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

Wavelet analysis plays important roles in both pure and applied mathematics such as signal processing, image compress, numerical solution, and local fractional calculus [

In practice, it usually happens that getting the direct sample from a random variable is impossible. In this paper, we want to consider the true density function

In many cases, a linear

Size-biased data arise when an observation depends on samples magnitude. Several examples of model (

The estimation problem for biased data (

In 2010, Ramírez and Vidakovic [

The current paper is organized as follows. In Section

In this section, we will recall some useful and well-known concepts and lemmas.

In order to construct a wavelet basis, we need a structure in

A multiresolution analysis (MRA) of

there exists a function

With the standard notation

As usual,

Let

If the scaling function

Letting

Between the different Besov spaces, the following embedding conclusions are established [

where

A scaling function

One of advantages of wavelets is that they can characterize Besov spaces.

Let

The notation

In this paper, the Besov balls

In this section, we will give a linear estimator for density derivatives

The linear wavelet estimator of the derivative of a density

The following inequalities play important roles in this paper.

Let

About the defined coefficients in (

If

By the definitions of

To estimate

if

if

To estimate the term

for

for

Let scaling function

Firstly, using triangular inequality and convexity inequality, we decompose

When

When

When

Theorem

This section is devoted to showing that the linear estimator defined in (

Let

Let

Based on the above lemmas, we have the following lower bound estimation.

Let

(i) Firstly, we prove

Recall that

Next, we prove

Taking