Summary
We study the estimation of a density and a hazard rate function based on censored data by the kernel smoothing method. Our technique is facilitated by a recent result of Lo and Singh (1986) which establishes a strong uniform approximation of the Kaplan-Meier estimator by an average of independent random variables. (Note that the approximation is carried out on the original probability space, which should be distinguished from the Hungarian embedding approach.) Pointwise strong consistency and a law of iterated logarithm are derived, as well as the mean squared error expression and asymptotic normality, which is obtain using a more traditional method, as compared with the Hajek projection employed by Tanner and Wong (1983).
Article PDF
Similar content being viewed by others
References
Barlow, R.E., Proschan, F.: Statistical theory of reliability and life testing: probability models. Holt, Rinehart and Winston, New York 1975
Blum, J.R., Susarla, V.: Maximal deviation theory of density and failure rate function estimates based on censored data. In: Krishnaiah, P.R. (ed.) Mult. Analysis vol. V, pp. 213–222 New York: North Holland, 1980
Breslow, N., Crowley, J.: A large sample study of the life table and product-limit estimates under random censorship. Ann. Stat.2, 437–443 (1974)
Burke, M.D.: Approximations of some hazard rate estimators in a competing risks model. Stoch. Processes Appl.14, 157–174 (1983)
Burke, M.D., Csörgő, S., Horváth, L.: A correction to and an improvement of strong approximation of some biometric estimates under random censorship. Probab. Th. Rel. Fields79, 51–57 (1988)
Burke, M.D., Horváth, L.: Density and failure rate estimation in a competing risks model. Sankhyā Ser. A.46, 135–154 (1984)
Földes, A., Rejtö, L., Winter, B.B.: Strong consistency properties of nonparametric estimators for randomly censored data. II: Estimation of density and failure rate. Period. Math. Hung.12, 15–29 (1981)
Hajek, J.: Asymptotic normality of simple linear statistics under alternatives. Ann. Math. Stat.39, 325–346 (1968)
Hall, P.: Laws of the iterated logarithm for nonparametric density estimators. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 47–61 (1981)
Kaplan, E.L., Meier, P.: Nonparametric estimation from incomplete observations. J. Am. Stat. Assoc.53, 457–481 (1958)
Liu, R.Y.C., Van Ryzin, J.: A histogram estimator of the hazard rate with censored data. Ann. Stat.13, 592–605 (1985)
Lo, S.H., Mack, Y.P., Wang, J.L.: Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator. Tech. Rep. No. 64, University of California, Davis. (1985)
Lo, S.-H., Singh, K.: The product-limit estimator and the bootstrap: Some asymptotic representations. Probab. Th. Rel. Fields71, 455–465 (1986)
Mack, Y.P., Rosenblatt, M.: Multivariatek-nearest neighbor density estimates. J. Multivariate. Anal.9, 1–15 (1979)
Mileniczuk, J.: Some asymptotic properties of kernel estimators of a density function in case of censored data. Ann. Stat.13, 766–773 (1986)
Padgett, W.J., McNichols, D.T.: Nonparametric density estimation from censored data. Commun. Stat. Theory Methods13, 1581–1613 (1984)
Ramlau-Hansen, H.: Smoothing counting process intensities by means of kernel functions. Ann. Stat.11, 453–466 (1983)
Rice, J., Rosenblatt, M.: Estimation of the log survivor function and hazard function. Sankhya Ser. A38, 60–78 (1976)
Singpurwalla, N.D., Wong, M.-Y.: Kernel estimators of the failure rate function and density function: An analogy. J. Am. Stat. Assoc.78, 478–481 (1983)
Tanner, M.A.: A note of the variable kernel estimator of the hazard function from randomly censored data. Ann. Stat.11, 994–998 (1983)
Tanner, M.A., Wong, W.H.: The estimation of the hazard function from randomly censored data by the kernel method. Ann. Stat.11, 989–993 (1983)
Walter, G., Blum, J.R.: Probability density estimation using delta sequences. Ann. Stat.7, 328–340 (1979)
Watson, G.S., Leadbetter, M.R.: Hazard analysis. I. Biometrika51, 175–184 (1964a)
Watson, G.S., Leadbetter, M.R.: Hazard analysis. II. Sankhyā Ser. A26, 101–116 (1964 b)
Yandell, B.S.: Nonparametric inference for rates and densities with censored serial data. Ph.D. dissertation, Biostatistics Program, University of California, Berkeley (1981)
Yandell, B.S.: Nonparametric inference for rates and densities with censored survival data. Ann. Stat.11, 1119–1135 (1983)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lo, S.H., Mack, Y.P. & Wang, J.L. Density and hazard rate estimation for censored data via strong representation of the Kaplan-Meier estimator. Probab. Th. Rel. Fields 80, 461–473 (1989). https://doi.org/10.1007/BF01794434
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01794434