Statistical inference in functional quadratic expectile regression model

Abstract

The functional quadratic regression model assumes a polynomial, rather than linear relationship between the scalar response variable and a functional predictor variable. This paper focuses on the statistical inference tailored for the functional quadratic expectile regression model. Functional coefficients are approximated by the functional principal component basis functions, and asymptotic properties of estimators are derived under some mild conditions. Furthermore, to inspect the effect of the functional quadratic term on the response variable, we develop an expectile rank score test and establish its asymptotic property. Simulations are conducted to assess the empirical performance of the proposed estimation methods and test statistic. Results indicate that the proposed estimators are comparable to competing estimation methods and the newly proposed expectile rank score test is effective. Finally, the advantages of our methodologies are illustrated using a real-data example.

Keywords

• expectile regression
• expectile rank score test
• functional quadratic regression model
• functional principal component analysis

Supporting Institution

Shanxi Normal University

Project Number

12401356,2022LY089,JYCJ2022004,2025XS110.

Ethical Statement

This manuscript is original work not previously published and is not being considered elsewhere.

Thanks

This work is partially supported by National Natural Science Foundation of China (12401356), National Statistical Science Research Project of China (2022LY089), the Natural Science Foundation of Shanxi normal University (JYCJ2022004) and Postgraduate Education Innovation Program of Shanxi Province (2025XS110).

References

1. [1] J. O. Ramsay and B. W. Silverman, Applied Functional Fata Analysis: Methods and Case Studies, Springer, New York, 2002.
2. [2] J. O. Ramsay and B. W. Silverman, Functional Data Analysis, Springer, New York, 2005.
3. [3] H. Cardot, F. Ferraty and P. Sarda, Functional linear model, Statist. Probab. Lett. 45 (1), 11–22, 1999.
4. [4] H. Cardot, F. Ferraty, A. Mas and P. Sarda, Testing hypotheses in the functional linear model, Scand. J. Stat. 30 (1), 241–255, 2003.
5. [5] K. Chen and H.-G. Müller, Conditional quantile analysis when covariates are functions, with application to growth data, J. R. Stat. Soc. Ser. B. 74 (1), 67–89, 2012.
6. [6] K. Kato, Estimation in functional linear quantile regression, Ann. Stat. 40 (6), 3108– 3136, 2012.
7. [7] T. T. Cai and P. Hall, Prediction in functional linear regression, Ann. Stat. 34 (5), 2159–2179, 2006.
8. [8] T. T. Cai and M. Yuan, Minimax and adaptive prediction for functional linear regression, J. Amer. Stat. Assoc. 107 (499), 1201–1216, 2012.

9. [9] P. Hall and J. L. Horowitz, Methodology and convergence rates for functional linear regression, Ann. Stat. 35 (1), 70–91, 2007.
10. [10] Y. Li and T. Hsing, On rates of convergence in functional linear regression, J. Multivar. Anal. 98 (9), 1782–1804, 2007.
11. [11] B. Liu, L. Wang and J. Cao, Estimating functional linear mixed-effects regression models, Comput. Stat. Data Anal. 106, 153–164, 2017.
12. [12] 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.
13. [13] H. Yeon, X. Dai and D. J. Nordman, Bootstrap inference in functional linear regression models with scalar response, Bernoulli 29 (4), 2599–2626, 2023.
14. [14] F. Yao and H.-G. Müllerller, Functional quadratic regression, Biometrika. 97 (1), 49–64, 2010.
15. [15] L. Horváth and R. Reeder, A test of significance in functional quadratic regression, Bernoulli. 19 (5A), 2120–2151, 2013.
16. [16] G. Boente and D. Parada, Robust estimation for functional quadratic regression models, Comput. Stat. Data Anal. 187, 107798, 2023.
17. [17] G. Shi, T. Xie and Z. Zhang, Statistical inference for the functional quadratic quantile regression model, Metrika. 83 (8), 937–960, 2020.
18. [18] W. K. Newey and J. L. Powell, Asymmetric least squares estimation and testing, Econometrica. 55, 819–847, 1987.
19. [19] J. Taylor, Estimating Value at Risk and Expected Shortfall using expectiles, J. Financ. Econom. 6 (2), 231–252, 2008.
20. [20] J. Cai and C. Weng, Optimal reinsurance with expectile, Scand. Actuar. J. 2016 (7), 624–645, 2016.
21. [21] F. Bellini and E. Di Bernardino, Risk management with expectiles, Eur. J. Financ. 23 (6), 487–506, 2017.
22. [22] J. Zhou, P. Mennig, D. Zhou and J. Sauer, Shadow prices of agrochemicals in the Chinese farming sector: A convex expectile regression approach, J. Environ. Manage, 366, 121518, 2024.
23. [23] H. Holzmann and B. Klar, Expectile asymptotics, Electron. J. Stat. 10 (2), 2355–2371, 2016.
24. [24] R. Man, K. M. Tan, Z. Wang and W.-X. Zhou, Retire: Robust expectile regression in high dimensions, J. Econom. 239 (2), 105459, 2024.
25. [25] P. Burdejová and W. K. Härdle, Dynamic semi-parametric factor model for functional expectiles, Comput. Stat. 34 (2), 489–502, 2019.
26. [26] M. Mohammedi, S. Bouzebda and A. Laksaci, The consistency and asymptotic normality of the kernel type expectile regression estimator for functional data, J. Multivar. Anal. 181, 104673, 2021.
27. [27] I. M. Almanjahie, S. Bouzebda, Z. C. Elmezouar and A. Laksaci, The functional kNN estimator of the conditional expectile: Uniform consistency in number of neighbors, Statist. Risk Model. 87,1007–1035, 2024.
28. [28] I. M. Almanjahie, S. Bouzebda, Z. Kaid and A. Laksaci, Nonparametric estimation of expectile regression in functional dependent data, J. Nonparametr. Stat. 34 (1), 250–281, 2022.
29. [29] J. Xiao, P. Yu, X. Song and Z. Zhang, Statistical inference in the partial functional linear expectile regression model, Sci. China Math. 65 (12), 2601–2630, 2022.
30. [30] P. Yu, X. Song and J. Du, Composite expectile estimation in partial functional linear regression model, J. Multivar. Anal. 203, 105343, 2024.
31. [31] F. Scheipl and S. Greven, Identifiability in penalized function-on-function regression models, Electron. J. Stat. 10, 495–526, 2016.
32. [32] Y. Gu and H. Zou, High-dimensional generalizations of asymmetric least squares regression and their applications, Ann. Stat. 44 (6), 2661–2694, 2016.
33. [33] L. Liao, C. Park and H. Choi, Penalized expectile regression: an alternative to penalized quantile regression, Ann. Inst. Stat. Math. 71 (2), 409–438, 2019.
34. [34] X. Cai, L. Xue and J. Cao, Variable selection for multiple function-on-function linear regression, Statist. Sin. 32 (3), 1435–1465, 2022.
35. [35] F. Sobotka and T. Kneib, Geoadditive expectile regression, Comput. Stat. Data Anal. 56 (4), 755–767, 2012.
36. [36] C. Gutenbrunner, J. Jurečková, R. Koenker and S. Portnoy, Tests of linear hypotheses based on regression rank scores, J. Nonparametr. Stat. 2 (4), 307–331, 1993.
37. [37] H. J. Wang, Z. Zhu and J. Zhou, Quantile regression in partially linear varying coefficient models, Ann. Stat. 37, 3841–3866, 2009.
38. [38] H. Zhu, T. Zhang, Y. Zhang and H. Lian, Estimation and inference in functional varying-coefficient single-index quantile regression models, J. Nonparametr. Stat. 36 (3), 643–672, 2024.
39. [39] Z. Chen, V. X. Cheng and X. Liu, Hypothesis testing on high dimensional quantile regression, J. Econom. 238 (1), 105525, 2024.
40. [40] F. Zhang and Q. Li, A continuous threshold expectile model, Comput. Stat. Data Anal. 116, 49–66, 2017.
41. [41] F. Ferraty and P. Vieu, Nonparametric Functional Data Analysis: Theory and Practice, Springer, 2006.
42. [42] H. Cardot, C. Crambes and P. Sarda, Quantile regression when the covariates are functions, J. Nonparametr. Stat. 17 (7), 841–856, 2005.