Optimal Wavelet Estimation of Density Derivatives for Size-Biased Data

Abstract

A perfect achievement has been made for wavelet density estimation by Dohono et al. in 1996, when the samples without any noise are independent and identically distributed (i.i.d.). But in many practical applications, the random samples always have noises, and estimation of the density derivatives is very important for detecting possible bumps in the associated density. Motivated by Dohono's work, we propose new linear and nonlinear wavelet estimators ${\widehat{f}}_{\text{lin}}^{(m)},{\widehat{f}}_{\text{non}}^{(m)}$ for density derivatives $f^{(m)}$ when the random samples have size-bias. It turns out that the linear estimation $E(\parallel {\widehat{f}}_{\text{lin}}^{(m)}-f^{(m)}{\parallel }_p)$ for $f^{(m)}\in B_{r,q}^s(A,L)$ attains the optimal covergence rate when $r\ge p$, and the nonlinear one $E(\parallel {\widehat{f}}_{\text{lin}}^{(m)}-f^{(m)}{\parallel }_p)$ does the same if .