Abstract
Censored data with functional predictors often emerge in many fields such as biology, neurosciences and so on. Many efforts on functional data analysis (FDA) have been made by statisticians to effectively handle such data. Apart from mean-based regression, quantile regression is also a frequently used technique to fit sample data. To combine the strengths of quantile regression and classical FDA models and to reveal the effect of the functional explanatory variable along with nonfunctional predictors on randomly censored responses, the focus of this paper is to investigate the semi-functional partial linear quantile regression model for data with right censored responses. An inverse-censoring-probability-weighted three-step estimation procedure is proposed to estimate parametric coefficients and the nonparametric regression operator in this model. Under some mild conditions, we also verify the asymptotic normality of estimators of regression coefficients and the convergence rate of the proposed estimator for the nonparametric component. A simulation study and a real data analysis are carried out to illustrate the finite sample performances of the estimators.
Similar content being viewed by others
References
Aneiros-Pérez, G., Cao, R., Fraiman, R., Genest, C., Vieu, P.: Recent advances in functional data analysis and high-dimensional statistics. J. Multivar. Anal. 170, 3–9 (2019)
Aneiros-Pérez, G., Vieu, P.: Automatic estimation procedure in partial linear model with functional data. Stat. Pap. 52(4), 751–771 (2011)
Bongiorno, E.G., Salinelli, E., Goia, A., Vieu, P.: Contributions in infinite-dimensional statistics and related topics. Societa Editrice Esculapio, Bologna (2014)
Bravo, F.: Semiparametric quantile regression with random censoring. Ann. Inst. Stat. Math. 72(1), 265–295 (2020)
Bang, H., Tsiatis, A.A.: Median regression with censored cost data. Biometrics 58(3), 643–649 (2002)
Chaouch, M., Khardani, S.: Randomly censored quantile regression estimation using functional stationary ergodic data. J. Nonparametric Stat. 27(1), 65–87 (2015)
Chen, K., Muller, H.G.: Conditional quantile analysis when covariates are functions, with application to growth data. J. Royal Stat. Soc. : Ser. B 74(1), 67–89 (2012)
Ding, H., Lu, Z., Zhang, J., Zhang, R.: Semi-functional partial linear quantile regression. Stat. Probab. Lett. 142, 92–101 (2018)
Du, J., Zhang, Z., Xu, D.: Estimation for the censored partially linear quantile regression models. Commun. Stat. -Simul. Comput. 47(8), 2393–2408 (2018)
Ferraty, F., Laksaci, A., Tadj, A., Vieu, P.: Rate of uniform consistency for nonparametric estimates with functional variables. J. Stat. Plan. Inference 140(2), 335–352 (2010)
Ferraty, F., Sued, M., Vieu, P.: Mean estimation with data missing at random for functional covariables. Statistics 47(4), 688–706 (2013)
Ferraty, F., Vieu, P.: Nonparametric Functional Data Analysis: Theory and Practice. Springer Science Business Media, Cham (2006)
Feng, H., Luo, Q.: A weighted quantile regression for nonlinear models with randomly censored data. Commun. Stat. -Theory Methods 50(18), 4167–4179 (2021)
Horvath, L., Kokoszka, P.: Inference for Fctional Data with Applications. Springer, New York (2012)
Hsing, T., Eubank, R.: Theoretical Foundations of Functional Data Analysis, with An Introduction to Linear Operators. Wiley, London (2015)
Jiang, F., Cheng, Q., Yin, G., Shen, H.: Functional censored quantile regression. J. Am. Stat. Assoc. 115(530), 931–944 (2020)
Kai, B., Li, R., Zou, H.: New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. Ann. Stat. 39(1), 305 (2011)
Kaplan, E.L., Meier, P.: Nonparametric estimation from incomplete observations. J. Am. Stat. Assoc. 53(282), 457–481 (1958)
Kong, D., Ibrahim, J.G., Lee, E., Zhu, H.: FLCRM: functional linear cox regression model. Biometrics 74(1), 109–117 (2018)
Kato, K.: Estimation in functional linear quantile regression. Ann. Stat. 40(6), 3108–3136 (2012)
Knight, K.: Limiting distributions for L1 regression estimators under general conditions. Ann. Stat. 26, 755–770 (1998)
Koenker, R., Bassett, G., Jr.: Regression quantiles. Econom.: J. Econom. Soc. 46, 33–50 (1978)
Kraus, D.: Components and completion of partially observed functional data. J. Royal Stat. Soc.: Ser. B Stat. Methodol. 77, 777–801 (2015)
Ling, N., Liang, L., Vieu, P.: Nonparametric regression estimation for functional stationary ergodic data with missing at random. J. Stat. Plan. Inference 162, 75–87 (2015)
Ling, N., Vieu, P.: Nonparametric modelling for functional data: selected survey and tracks for future. Statistics 52(4), 934–949 (2018)
Liu, H., Yang, H., Xia, X.: Robust estimation and variable selection in censored partially linear additive models. J. Korean Stat. Soc. 46(1), 88–103 (2017)
Lu, Y., Du, J., Sun, Z.: Functional partially linear quantile regression model. Metrika 77(2), 317–332 (2014)
Portnoy, S.: Censored regression quantiles. J. Am. Stat. Assoc. 98(464), 1001–1012 (2003)
Ramsay, J.O., Silverman, B.W.: Functional Data Analysis. Springer, New York (2005)
Ramsay, J.O., Silverman, B.W.: Applied Functional Data Analysis: Methods and Case Studies. Springer, New York (2002)
Sang, P., Cao, J.: Functional single-index quantile regression models. Stati. Comput. 30, 1–11 (2020)
Shi, G.M., Zhang, Z.Z., Xie, T.F.: Estimation of functional partially linear quantile regression model with censored responses. Math. Pract. Theory 51(3), 152–166 (2021)
Shows, J.H., Lu, W., Zhang, H.H.: Sparse estimation and inference for censored median regression. J. Stat. Plan. Inference 140(7), 1903–1917 (2010)
Tang, L., Zhou, Z., Wu, C.: Weighted composite quantile estimation and variable selection method for censored regression model. Stat. Probab. Lett. 82(3), 653–663 (2012)
Tang, Q., Cheng, L.: Partial functional linear quantile regression. Sci. China Math. 57(12), 2589–2608 (2014)
Van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (1998)
Wang, H.J., Wang, L.: Locally weighted censored quantile regression. J. Am. Stat. Assoc. 104(487), 1117–1128 (2009)
Xiao, J., Xie, T., Zhang, Z.: Estimation in partially observed functional linear quantile regression. J. Syst. Sci. Complex. 35, 1–29 (2021)
Xu, D., Du, J.: Nonparametric quantile regression estimation for functional data with responses missing at random. Metrika 83(8), 977–990 (2020)
Yu, D., Kong, L., Mizera, I.: Partial functional linear quantile regression for neuroimaging data analysis. Neurocomputing 195, 74–87 (2016)
Yu, P., Li, T., Zhu, Z., Zhang, Z.: Composite quantile estimation in partial functional linear regression model with dependent errors. Metrika 82(6), 633–656 (2019)
Acknowledgements
The authors would like to thank the Editor in Chief, the A.E and the two anonymous reviewers for their insightful comments and suggestions, which have led to a great improvement of this manuscript. This research is supported by the National Natural Science Foundation of China (Grant No. 72071068, Grant No. 11901286) and the Fundamental Research Funds for the Central Universities of China (Grant No. JZ2022HGQA0151).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ling, N., Yang, J., Yu, T. et al. Semi-Functional Partial Linear Quantile Regression Model with Randomly Censored Responses. Commun. Math. Stat. (2024). https://doi.org/10.1007/s40304-023-00377-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40304-023-00377-z
Keywords
- Functional data analysis
- Quantile regression
- Semi-functional partial linear model
- Random censorship
- Asymptotic properties