Nonparametric modal regression based on orthogonal series estimation

Zhongzheng Wang; Xiaobing Zhao; Zhongzheng Wang; Xiaobing Zhao

doi:10.3934/era.2026237

Electronic Research Archive

2026, Volume 34, Issue 8: 5328-5351. doi: 10.3934/era.2026237

Previous Article Next Article

Research article

Nonparametric modal regression based on orthogonal series estimation

Zhongzheng Wang ,
Xiaobing Zhao ^,

School of Data Sciences, Zhejiang University of Finance and Economics, Hangzhou 310018, China

Received: 19 May 2026 Revised: 12 June 2026 Accepted: 17 June 2026 Published: 23 June 2026

In this paper, we proposed an orthogonal series nonparametric modal regression (OSNMR) method for estimating the conditional mode under a unimodal setting. The unknown modal regression function was approximated by a truncated expansion of orthogonal basis functions, which represents the nonparametric component through a finite-dimensional coefficient vector. These coefficients were estimated using a kernel-density-based modal regression criterion and computed by the modal expectation-maximization (MEM) algorithm. Compared with local polynomial modal regression, the proposed method provides a global approximation and avoids repeated pointwise estimation. The orthogonal structure further separates coefficient estimation error from truncation approximation error, which makes the theoretical arguments clearer and more tractable. We established the convergence rate and asymptotic normality of the proposed estimator, and evaluated its finite-sample performance through simulations and a real data application.
- orthogonal series expansion,
- modal regression,
- MEM algorithm,
- kernel density estimation,
- mode
Citation: Zhongzheng Wang, Xiaobing Zhao. Nonparametric modal regression based on orthogonal series estimation[J]. Electronic Research Archive, 2026, 34(8): 5328-5351. doi: 10.3934/era.2026237

Related Papers:

Abstract

In this paper, we proposed an orthogonal series nonparametric modal regression (OSNMR) method for estimating the conditional mode under a unimodal setting. The unknown modal regression function was approximated by a truncated expansion of orthogonal basis functions, which represents the nonparametric component through a finite-dimensional coefficient vector. These coefficients were estimated using a kernel-density-based modal regression criterion and computed by the modal expectation-maximization (MEM) algorithm. Compared with local polynomial modal regression, the proposed method provides a global approximation and avoids repeated pointwise estimation. The orthogonal structure further separates coefficient estimation error from truncation approximation error, which makes the theoretical arguments clearer and more tractable. We established the convergence rate and asymptotic normality of the proposed estimator, and evaluated its finite-sample performance through simulations and a real data application.

References

[1]	M. J. Lee, Mode regression, J. Econom., 42 (1989), 337–349. https://doi.org/10.1016/0304-4076(89)90057-2 doi: 10.1016/0304-4076(89)90057-2
[2]	W. Yao, B. G. Lindsay, R. Li, Local modal regression, J. Nonparametric Stat., 24 (2012), 647–663. https://doi.org/10.1080/10485252.2012.678848 doi: 10.1080/10485252.2012.678848
[3]	W. Yao, L. Li, A new regression model: Modal linear regression, Scand. J. Stat., 41 (2014), 656–671. https://doi.org/10.1111/sjos.12054 doi: 10.1111/sjos.12054
[4]	Y. C. Chen, C. R. Genovese, R. J. Tibshirani, L. Wasserman, Nonparametric modal regression, Ann. Stat., 44 (2016), 489–514. https://doi.org/10.1214/15-AOS1373 doi: 10.1214/15-AOS1373
[5]	Y. C. Chen, Modal regression using kernel density estimation: A review, Wiley Interdiscip. Rev.-Comput. Stat., 10 (2018), e1431. https://doi.org/10.1002/wics.1431 doi: 10.1002/wics.1431
[6]	T. Wang, Non-parametric estimator for conditional mode with parametric features, Oxf. Bull. Econ. Stat., 86 (2024), 44–73. https://doi.org/10.1111/obes.12577 doi: 10.1111/obes.12577
[7]	J. Einbeck, G. Tutz, Modelling beyond regression functions: An application of multimodal regression to speed–flow data, J. R. Stat. Soc. Ser. C-Appl. Stat., 55 (2006), 461–475. https://doi.org/10.1111/j.1467-9876.2006.00547.x doi: 10.1111/j.1467-9876.2006.00547.x
[8]	L. Yang, W. Yuan, S. Wang, Modal regression models based on B-splines, Comput. Stat., 40 (2025), 225–248. https://doi.org/10.1007/s00180-024-01487-0 doi: 10.1007/s00180-024-01487-0
[9]	A. Ullah, T. Wang, W. Yao, Nonlinear modal regression for dependent data with application for predicting COVID-19, J. R. Stat. Soc. Ser. A-Stat. Soc., 185 (2022), 1424–1453. https://doi.org/10.1111/rssa.12849 doi: 10.1111/rssa.12849
[10]	X. Jing, J. S. Cho, Forecasting the confirmed COVID-19 cases using modal regression, J. Forecast., 44 (2025), 1578–1601. https://doi.org/10.1002/for.3261 doi: 10.1002/for.3261
[11]	H. Yuan, S. Xiang, W. Yao, A new bandwidth selection method for nonparametric modal regression based on generalized hyperbolic distributions, Comput. Stat., 39 (2024), 1729–1746. https://doi.org/10.1007/s00180-023-01435-4 doi: 10.1007/s00180-023-01435-4
[12]	D. J. Henderson, C. F. Parmeter, R. R. Russell, Modes, weighted modes, and calibrated modes: Evidence of clustering using modality tests, J. Appl. Econom., 23 (2008), 607–638. https://doi.org/10.1002/jae.1023 doi: 10.1002/jae.1023
[13]	Z. C. Han, Y. Y. Zhao, Nonparametric modal regression with mixed variables and application to analyze the GDP data, J. Comput. Appl. Math., 445 (2024), 115841. https://doi.org/10.1016/j.cam.2024.115841 doi: 10.1016/j.cam.2024.115841
[14]	M. J. Lee, Quadratic mode regression, J. Econom., 57 (1993), 1–19. https://doi.org/10.1016/0304-4076(93)90056-B doi: 10.1016/0304-4076(93)90056-B
[15]	G. Kemp, J. Santos Silva, Regression towards the mode, J. Econom., 170 (2012), 92–101. https://doi.org/10.1016/j.jeconom.2012.03.002 doi: 10.1016/j.jeconom.2012.03.002
[16]	H. Zhou, X. Huang, Nonparametric modal regression in the presence of measurement error, Electron. J. Stat., 10 (2016), 3579–3620. https://doi.org/10.1214/16-EJS1210 doi: 10.1214/16-EJS1210
[17]	X. Li, X. Huang, Linear mode regression with covariate measurement error, Can. J. Stat., 47 (2019), 262–280. https://doi.org/10.1002/cjs.11492 doi: 10.1002/cjs.11492
[18]	S. Xiang, W. Yao, Nonparametric statistical learning based on modal regression, J. Comput. Appl. Math., 409 (2022), 114130. https://doi.org/10.1016/j.cam.2022.114130 doi: 10.1016/j.cam.2022.114130
[19]	H. Ota, K. Kato, S. Hara, Quantile regression approach to conditional mode estimation, Electron. J. Stat., 13 (2019), 3120–3160. https://doi.org/10.1214/19-EJS1607 doi: 10.1214/19-EJS1607
[20]	Y. Feng, J. Fan, J. A. K. Suykens, A statistical learning approach to modal regression, J. Mach. Learn. Res., 21 (2020), 1–35. https://doi.org/10.5555/3455716.3455718 doi: 10.5555/3455716.3455718
[21]	J. M. Krief, Semi-linear mode regression, Econom. J., 20 (2017), 149–167. https://doi.org/10.1111/ectj.12088 doi: 10.1111/ectj.12088
[22]	A. Ullah, T. Wang, W. Yao, Modal regression for fixed effects panel data, Empirical Econ., 60 (2021), 261–308. https://doi.org/10.1007/s00181-020-01999-w doi: 10.1007/s00181-020-01999-w
[23]	A. Ullah, T. Wang, W. Yao, Semiparametric partially linear varying coefficient modal regression, J. Econom., 235 (2023), 1001–1026. https://doi.org/10.1016/j.jeconom.2022.09.002 doi: 10.1016/j.jeconom.2022.09.002
[24]	T. Wang, W. Yao, Nonparametric spatial mode-oriented regression, Electron. J. Stat., 19 (2025), 4256–4311. https://doi.org/10.1214/25-EJS2427 doi: 10.1214/25-EJS2427
[25]	T. Wang, W. Yao, Online kernel-based mode learning, J. Comput. Graph. Stat., 34 (2025), 1498–1512. https://doi.org/10.1080/10618600.2025.2459287 doi: 10.1080/10618600.2025.2459287
[26]	S. Xiang, W. Yao, X. Cui, Advances in modal regression: From theoretical foundations to practical implementations, Wiley Interdiscip. Rev.-Comput. Stat., 18 (2026), e70061. https://doi.org/10.1002/wics.70061 doi: 10.1002/wics.70061
[27]	C. Dong, J. Gao, Modern Series Methods in Econometrics and Statistics, Springer, Switzerland, 2025. https://doi.org/10.1007/978-981-96-2822-3
[28]	J. Li, S. Ray, B. G. Lindsay, A nonparametric statistical approach to clustering via mode identification, J. Mach. Learn. Res., 8 (2007), 1687–1723. https://doi.org/10.5555/1314498.1314555 doi: 10.5555/1314498.1314555
[29]	H. Zhou, X. Huang, Bandwidth selection for nonparametric modal regression, Commun. Stat.-Simul. Comput., 48 (2019), 968–984. https://doi.org/10.1080/03610918.2017.1402044 doi: 10.1080/03610918.2017.1402044
[30]	J. Z. Huang, C. O. Wu, L. Zhou, Varying-coefficient models and basis function approximations for the analysis of repeated measurements, Biometrika, 89 (2002), 111–128. https://doi.org/10.1093/biomet/89.1.111 doi: 10.1093/biomet/89.1.111
[31]	W. Zhao, R. Zhang, J. Liu, Y. Lv, Robust and efficient variable selection for semiparametric partially linear varying coefficient model based on modal regression, Ann. Inst. Stat. Math., 66 (2014), 165–191. https://doi.org/10.1007/s10463-013-0410-4 doi: 10.1007/s10463-013-0410-4
[32]	W. K. Newey, D. McFadden, Large sample estimation and hypothesis testing, in Handbook of Econometrics, Elsevier, Amsterdam, (1994), 2111–2245. https://doi.org/10.1016/S1573-4412(05)80005-4

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)