Abstract
This paper is intended as an investigation of parametric estimation for the randomly right censored data. In parametric estimation, the Kullback-Leibler information is used as a measure of the divergence of a true distribution generating a data relative to a distribution in an assumed parametric model M. When the data is uncensored, maximum likelihood estimator (MLE) is a consistent estimator of minimizing the Kullback-Leibler information, even if the assumed model M does not contain the true distribution. We call this property minimum Kullback-Leibler information consistency (MKLI-consistency). However, the MLE obtained by maximizing the likelihood function based on the censored data is not MKLI-consistent. As an alternative to the MLE, Oakes (1986, Biometrics, 42, 177–182) proposed an estimator termed approximate maximum likelihood estimator (AMLE) due to its computational advantage and potential for robustness. We show MKLI-consistency and asymptotic normality of the AMLE under the misspecification of the parametric model. In a simulation study, we investigate mean square errors of these two estimators and an estimator which is obtained by treating a jackknife corrected Kaplan-Meier integral as the log-likelihood. On the basis of the simulation results and the asymptotic results, we discuss comparison among these estimators. We also derive information criteria for the MLE and the AMLE under censorship, and which can be used not only for selecting models but also for selecting estimation procedures.
Similar content being viewed by others
References
Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle, 2nd International Symposium on Information Theory (eds. B. N. Petrov, and F. Csaki), 267-281, Akademiai Kiado, Budapest. (Reproduced (1992) Breakthroughs in Statistics (eds. S. Kotz and N. L. Johnson), 1, 610–624, Springer, New York.)
Andersen, P. K., Borgan, O., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes, Springer, New York.
Cramér, H. (1946). Mathematical Methods of Statistics, Princeton University Press, Princeton, N. J.
Kalbfleisch, J. D. and Prentice, R. L. (1980). The Statistical Analysis of Failure Time Data, Wiley, New York.
Kaplan, E. L. and Meier, P. (1958). Non-parametric estimation from incomplete observations, J. Amer. Statist. Assoc., 53, 457-481.
Konishi, S. and Kitagawa, G. (1996). Generalized information criteria in model selection, Biometrika, 83, 875-890.
Lehmann, E. L. (1983). Theory of Point Estimation, Wiley, New York.
Mauro, D. (1985). A combinatoric approach to the Kaplan-Meier estimation, Ann. Statist., 13, 142-149.
Miller, R. G., Jr. (1983). “What Price Kaplan-Meier?”, Biometrics, 39, 1077-1081.
Oakes, D. (1986). An approximate likelihood procedure for censored data, Biometrics, 42, 177-182.
Stute, W. (1994). The bias of Kaplan-Meier integrals, Scand. J. Statist., 21, 475-484.
Stute, W. (1995). The central limit theorem under random censorship, Ann. Statist., 23, 422-439.
Stute, W. and Wang, J. L. (1993). The strong law under random censorship, Ann. Statist., 21, 1591-1607.
Stute, W. and Wang, J. L. (1994). The jackknife estimate of a Kaplan-Meier integral, Biometrika, 81, 602-606.
Takeuchi, K. (1974). Toukeiteki Suitei no Zenkin Riron, Kyoiku-Shuppan, Japan (in Japanese).
Takeuchi, K. (1976). Distribution of information statistics and criteria for adequacy of models, Mathematical Sciences, 153, 12-18 (in Japanese).
Wald, A. (1949). Note on the consistency of the maximum likelihood estimate, Ann. Math. Statist., 20, 595-601.
Wang, J. L. (1995). M-estimators for censored data: strong consistency, Scand. J. Statist., 22, 197-205.
Author information
Authors and Affiliations
About this article
Cite this article
Suzukawa, A., Imai, H. & Sato, Y. Kullback-Leibler Information Consistent Estimation for Censored Data. Annals of the Institute of Statistical Mathematics 53, 262–276 (2001). https://doi.org/10.1023/A:1012414621245
Issue Date:
DOI: https://doi.org/10.1023/A:1012414621245