62034 [33] Vardi, Y. (1985). Empirical distributions in selection bias models. Ann. Statist. 13 178-203. · Zbl 0578.62047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.