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GLOBAL BAHADUR REPRESENTATION FOR NONPARAMETRIC CENSORED REGRESSION QUANTILES AND ITS APPLICATIONS

Published online by Cambridge University Press:  25 February 2013

Efang Kong ,
Oliver Linton  and
Yingcun Xia
Efang Kong*
Affiliation:
University of Kent at Canterbury
Oliver Linton
Affiliation:
University of Cambridge
Yingcun Xia
Affiliation:
Nanjing University and National University of Singapore
*
*Address correspondence to Efang Kong, School of Mathematics, Statistics and Actuarial Science, University of Kent at Canterbury, UK; e-mail: [email protected].
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Abstract

This paper is concerned with the nonparametric estimation of regression quantiles of a response variable that is randomly censored. Using results on the strong uniform convergence rate of U-processes, we derive a global Bahadur representation for a class of locally weighted polynomial estimators, which is sufficiently accurate for many further theoretical analyses including inference. Implications of our results are demonstrated through the study of the asymptotic properties of the average derivative estimator of the average gradient vector and the estimator of the component functions in censored additive quantile regression models.

Type
ARTICLES
Information
Econometric Theory , Volume 29 , Issue 5 , October 2013 , pp. 941 - 968
Copyright
Copyright © Cambridge University Press 2013 

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Footnotes

The authors thank Professor Arthur Lewbel (co-editor) and three referees for helpful comments. Xia’s work is partially supported by a grant of NUS R-155-000-121-112.

References

REFERENCES

Arcones, M.A. (1995) A Bernstein-type inequality for U-statistics and U-processes. Statistics and Probability Letters 22, 239–247.CrossRefGoogle Scholar
Babu, G. (1989) Strong representations for LAD estimators in linear models. Probability Theory and Related Fields 83, 547–558.CrossRefGoogle Scholar
Bahadur, R.R. (1966) A note on quantiles in large samples. Annals of Mathematical Statistics 37, 577–580.CrossRefGoogle Scholar
Bang, H. & Tsiatis, A.A. (2002) Median regression with censored cost data. Biometrics 58, 643–649.CrossRefGoogle ScholarPubMed
Belloni, A., Chernozhukov, V., & Fernández-Val, I. (2011) Conditional Quantile Processes Based on Series or Many Regressors. CEMMAP working paper cwp1911.CrossRefGoogle Scholar
Bickel, P. & Rosenblatt, M. (1973) On some global measures of the deviation of density function estimators. Annals of Statistics 1, 1071–1095.CrossRefGoogle Scholar
Buchinsky, M. (1994) Changes in the U.S. wage structure 1963-1987: Application of quantile regression. Econometrica 62, 405–458.CrossRefGoogle Scholar
Buckley, J. & James, I. (1979) Linear regression with censored data. Biometrika 66, 429–436.CrossRefGoogle Scholar
Carroll, R.J. (1978) On almost sure expansions for M-estimates. Annals of Statistics 6, 314–318.CrossRefGoogle Scholar
Chaudhuri, P. (1991a) Global nonparametric estimation of conditional quantile functions and their derivatives. Journal of Multivariate Analysis 39, 246–269.CrossRefGoogle Scholar
Chaudhuri, P. (1991b) Nonparametric estimates of regression quantiles and their local Bahadur representation. Annals of Statistics 19, 760–777.CrossRefGoogle Scholar
Chaudhuri, P., Doksum, K., & Samarov, A. (1997) On average derivative quantile regression. Annals of Statistics 25, 715–744.Google Scholar
Chen, S., Dahl, G.B., & Khan, S. (2005) Nonparametric identification and estimation of a censored location-regression model. Journal of the American Statistical Association 100, 212–221.CrossRefGoogle Scholar
Dabrowska, D.M. (1992) Nonparametric quantile regression with censored data. Sankhyā Series A 54, 252–259.Google Scholar
De Gooijer, J.G. & Zerom, D. (2003) On additive conditional quantiles with high dimensional covariates. Journal of the American Statistical Association 98, 135–146.CrossRefGoogle Scholar
de Uña-Álvarez, J. & Roca-Pardiñasa, J. (2009) Additive models in censored regression. Computational Statistics and Data Analysis 53, 3490–3501.CrossRefGoogle Scholar
Efron, B. (1967) The two-sample problem with censored data. In Le Cam, L. & Neyman, J. (eds.), Proceedings of the Fifth Berkeley Symposium in Mathematical Statistics, IV, pp. 831–853. Prentice-Hall.Google Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modeling and Its Applications. Chapman and Hall.Google Scholar
Gonzalez-Manteiga, W. & Cadarso-Suarez, C. (1994) Asymptotic properties of a generalized Kaplan-Meier estimator with some applications. Journal of Nonparametric Statistics 4, 65–78.CrossRefGoogle Scholar
Guerre, E. & Sabbah, C. (2012) Uniform bias study and Bahadur representation for local polynomial estimators of the conditional quantile function. Econometric Theory 28, 87–129.CrossRefGoogle Scholar
Härdle, W. & Stoker, T.M. (1989) Investigating smooth multiple regression by the method of average derivatives. Journal of American Statistical Association 84, 986–995.Google Scholar
He, X. & Shao, Q.M. (1996) A general Bahadur representation of M-estimators and its application to linear regression with nonstochastic designs. Annals of Statistics 24, 2608–2630.CrossRefGoogle Scholar
Hengartner, N. & Sperlich, S. (2005) Rate optimal estimation with the integration method in the presence of many covariates. Journal of Multivariate Analysis 95, 246–272.CrossRefGoogle Scholar
Heuchenne, C. & Van Keilegom, I. (2007a) Location estimation in nonparametric regression with censored data. Journal of Multivariate Analysis 98, 1558–1582.Google Scholar
Heuchenne, C. & Van Keilegom, I. (2007b) Nonlinear regression with censored data. Technometrics 49, 34–44.CrossRefGoogle Scholar
Heuchenne, C. & Van Keilegom, I. (2008) Estimation in nonparametric location-scale regression models with censored data. Annals of the Institute of Statistical Mathematics 62, 439–463.CrossRefGoogle Scholar
Honoré, B., Khan, S., & Powell, J. (2002) Quantile regression under random censoring. Journal of Econometrics 109, 67–105.CrossRefGoogle Scholar
Horowitz, J. & Lee, S. (2005) Nonparametric estimation of an additive quantile regression model. Journal of the American Statistical Association 100, 1238–1249.CrossRefGoogle Scholar
Huber, P.J. (1981) Robust Statistics. Wiley.CrossRefGoogle Scholar
Johnston, G. (1982) Probabilities of maximal deviations of nonparametric regression function estimates. Journal of Multivariate Analysis 12, 402–414.CrossRefGoogle Scholar
Koenker, R. (2005) Quantile Regression. Cambridge University Press.CrossRefGoogle Scholar
Koenker, R. & Bassett, G. (1978) Regression quantiles. Econometrica 46, 33–50.CrossRefGoogle Scholar
Koenker, R. & Bilias, Y. (2001) Quantile regression for duration data: A reappraisal of the Pennsylvania reemployment bonus experiments. Empirical Economics 26, 199–220.CrossRefGoogle Scholar
Kong, E., Linton, O., & Xia, Y. (2010) Uniform Bahadur representation for local polynomial estimates of M-regression and its application to the additive model. Econometric Theory 26, 1529– 1564.CrossRefGoogle Scholar
Kong, E. & Xia, Y. (2012) A single-index quantile regression model and its estimation. Econometric Theory 28, 730–768.CrossRefGoogle Scholar
Koul, H., Susarla, V., & Van Ryzin, J. (1981) Regression analysis with randomly right censored data. Annals of Statistics 42, 1276–1288.CrossRefGoogle Scholar
Lewbel, A. & Linton, O. (2002) Nonparametric censored and truncated regression. Econometrica 70, 765–779.CrossRefGoogle Scholar
Linton, O. (2001) Estimating additive nonparametric models by partial Lq norm: The curse of fractionality. Econometric Theory 17, 1037–1050.CrossRefGoogle Scholar
Linton, O. & Härdle, W. (1996) Estimation of additive regression models with known links. Biometrika 83, 529–540.CrossRefGoogle Scholar
Linton, O., Mammen, E., Nielsen, J.P., & Van Keilegom, I. (2011) Nonparametric regression with filtered data. Bernoulli 17, 60–87.CrossRefGoogle Scholar
Lu, X. & Cheng, T.L. (2007) Randomly censored partially linear single-index models. Journal of Multivariate Analysis 98, 1895–1922.Google Scholar
Mack, Y.P. & Silverman, B.W. (1982) Weak and strong uniform consistency of kernel regression estimates. Probability Theory and Related Fields 61, 405–415.Google Scholar
Martinsek, A. (1989) Almost Sure Expansion for M-estimators and S-estimators in Regression. Technical Report 25, Dept. Statistics, University of Illinois.Google Scholar
Masry, E. (1996) Multivariate local polynomial regression for time series: Uniform strong convergence and rates. Journal of Time Series Analysis 17, 571–599.CrossRefGoogle Scholar
Nolan, D. & Pollard, D. (1987) U-processes: Rates of convergence. Annals of Statistics 15, 780–799.CrossRefGoogle Scholar
Pakes, A. & Pollard, D. (1989) Simulation and the asymptotics of optimization estimators. Econometrica 57, 1027–1057.CrossRefGoogle Scholar
Peng, L. & Huang, Y. (2008) Survival analysis with quantile regression models. Journal of the American Statistical Association 103, 637–649.Google Scholar
Pollard, D. (1991) Asymptotics for least absolute deviation regression estimators. Econometric Theory 7, 186–199.CrossRefGoogle Scholar
Portnoy, S. (1997) Local asymptotics for quantile smoothing splines. Annals of Statistics 25, 414–434.Google Scholar
Portnoy, S. (2003) Censored regression quantiles. Journal of the American Statistical Association 98, 1001–1012.CrossRefGoogle Scholar
Ritov, Y. (1990) Estimation in a linear regression model with censored data. Annals of Statistics. 18, 27–41.CrossRefGoogle Scholar
Serfling, R. (1980) Approximation Theorems of Mathematical Statistics. Wiley.CrossRefGoogle Scholar
Spierdijk, L. (2008) Nonparametric conditional hazard rate estimation: A local linear approach. Computational Statistics and Data Analysis 52, 2419–2434.CrossRefGoogle Scholar
Stone, C.J. (1980) Optimal rates of convergence for nonparametric estimators. Annals of Statistics 8, 1348–1360.CrossRefGoogle Scholar
Van Keilegom, I. & Veraverbeke, N. (1998) Bootstrapping quantiles in a fixed design regression model with censored data. Journal of Statistical Planning and Inference 69, 115–131.CrossRefGoogle Scholar
Wang, H. & Wang, L. (2009) Locally weighted censored quantile regression. Journal of the American Statistical Association 104, 1117–1128.Google Scholar
Wang, Y., He, S., Zhu, L., & Yuen, K.C. (2007) Asymptotics for a censored generalized linear model with unknown link function. Probability Theory and Its Related Fields 138, 235–267.CrossRefGoogle Scholar
Wu, T.Z., Yu, K., & Yu, Y. (2010) Single-index quantile regression. Journal of Multivariate Analysis 101, 1607–1621.Google Scholar
Wu, W.B. (2005) On the Bahadur representation of sample quantiles for dependent sequences. Annals of Statistics 33, 1934–1963.CrossRefGoogle Scholar
Xia, Y., Zhang, D., & Xu, J. (2010) Dimension reduction and semi-parametric estimation of survival models. Journal of the American Statistical Association 105, 278–290.CrossRefGoogle Scholar
Ying, Z., Jung, S.H., & Wei, L.J. (1995) Survival analysis with median regression models. Journal of the American Statistical Association 90, 178–184.CrossRefGoogle Scholar
Yu, K. & Lu, Z. (2004) Local linear additive quantile regression. Scandinavian Journal of Statistics 31, 333–346.CrossRefGoogle Scholar
Zhou, Z. & Wu, W. (2009) Local linear quantile estimation for non-stationary time series. Annals of Statistics 37, 2696–2729.CrossRefGoogle Scholar