Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University 10.15672/hujms.1608956 Large and Complex Data Theory Büyük ve Karmaşık Veri Teorisi Smoothed least absolute deviation estimation in functional linear model https://orcid.org/0009-0007-4943-4319 He Yanfei Shanxi Normal University, https://orcid.org/0000-0002-7002-3211 Yu Ping Shanxi Normal University https://orcid.org/0009-0003-9434-2924 Shi Jianhong Shanxi Normal University https://orcid.org/0009-0005-3115-9334 Xuan Wenhui Shanxi Normal University 06 24 2025 54 3 1107 1127 12 28 2024 05 08 2025 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

The functional linear model extends classical regression by modeling scalar responses as functions of stochastic processes. This paper proposes a novel convolution-type smoothed least absolute deviation estimator that addresses the non-smoothness and strict convexity challenges of conventional least absolute deviation estimation. By approximating both the predictor variable and slope function via functional principal component basis expansions, we develop a robust estimator with strong theoretical guarantees. Under mild regularity conditions, we establish the estimator's consistency aligning with the least absolute deviation estimator as the bandwidth vanishes and derive the convergence rate for the prediction error. Simulation studies demonstrate that the proposed smoothed least absolute deviation estimator outperforms conventional estimation methods--including ordinary least squares, standard least absolute deviation, spline-based regression, penalized spline smoothing, and Bayesian estimation, particularly in scenarios involving heavy-tailed error distributions, outlier contamination, and heteroscedasticity. Applications to the Berkeley Growth Study and the Capital Bike Share dataset further validate its practical utility.

 Convolution-type smoothed least absolute deviation functional principal component analysis least absolute deviation robust estimation This work is supported by National Natural Science Foundation of China (12071267,12401356), Natural Science Foundation of Shanxi Province (202203021222223), National Statistical Science Research Project of China (2022LY089) and the Natural Science Foundation of Shanxi normal University (JYCJ2022004). 12071267,12371272,12401356,202203021222223,2022LY08,JYCJ2022004. [1] J.O. Ramsay and B.W. Silverman, Functional Data Analysis, Springer, New York, 2005. [2] P. Yushkevich, S.M. Pizer and S. Joshi et al., Intuitive, localized analysis of shape variability. Inf. Process. Med. Imaging, Springer, Berlin, 402–408, 2001. [3] N. Locantore, J.S Marron and D.G. Simpson et al., Robust principal component analysis for functional data, Test. 8, 1–73, 1999. [4] J.O. Ramsay and B.W. Silverman, Applied functional data analysis: methods and case studies, Springer, New York, 2002. [5] F. Ferraty, Nonparametric functional data analysis, Springer, New York, 2006. [6] L. Horváth and P. Kokoszka, Inference for functional data with applications, Springer, New York, 2012. [7] T. Hsing and R. Eubank, Theoretical foundations of functional data analysis, with an introduction to linear operators, John Wiley & Sons, Vol. 997, 2015. [8] P. Kokoszka and M. Reimherr, Introduction to functional data analysis, Chapman and Hall/CRC, New York, 2017. [9] C.M. Crainiceanu, J. Leroux and E. Cui, Functional Data Analysis with R, Chapman and Hall/CRC, New York, 2024. [10] H. Cardot, F. Ferraty and P. Sarda, Functional linear model, Stat. Probab. Lett. 45 (1), 11–22, 1999. [11] T.T. Cai and P. Hall, Prediction in functional linear regression, Ann. Stat. 34 (5), 2159–2179, 2006. [12] P. Hall and J.L. Horowitz, Methodology and convergence rates for functional linear regression, Ann. Stat. 35 (1), 70–91, 2007. [13] A.R. Rao and M. Reimherr, Nonlinear functional modeling using neural networks, J. Comput. Graph. Stat. 32 (4), 1248–1257, 2023. [14] H. Yeon, X. Dai and D.J. Nordman, Bootstrap inference in functional linear regression models with scalar response, Bernoulli. 29 (4), 2599–2626, 2023. [15] I. Kalogridis and S. Van Aelst, Robust penalized estimators for functional linear regression, J. Multivar. Anal. 194, 105104, 2023. [16] S. Gurer, H.L. Shang and A. Mandal et al., Locally sparse and robust partial least squares in scalar-on-function regression, Stat. Comput. 34 (5), 150, 2024. [17] J. Liu and L. Shi, Statistical Optimality of Divide and Conquer Kernel-based Functional Linear Regression, J. Mach. Learn. Res. 25 (155), 1–56, 2024. [18] Q. Tang and L. Cheng, Partial functional linear quantile regression, Sci. China Math. 57, 2589–2608, 2014. [19] P. Yu, X. Song and J. Du, Composite expectile estimation in partial functional linear regression model, J. Multivar. Anal. 203, 105343, 2024. [20] R. Ghosal, Hypothesis Testing and Variable Selection in Functional Concurrent Regression Model, PhD thesis, NC State Univ, 2019. [21] J. Martínez, Á. Saavedra and P. J. García-Nieto et al., Air quality parameters outliers detection using functional data analysis in the Langreo urban area (Northern Spain), Appl. Math. Comput. 241, 1–10, 2014. [22] D. Pollard, Asymptotics for least absolute deviation regression estimators, Econom. Theory. 7 (2), 186–199, 1991. [23] R. Koenker and Jr.G. Bassett, Regression quantiles, Econometrica. 46 (1), 33–50. 1978. [24] M. Fernandes, E. Guerre and E. Horta, Smoothing quantile regressions, J. Bus. Econ. Stat. 39 (1), 338–357, 2021. [25] K.M Tan, L. Wang, and W.X. Zhou, High-dimensional quantile regression: Convolution smoothing and concave regularization, J. R. Stat. Soc. B. 84 (1), 205–233, 2022. [26] X. He, X. Pan, K.M. Tan and W.X. Zhou, Smoothed quantile regression with largescale inference, J. Econom. 232 (2), 367–388, 2023. [27] H. Cardot, C. Crambes and P. Sarda, Quantile regression when the covariates are functions, J. Nonparametr. Stat. 17, 841–856, 2005. [28] C. Crambes, A. Kneip and P. Sarda, Smoothing splines estimators for functional linear regression, Ann. Stat. 37, 35–72, 2009. [29] J. Zhang, J. Cao and L. Wang, Robust Bayesian functional principal component analysis, Stat. Comput. 35, 46, 2025. [30] L. Zhou, B. Wang and H. Zou, Sparse convoluted rank regression in high dimensions, J. Amer. Statist. Assoc. 119 (546), 1500–1512, 2024. [31] Z. Wang, Y. Bai and W.K. Härdle et al., Smoothed quantile regression for partially functional linear models in high dimensions, Biom. J. 65 (7), 2200060, 2023. [32] P. Yu, Z. Zhang and J. Du, A test of linearity in partial functional linear regression, Metrika. 79, 953–969, 2016. [33] J. Xiao, P. Yu and X. Song et al., Statistical inference in the partial functional linear expectile regression model, Sci. China Math. 65 (12), 2601–2630, 2022. [34] K. Lange, D.R. Hunter and I. Yang, Optimization transfer using surrogate objective functions, J. Comput. Graph. Stat. 9 (1), 1–20, 2000. [35] P. H. C. Eilers and B. D. Marx, Flexible smoothing with B-splines and penalties, Statist. Sci. 11(2), 89–121, 1996. [36] G. Boente, M. Salibian-Barrera and P. Vena, Robust estimation for semi-functional linear regression models, Comput. Stat. Data Anal. 152, 107041, 2020. [37] R. Tuddenham and M. Snyder, Physical growth of California boys and girls from birth to eighteen years, Univ. Calif. Publ. Child Dev. 1 (2), 183–364, 1954. [38] J. M. Tanner, Growth at Adolescence, Blackwell Scientific Publications, Oxford, 1962. [39] J.S. Kim, A.M. Staicu and A. Maity et al., Additive function-on-function regression, J. Comput. Graph. Stat. 27 (1), 234–244, 2018.