Abstract
Donoho et al. in 1996 have made almost perfect achievements in wavelet estimation for a density function f in Besov spaces B s r,q (ℝ). Motivated by their work, we define new linear and nonlinear wavelet estimators f lin n,m , f non n,m for density derivatives f (m). It turns out that the linear estimation E(‖f lin n,m − f (m)‖p) for f (m) ∈ B s r,q (ℝ) attains the optimal when r ⩾ p, and the nonlinear one E(‖f non n,m − f (m)‖ p ) does the same if \( r \leqslant \frac{p} {{2(s + m) + 1}} \). In addition, our method is applied to Sobolev spaces with non-negative integer exponents as well.
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Liu, Y., Wang, H. Wavelet estimations for density derivatives. Sci. China Math. 56, 483–495 (2013). https://doi.org/10.1007/s11425-012-4493-9
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DOI: https://doi.org/10.1007/s11425-012-4493-9