Abstract
Currently, working with partially observed functional data has attracted a greatly increasing attention, since there are many applications in which each functional curve may be observed only on a subset of a common domain, and the incompleteness makes most existing methods for functional data analysis ineffective. In this paper, motivated by the appealing characteristics of conditional quantile regression, the authors consider the functional linear quantile regression, assuming the explanatory functions are observed partially on dense but discrete point grids of some random subintervals of the domain. A functional principal component analysis (FPCA) based estimator is proposed for the slope function, and the convergence rate of the estimator is investigated. In addition, the finite sample performance of the proposed estimator is evaluated through simulation studies and a real data application.
Similar content being viewed by others
References
Koenker R W, Bassett G, and Jan N, Regression quantiles, Econornetrica, 1978, 46(1): 33–50.
Koenker R W, Quantile Regression (Econometric Society Monographs), Cambridge University Press, Cambridge, 2005.
Ramsay J and Silverman B, Funtional Data Analysis, Springer, New York, 2005.
Ferraty F and Vieu P, Nonparametric Functional Data Analysis: Theory and Practice, Springer, New York, 2006.
Ferraty F and Romain Y, The Oxford Handbook of Functional Data Analaysis, Oxford University Press, Oxford, 2011.
Horváth L and Kokoszka P, Inference for Functional Data with Applications, Springer, New York, 2012.
Cai T and Hall P, Prediction in functional linear regression, Annals of Statistics, 2006, 34(5): 2159–2179.
Hall P and Horowitz J L, Methodology and convergence rates for functional linear regression, Annals of Statistics, 2007, 35(1): 70–91.
Yuan M and Cai T, A reproducing kernel Hilbert space approach to functional linear regression, Annals of Statistics, 2010, 38(6): 3412–3444.
Delaigle A and Hall P, Methodology and theory for partial least squares applied to functional data, Annals of Statistics, 2012, 40(1): 322–352.
Zhao Y, Wavelet-based lasso in functional linear regression, Dissertation Abstracts International: Section B: The Sciences and Engineering, 2012, 21(3): 600–617.
Zhao Y, Chen H, and Ogden R T, Wavelet-based weighted lasso and screening approaches in functional linear regression, Journal of Computational and Graphical Statistics, 2015, 24(3): 655–675.
Cardot H, Crambes C, and Sarda P, Quantile regression when the covariates are functions, Journal of Nonparametric Statistics, 2005, 17(7): 841–856.
Chen K and Müller H, Conditional quantile analysis when covariates are functions, with application to growth data, Journal of the Royal Statistical Society, 2012, 74(1): 67–89.
Kato K, Estimation in functional linear quantile regression, The Annals of Statistics, 2012, 40(6): 3108–3136.
Tang Q G and Cheng L S, Partial functional linear quantile regression, Science China Mathematics, 2014, 57(12): 2589–2608.
Yu P, Zhang Z, and Du J, A test of linearity in partial functional linear regression, Metrika, 2016, 79(8): 953–969.
Yao F, Sue-Chee S, and Wang F, Regularized partially functional quantile regression, Journal of Multivariate Analysis, 2017, 156: 39–56.
Ma H, Li T, Zhu H, et al., Quantile regression for functional partially linear model in ultra-high dimensions, Computational Statistics and Data Analysis, 2019, 129: 135–147.
Bugni F A, Specification test for missing functional data, Econometric Theory, 2012, 28(5): 959–1002.
Delaigle A and Hall P, Classification using censored functional data, Journal of the American Statistical Association, 2013, 108(504): 1269–1283.
Liebl D, Modeling and forecasting electricity spot prices: A functional data perspective, Annals of Applied Statistics, 2013, 7(3): 1562–1592.
Gellar J E, Colantuoni E, Needham D M, et al., Variable-domain functional regression for modeling ICU data, Journal of the American Statistical Association, 2014, 109(508): 1425–1439.
Goldberg Y, Ritov Y, and Mandelbaum A, Predicting the continuation of a function with applications to call center data, Journal of Statistical Planning and Inference, 2014, 147: 53–65.
Kraus D, Components and completion of partially observed functional data, Journal of the Royal Statistical Society, Series B: Statistical Methodology, 2015, 77(4): 777–801.
Delaigle A and Hall P, Approximating fragmented functional data by segments of Markov chains, Biometrika, 2016, 103(4): 779–799.
Gromenko O, Kokoszka P, and Sojka J, Evaluation of the cooling trend in the ionosphere using functional regression with incomplete curves, Annals of Applied Statistics, 2017, 11(2): 898–918.
Dawson M and Müller H G, Dynamic modeling of conditional quantile trajectories, with application to longitudinal snippet data, Journal of the American Statistical Association, 2018, 113(524): 1612–1624.
Kraus D and Stefanucci M, Classification of functional fragments by regularized linear classifiers with domain selection, Biometrika, 2019, 106(1): 161–180.
Descary M H and Panaretos V M, Recovering covariance from functional fragments, Biometrika, 2019, 106(1): 145–160.
Yao F, Müller H G, and Wang J L, Functional data analysis for sparse longitudinal data, Journal of the American Statistical Association, 2005, 100(470): 577–590.
Rice J A and Silverman B W, Estimating the mean and covariance structure nonparametrically when the data are curves, Journal of the Royal Statistical Society: Series B (Methodological), 1991, 53(1): 233–243.
Febrero-bande F, Statistical computing in functional data analysis, Journal of Statistical Softaware, 2012, 51(4): 1–28.
Aneiros-Pérez G and Vieu P, Semi-functional partial linear regression, Statistics & Probability Letters, 2006, 76(11): 1102–1110.
van der Vaart Aad W and Wellner Jon A, Weak Convergence and Empirical Processes, Springer, New York, 1996.
Bosq D, Linear Processes in Function Spaces, Springer, New York, 2000.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the National Natural Science Foundation of China under Grant No. 11771032.
This paper was recommended for publication by Editor TANG Niansheng.
Rights and permissions
About this article
Cite this article
Xiao, J., Xie, T. & Zhang, Z. Estimation in Partially Observed Functional Linear Quantile Regression. J Syst Sci Complex 35, 313–341 (2022). https://doi.org/10.1007/s11424-020-0019-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-020-0019-7