Abstract
We introduce a new estimator for the conditional distribution functions under the proportional hazards model of random censorship. Such estimator generalizes the one proposed by Abdushkurov, Chen and Lin when covariates are present. Asymptotic theory is given for this estimator. First, we established the strong consistency, and also obtain the rate of this convergence. Then, an asymptotic representation for the conditional distribution function estimator leads us to derive its asymptotic normality. The practical performance of the estimation procedure is illustrated on a real data set. Finally, as a further application of the new estimator, some functionals of interest in survival exploratory analysis are brieflys discussed.
Similar content being viewed by others
References
Abdushukurov, A.A. (1984). On some estimates of the distribution function under random censorship. In Conference of Young Scientists. Mathematical Institute of the Academic Sciences, Uzbek SSR, Tashkent. VINITI No. 8756-V (in Russian).
Aerts, M., P. Janssen and N. Veraverbeke (1994). Asymptotic theory for regression quantile estimators in the heteroscedastic regression model. InAsymptotic Statistics (P. Mandl and M. Husková ed) Physica-Verlag, Heidelberg, 151–161.
Beran, R. (1981). Nonparametric regression with randomly censored survival data. Technical Report, University of California, Berkeley.
Cheng, P.E. and G.D. Lin (1984). Maximum likelihood estimation of survival function under the Koziol-Green proportional hazard model. Technical Report B-84-5, Institute of Statistics, Academia Sinica, Taipei, Taiwan.
Cheng, P.E. and G.D. Lin (1987). Maximum likelihood estimation of a survival function under the Koziol-Green proportional hazards models.Statistics and Probability Letters,5, 75–80.
Gsörgö, S. (1988). Estimation in the proportional hazards model of random censorship.Statistics,19, 437–463.
Csörgő, S. and J. Miolniczuk (1988). Density estimation in the simple proportional hazards model.Statistics and Probability Letters,6, 419–426.
Dabrowska, D. M. (1987). Non-parametric regression with censored survival data.Scandinavian Journal of Statistics,14, 181–197.
Dabrowska, D.M. (1989). Uniform consistency of the kernel conditional Kaplan-Meier estimate.Annals of Statistics,17, 1157–1167.
Dabrowska, D.M. (1992a). Variable bandwidth conditional Kaplan-Meiers estimate.Scandinavian Journal of Statistics,19, 351–361.
Dabrowska, D.M. (1992b). Nonparametric quantiles regression with censored data.Sankhya. A,54, 252–259.
De Uña, J., W. González-Manteiga and C. Cadarso-Suarez (1997). Bootstrap selection of the smoothing parameter in density estimation under the Koziol-Green model. In:L 1-Statistical Procedures and Related Topics (Y. Dodge ed.) IMS Lecture Notes Monograph Series, vol. 31, Hayward, California, 385–398.
Fleming, T.R. and D.P. Harrington (1991).Counting Processes and Survival Analysis. Wiley-Interscience.
Földes, A. and L. Rejtő (1981). A LIL type result for the product limit estimator.Z. Wahrscheinlichkeitscheorie Verw. Gebiete,56, 75–86.
Gasser, T. and H.G. Müller (1984). Estimating regression functions and their derivatives by the kernel method.Scandinavian Journal of Statistics,11, 171–185.
Gentleman, R. and J. Crowley (1991). Graphical models for censored data.Journal of the American Statistical Association,86, 678–683.
Georgiev A.A. (1989). Asymptotic properties of the multivariate nadaraya-Watson regression function estimate: the fixed design case.Statistics and Probability Letters,7, 35–40.
Ghorai, J.K. and L.M. pattanaik (1993). Asymptoptically optimal bandwidth selection of the kernel density estimator under the proportional hazards model.Communications in Statistics-Theory and Methods,22 1383–1401.
Gijbels, I. and V.K. Klonias (1981). Density estimation under the Koziol-Green model of censoring by penalized likelihood methods.Canadian Journal of Statistics,19, 23–38.
Gijbels, I. and N. Veraverbeke (1989). Quantile estimation in the proportional hazards model of random censorship.Communications in Statistics-Theory and Methods,18, 1645–1663.
González-Manteiga, W. and C. Cadarso-Suárcz (1991). Linear regression with randomly right-censored data using prior nonparametric estimation. InNonparametric Functional Estimation and Related Topics (G. Roussas ed.) Kluwer Academic Press, 315–328.
González-Manteiga, W. and C. Cadarso-Suzárez (1994). Asympotic properties of a generalized Kaplan-Meier estimator with some application.Journal of Nonparametric Statistics,4, 65–78.
Henze, N. (1993). A quick omnibus test for the proportional hazards model of random Censorship.Statistics,24, 253–263.
Herbst, T. (1992). Test of fit with the Koziol-Green model for random censorship.Statistics and Descisions,10, 163–171.
Koziol, J.A. and S.B. Green (1976). A Cramér-von Mises statistic for randomly censored data.Biometrika,63, 465–474.
Lo, S.-H. and K. Singh (1986). The product-limit estimator and the bootstrap: some asymptotic representations.Probability Theory and Related Fields,71, 455–465.
Serfling, R.J. (1980).Approximation Theorems of Mathematical Statistics. Wiley, New York.
Van Keilegom, I. and N. Veraverbeke (1996). Uniform strong convergence results for the conditional Kaplan-Meier estimator and its quantiles.Communications in Statistics-Theory and Methods 25, 2251–2265.
Van Keilegom, I. and N. Veraverbeke (1997). Estimation and bootstrap with censored data in fixed design nonparametric regression.Annals of the Institute of Statistical Mathematics,49, 467–491.
Van Keilegom, I. and G. Veraverbeke (1998). Bootstrapping quantiles in a fixed design regression model with censored data.Journal of Statistical Planning and Inference,69, 115–131.
Author information
Authors and Affiliations
Additional information
For the second author, this research was supported by a Human Capital postdoc fellowship, under the Programme ERB-CHRX-CT 940693.
Rights and permissions
About this article
Cite this article
Veraverbeke, N., Cadarso-Suárez, C. Estimation of the conditional distribution in a conditional Koziol-green model. Test 9, 97–122 (2000). https://doi.org/10.1007/BF02595853
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02595853