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Estimation in Partially Observed Functional Linear Quantile Regression

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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.

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References

  1. Koenker R W, Bassett G, and Jan N, Regression quantiles, Econornetrica, 1978, 46(1): 33–50.

    Article  MathSciNet  Google Scholar 

  2. Koenker R W, Quantile Regression (Econometric Society Monographs), Cambridge University Press, Cambridge, 2005.

    Book  Google Scholar 

  3. Ramsay J and Silverman B, Funtional Data Analysis, Springer, New York, 2005.

    Book  Google Scholar 

  4. Ferraty F and Vieu P, Nonparametric Functional Data Analysis: Theory and Practice, Springer, New York, 2006.

    MATH  Google Scholar 

  5. Ferraty F and Romain Y, The Oxford Handbook of Functional Data Analaysis, Oxford University Press, Oxford, 2011.

    Google Scholar 

  6. Horváth L and Kokoszka P, Inference for Functional Data with Applications, Springer, New York, 2012.

    Book  Google Scholar 

  7. Cai T and Hall P, Prediction in functional linear regression, Annals of Statistics, 2006, 34(5): 2159–2179.

    MathSciNet  MATH  Google Scholar 

  8. Hall P and Horowitz J L, Methodology and convergence rates for functional linear regression, Annals of Statistics, 2007, 35(1): 70–91.

    MathSciNet  MATH  Google Scholar 

  9. Yuan M and Cai T, A reproducing kernel Hilbert space approach to functional linear regression, Annals of Statistics, 2010, 38(6): 3412–3444.

    Article  MathSciNet  Google Scholar 

  10. Delaigle A and Hall P, Methodology and theory for partial least squares applied to functional data, Annals of Statistics, 2012, 40(1): 322–352.

    MathSciNet  MATH  Google Scholar 

  11. Zhao Y, Wavelet-based lasso in functional linear regression, Dissertation Abstracts International: Section B: The Sciences and Engineering, 2012, 21(3): 600–617.

    MathSciNet  Google Scholar 

  12. 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.

    Article  MathSciNet  Google Scholar 

  13. Cardot H, Crambes C, and Sarda P, Quantile regression when the covariates are functions, Journal of Nonparametric Statistics, 2005, 17(7): 841–856.

    Article  MathSciNet  Google Scholar 

  14. 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.

    Article  MathSciNet  Google Scholar 

  15. Kato K, Estimation in functional linear quantile regression, The Annals of Statistics, 2012, 40(6): 3108–3136.

    Article  MathSciNet  Google Scholar 

  16. Tang Q G and Cheng L S, Partial functional linear quantile regression, Science China Mathematics, 2014, 57(12): 2589–2608.

    Article  MathSciNet  Google Scholar 

  17. Yu P, Zhang Z, and Du J, A test of linearity in partial functional linear regression, Metrika, 2016, 79(8): 953–969.

    Article  MathSciNet  Google Scholar 

  18. Yao F, Sue-Chee S, and Wang F, Regularized partially functional quantile regression, Journal of Multivariate Analysis, 2017, 156: 39–56.

    Article  MathSciNet  Google Scholar 

  19. 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.

    Article  MathSciNet  Google Scholar 

  20. Bugni F A, Specification test for missing functional data, Econometric Theory, 2012, 28(5): 959–1002.

    Article  MathSciNet  Google Scholar 

  21. Delaigle A and Hall P, Classification using censored functional data, Journal of the American Statistical Association, 2013, 108(504): 1269–1283.

    Article  MathSciNet  Google Scholar 

  22. Liebl D, Modeling and forecasting electricity spot prices: A functional data perspective, Annals of Applied Statistics, 2013, 7(3): 1562–1592.

    Article  MathSciNet  Google Scholar 

  23. 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.

    Article  MathSciNet  Google Scholar 

  24. 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.

    Article  MathSciNet  Google Scholar 

  25. 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.

    Article  MathSciNet  Google Scholar 

  26. Delaigle A and Hall P, Approximating fragmented functional data by segments of Markov chains, Biometrika, 2016, 103(4): 779–799.

    Article  MathSciNet  Google Scholar 

  27. 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.

    Article  MathSciNet  Google Scholar 

  28. 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.

    Article  MathSciNet  Google Scholar 

  29. Kraus D and Stefanucci M, Classification of functional fragments by regularized linear classifiers with domain selection, Biometrika, 2019, 106(1): 161–180.

    Article  MathSciNet  Google Scholar 

  30. Descary M H and Panaretos V M, Recovering covariance from functional fragments, Biometrika, 2019, 106(1): 145–160.

    Article  MathSciNet  Google Scholar 

  31. 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.

    Article  MathSciNet  Google Scholar 

  32. 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.

    MathSciNet  MATH  Google Scholar 

  33. Febrero-bande F, Statistical computing in functional data analysis, Journal of Statistical Softaware, 2012, 51(4): 1–28.

    Google Scholar 

  34. Aneiros-Pérez G and Vieu P, Semi-functional partial linear regression, Statistics & Probability Letters, 2006, 76(11): 1102–1110.

    Article  MathSciNet  Google Scholar 

  35. van der Vaart Aad W and Wellner Jon A, Weak Convergence and Empirical Processes, Springer, New York, 1996.

    Book  Google Scholar 

  36. Bosq D, Linear Processes in Function Spaces, Springer, New York, 2000.

    Book  Google Scholar 

Download references

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Correspondence to Zhongzhan Zhang.

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.

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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

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  • DOI: https://doi.org/10.1007/s11424-020-0019-7

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