Abstract
Consider a regression model in which the responses are subject to random right censoring. In this model, Beran studied the nonparametric estimation of the conditional cumulative hazard function and the corresponding cumulative distribution function. The main idea is to use smoothing in the covariates. Here we study asymptotic properties of the corresponding hazard function estimator obtained by convolution smoothing of Beran's cumulative hazard estimator. We establish asymptotic expressions for the bias and the variance of the estimator, which together with an asymptotic representation lead to a weak convergence result. Also, the uniform strong consistency of the estimator is obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beran, R. (1981). Nonparametric regression with randomly censored survival data, Tech. Report, University of California, Berkeley.
Cox, D. R. (1972). Regression models and life-tables, J. Roy. Statist. Soc. Ser. B, 34, 187-202.
Dabrowska, D. M. (1987). Non-parametric regression with censored survival time data, Scand. J. Statist., 14, 181-197.
Dabrowska, D. M. (1989). Uniform consistency of the kernel conditional Kaplan-Meier estimate, Ann. Statist., 17, 1157-1167.
Diehl, S. and Stute, W. (1988). Kernel density and hazard function estimation in the presence of censoring, J. Multivariate Anal., 25, 299-310.
Li, G. (1997). Optimal rate local smoothing in a multiplicative intensity counting process model, Math. Methods Statist., 6, 224-244.
Li, G. and Doss, H. (1995). An approach to nonparametric regression for life history data using local linear fitting, Ann. Statist., 23, 787-823.
Lo, S.-H., Mack, Y. P. and Wang, J.-L. (1989). Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator, Probab. Theory Related Fields, 80, 461-473.
McKeague, I. W. and Utikal, K. J. (1990). Inference for a nonlinear counting process regression model, Ann. Statist., 18, 1172-1187.
McNichols, D. T. and Padgett, W. J. (1985). Nonparametric methods for hazard rate estimation from right-censored samples, Journal of the Chinese Statistical Association, 23, 1-15.
Müller, H.-G. and Wang, J.-L. (1990). Locally adaptive hazard smoothing, Probab. Theory Related Fields, 85, 523-538.
Nielsen, J. P. and Linton, O. B. (1995). Kernel estimation in a nonparametric marker dependent hazard model, Ann. Statist., 23, 1735-1748.
Ramlau-Hansen, H. (1983). Smoothing counting process intensities by means of kernel functions, Ann. Statist., 11, 453-466.
Rice, J. and Rosenblatt, M. (1976). Estimation of the log survivor function and hazard function, Sankhyā Ser. A, 38, 60-78.
Rosenblatt, M. (1971). Curve estimates, Ann. Math. Statist., 42, 1815-1842.
Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics, Wiley, New York.
Singpurwalla, N. D. and Wong, M.-Y. (1983). Estimation of the failure rate. A survey of nonparametric methods, part I: Non-bayesian methods, Comm. Statist. Theory Methods, 12, 559-588.
Tanner, M. A. and Wong, W. H. (1983). The estimation of the hazard function from randomly censored data by the kernel method, Ann. Statist., 11, 989-993.
van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes, Springer, New York.
Van Keilegom, I. and Veraverbeke, N. (1996). Uniform strong convergence results for the conditional Kaplan-Meier estimator and its quantiles, Comm. Statist. Theory Methods, 25, 2251-2265.
Van Keilegom, I. and Veraverbeke, N. (1997a). Weak convergence of the bootstrapped conditional Kaplan-Meier process and its quantile process, Comm. Statist. Theory Methods, 26, 853-869.
Van Keilegom, I. and Veraverbeke, N. (1997b). Estimation and bootstrap with censored data in fixed design nonparametric regression, Ann. Inst. Statist. Math., 49, 467-491.
Van Keilegom, I. and Veraverbeke, N. (1998). Bootstrapping quantiles in a fixed design regression model with censored data, J. Statist. Plann. Inference, 69, 115-131.
Van Keilegom, I. and Veraverbeke, N. (2001). Density and hazard estimation in censored regression models, Bernoulli, (under revision).
Watson, G. S. and Leadbetter, M. R. (1964a). Hazard analysis I, Biometrika, 51, 175-184.
Watson, G. S. and Leadbetter, M. R. (1964b). Hazard analysis II, Sankhyā Ser. A, 26, 110-116.
Yandell, B. S. (1983). Nonparametric inference for rates with censored survival data, Ann. Statist., 11, 1119-1135.
Author information
Authors and Affiliations
About this article
Cite this article
Van Keilegom, I., Veraverbeke, N. Hazard Rate Estimation in Nonparametric Regression with Censored Data. Annals of the Institute of Statistical Mathematics 53, 730–745 (2001). https://doi.org/10.1023/A:1014696717644
Issue Date:
DOI: https://doi.org/10.1023/A:1014696717644