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Abstract and Applied Analysis
Volume 2014, Article ID 512634, 13 pages
Research Article

Optimal Wavelet Estimation of Density Derivatives for Size-Biased Data

Department of Applied Mathematics, Beijing University of Technology, Beijing 100124, China

Received 10 November 2013; Accepted 17 December 2013; Published 30 January 2014

Academic Editor: D. Baleanu

Copyright © 2014 Jinru Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 for density derivatives when the random samples have size-bias. It turns out that the linear estimation for attains the optimal covergence rate when , and the nonlinear one does the same if .