Nonparametric derivative estimation with bimodal kernels under correlated errors

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.