On the estimation of density-weighted average derivative by wavelet methods under various dependence structures

The problem of estimating the density-weighted average derivative of a regression function is considered. We present a new consistent estimator based on a plug-in approach and wavelet projections. Its performances are explored under various dependence structures on the observations: the independent case, the ρ-mixing case and the α-mixing case. More precisely, denoting n the number of observations, in the independent case, we prove that it attains 1/n under the mean squared error, in the ρ-mixing case, 1/ √ n under the mean absolute error, and, in the α-mixing case, √ lnn/n under the mean absolute error. A short simulation study illustrates the theory.