Data-Driven Wavelet Estimations for Density Derivatives

This paper addresses the adaptive wavelet estimations for density derivatives by using data-driven methods. Based on the classical linear wavelet estimator of density derivatives, we provide a point-wise estimation under the local Hölder condition firstly. Moreover, we introduce a data-driven wavelet estimator for adaptivity and prove a point-wise oracle inequality, which does not require any assumption on the underlying function. Finally, by using the point-wise oracle inequality, the point-wise estimation under the local Hölder condition and \(L^p\) $L^p$ -risk ( \(1\le p<\infty \) $1\le p<\infty$ ) estimation on Besov spaces are investigated respectively.