Abstract
For the derivative estimation, nonparametric regression with unimodal kernels performs well under independent errors, while it breaks down under correlated errors. In this paper, we propose the local polynomial regression based on bimodal kernels for the derivative estimation under correlated errors. Unlike the conventional local polynomial estimator, the proposed estimator does not require any prior knowledge about the correlation structure of errors. For the proposed estimator, we deduce the main theoretical results, including the asymptotic bias, asymptotic variance, and asymptotic normality. Based on the asymptotic mean integrated squared error, we also provide a data-driven bandwidth selection criterion. Subsequently, we compare three popular bimodal kernels from the robustness and efficiency. Simulation studies show that the heavy-tailed bimodal kernel is more robust and efficient than the other two bimodal kernels and two popular unimodal kernels, especially for high-frequency oscillation functions. Finally, two real data examples are presented to illustrate the feasibility of the proposed estimator.
Similar content being viewed by others
References
Ahmed G, Al-Gasaymeh A, Mehmood T et al (2017) The global financial crisis and international trade. Asian Econ Financ Rev 7(6):600–610
Altman NS (1992) An iterated Cochrane–Orcutt procedure for nonparametric regression. J Stat Comput Simul 40(1–2):93–108
Altman N (2000) Krige, smooth, both or neither? (with discussion). Aust N Z J Stat 42(4):441–461
Avramidy IG, Barvinsky AO (1985) Asymptotic freedom in higher-derivative quantum gravity. Phys Lett B 159(4–6):269–274
Brabanter KD, Brabanter JD, Suykens JA, Moor BD (2011) Kernel regression in the presence of correlated errors. J Mach Learn Res 12:1955–1976
Brabanter KD, Brabanter JD, Gijbels I, Moor BD (2013) Derivative estimation with local polynomial fitting. J Mach Learn Res 14(1):281–301
Brabanter KD, Cao F, Gijbels I, Opsomer J (2018) Local polynomial regression with correlated errors in random design and unknown correlation structure. Biometrika 105(3):681–690
Calonico S, Cattaneo MD, Farrell MH (2018) On the effect of bias estimation on coverage accuracy in nonparametric inference. J Am Stat Assoc 113(522):767–779
Charnigo R, Hall B, Srinivasan C (2011) A generalized \(c_{p}\) criterion for derivative estimation. Technometrics 53(3):238–253
Cornish NJ, Littenberg TB (2007) Tests of Bayesian model selection techniques for gravitational wave astronomy. Phys Rev D 76(8):083006
Cressie N (2015) Statistics for spatial data. John Wiley & Sons, Hoboken
Delecroix M, Rosa A (1996) Nonparametric estimation of a regression function and its derivatives under an ergodic hypothesis. J Nonparametric Stat 6(4):367–382
Fan J, Gijbels I (1995) Data-driven bandwidth selection in local polynomial fitting: variable bandwidth and spatial adaptation. J R Stat Soc B 57(2):371–394
Fan J, Gijbels I (1996) Local polynomial modelling and its applications. Chapman & Hall, London
Francisco-Fernández M, Vilar-Fernández JM (2001) Local polynomial regression estimation with correlated errors. Commun Stat Theory Methods 30(7):1271–1293
Ganong P, Jäger S (2018) A permutation test for the regression kink design. J Am Stat Assoc 113(522):494–504
Gijbels I, Goderniaux AC (2005) Data-driven discontinuity detection in derivatives of a regression function. Commun Stat Theory Methods 33(4):851–871
Guo J, Liu Y, Wu X, Chen J (2022) Assessment of the impact of Fukushima nuclear wastewater discharge on the global economy based on GTAP. Ocean Coast Manag 228:106296
Hart JD (1991) Kernel regression estimation with time series errors. J R Stat Soc B 53(1):173–187
Ibn-Mohammed T, Mustapha K, Godsell J, Adamu Z, Babatunde K, Akintade D, Acquaye A, Fujii H, Ndiaye M, Yamoah F et al (2021) A critical analysis of the impacts of COVID-19 on the global economy and ecosystems and opportunities for circular economy strategies. Resour Conserv Recycl 164:105169
Kim TY, Park BU, Moon MS, Kim C (2009) Using bimodal kernel for inference in nonparametric regression with correlated errors. J Multivar Anal 100(7):1487–1497
Liu Y, Brabanter KD (2020) Smoothed nonparametric derivative estimation using weighted difference quotients. J Mach Learn Res 21(65):1–45
Liu S, Kong X (2022) A generalized correlated \(c_{p}\) criterion for derivative estimation with dependent errors. Comput Stat Data Anal 171:107473–107495
Liu S, Yang J (2023) Kernel regression for estimating regression function and its derivatives with unknown error correlations. Metrika. https://doi.org/10.1007/s00184-023-00901-9
Luo S, Zhu Y, Chen SX (2022) Episode based air quality assessment. Atmos Environ 285:119242
Müller HG, Stadtmüller U, Schmitt T (1987) Bandwidth choice and confidence intervals for derivatives of noisy data. Biometrika 74(4):743–749
Opsomer J, Wang Y, Yang Y (2001) Nonparametric regression with correlated errors. Stat Sci 16(2):134–153
Page GL, Rodríguez-Álvarez MX, Lee DJ (2020) Bayesian hierarchical modelling of growth curve derivatives via sequences of quotient differences. J R Stat Soc Ser C 69(2):459–481
Park C, Kim TY, Ha J, Luo Z, Hwang S (2015) Using a bimodal kernel for a nonparametric regression specification test. Stat Sin 25:1145–1161
Rondonotti V, Marron J, Park C (2007) Sizer for time series: a new approach to the analysis of trends. Electron J Stat 1:268–289
Simpkin AJ, Durban M, Lawlor DA (2018) Derivative estimation for longitudinal data analysis: examining features of blood pressure measured repeatedly during pregnancy. Stat Med 37(19):2836–2854
Stone CJ (1985) Additive regression and other nonparametric models. Ann Stat 13(2):689–705
Wang W, Lin L (2015) Derivative estimation based on difference sequence via locally weighted least squares regression. J Mach Learn Res 16(1):2617–2641
Wang K, Bichot CE, Li Y, Li B (2017) Local binary circumferential and radial derivative pattern for texture classification. Pattern Recogn 67:213–229
Wang W, Yu P, Lin L, Tong T (2019) Robust estimation of derivatives using locally weighted least absolute deviation regression. J Mach Learn Res 20(1):2157–2205
Wang W, Lu J, Tong T, Liu Z (2022) Debiased learning and forecasting of first derivative. Knowl Based Syst 236:107781
Zhou S, Wolfe DA (2000) On derivative estimation in spline regression. Stat Sin 10:93–108
Acknowledgements
Wang’s work was supported by National Natural Science Foundation of China (No.12071248 & 12271294). Zhao’s work was supported by National Natural Science Foundation of China (No.12171277).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kong, D., Zhao, S. & Wang, W. Nonparametric derivative estimation with bimodal kernels under correlated errors. Comput Stat (2023). https://doi.org/10.1007/s00180-023-01419-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00180-023-01419-4