Abstract
In this article the authors establish the Bahadur type representations for the kernel quantile estimator and the kernel estimator of the derivatives of the quantile function on the basis of left truncated and right censored data. Under suitable conditions, with probability one, the exact convergence rate of the remainder term in the representations is obtained. As a by-product, the LIL, the asymptotic normality for those kernel estimators are derived.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.R. Bahadur. A Note on Quantiles in Large Samples.Ann. Math. Statist., 1966, 37: 577–580
J. Kiefer. On Bahadur's Representation of Sample Quantiles.Ann. Math. Statist., 1967, 38: 1323–1342
M. Csörgö, P. Révész. Strong Approximations in Probability and Statistics. Academic press, New York, 1981
E. Parzen. Nonparametric Statistical Data Modeling.J. Amer. Statist. Assoc., 1979, 74: 105–131
M. Falk. Asymptotic Normality of the Kernel Quantile Estimators.Ann. Statist., 1985, 13: 410–416
S.J. Sheather, J.S. Marron. Kernel Quantile Estimators.J. Amer. Statist. Assoc., 1990, 85: 410–416
X. Xiang. On Bahadur Representation of Kernel Quantile Estimators.Scand. J. Statist., 1994, 21: 169–178
X. Xiang. Bahadur Representation of the Kernel Quantile Estimator under Random Censorship.J. Multivariate Anal., 1995, 54: 193–209
W.Y. Tsai, N.P. Jeweli, M.C. Wang. A Note on the Product Limit Estimator under Right Consoring and Left Truncation.Biometrika, 1987, 74: 883–886
I. Gijbels, J.L. Wang. Strong Representations of the Survival Function Estimator for Truncated and Censored Data with Applications.J. Multivariate Anal., 1993, 47: 210–229.
Zhou Yong. A Note on the TJW Product-limit, Estimator for Truncated and Censored Data.Statist. Probab. Lett., 1996, 12: 381–387
P. Hall. Laws of the Iterated Logarithm for Nonparametric Density Estimators.Z. Wahrsch. Verw. Gebiete, 1981, 56: 47–61
E. Csáki. Some Notes on the Law of the Iterated Logarithm for Empirical Distribution Function. In: Colloq. Math. Soc. János Bolyai, Vol. 11, Limit Theorems of Probability (P. Révész, ed.), North-Holland, Amsterdam, 1975, 47–57
G.R. Shorack, J.A. Wellner. Empirical Processes with Applications to Statistics. Wiley, New York, 1986
W. Stute. The Oscillation Behavior of Empirical Processes.Ann. Probab., 1982, 10: 86–107
M. Loeve. Probability Theory I, 4th Ed. Springer-Verlag, New York, 1977
E. Parzen. On Estimation of a Probability Density Function and Mode.Ann. Math. Statist., 1962, 33: 1065–1076
Author information
Authors and Affiliations
Additional information
This research is supported by the Postdoctoral Programe Foundation and the National Natural Sciences Foundation of China.
Rights and permissions
About this article
Cite this article
Liuquan, S., Zhongguo, Z. Bahadur representation of the kernel quantile estimator under truncated and censored data. Acta Mathematicae Applicatae Sinica 15, 257–268 (1999). https://doi.org/10.1007/BF02669830
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02669830