Abstract
In real applications, the correlated data are commonly encountered. To model such data, many techniques have been proposed. However, of the developed techniques, emphasis has been on the mean function estimation under correlated errors, with scant attention paid to the derivative estimation. In this paper, we propose the locally weighted least squares regression based on different difference quotients to estimate the different order derivatives under correlated errors. For the proposed estimators, we derive their asymptotic bias and variance with different covariance structure errors, which dramatically reduce the estimation variance compared with traditional methods. Furthermore, we establish their asymptotic normality for constructing confidence interval. Based on the asymptotic mean integrated squared error, we provide a data-driven tuning parameters selection criterion. Simulation studies show that the proposed method is more robust and efficient than four other popular methods. Finally, we illustrate the usefulness of the proposed method with a real data example.
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Acknowledgements
Wang’s work was supported by National Natural Science Foundation of China (No. 12071248). Zhao’s work was supported by National Natural Science Foundation of China (No. 12171277).
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42952_2023_240_MOESM1_ESM.pdf
The supplementary material contains the proofs of Theorems 1–5, theoretical properties when \(\rho = \pm \frac{1}{2}\) and some additional simulation results in Sect. 5. (PDF 501 KB)
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Kong, D., Shen, W., Zhao, S. et al. Robust and Efficient derivative estimation under correlated errors. J. Korean Stat. Soc. 53, 149–168 (2024). https://doi.org/10.1007/s42952-023-00240-5
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DOI: https://doi.org/10.1007/s42952-023-00240-5