L p $L^{p}$ convergence rate of wavelet estimators for the bias and multiplicative censoring model

Lin Hu,Junlian Xu,Linshan Li,Huiying Wang

doi:10.1186/s13660-015-0732-2

Abstract

Using wavelets methods, Abbaszadeh, Chesneau, Doosti studied the density estimation problem under bias and multiplicative censoring (Stat. Probab. Lett. 82:932-941, 2012), and obtains the convergence rate of wavelet estimators in $L^{2}$ norm for a density function in Besov space. This paper deals with $L^{p}$ risk estimation with $1\leq p<\infty$ based on wavelet bases. Motivated by the work of Youming and Junlian (2014), we construct new estimators: a linear one and a nonlinear adaptive one; an upper bound of wavelet estimators on $L^{p}$ risk for a density function in Besov space is provided, which generalizes Abbaszadeh et al.’s theorems. It turns out that the nonlinear adaptive estimator obtains faster rate of convergence than the linear one for $r< p$ .

Highlights

Introduction and preliminary1.1 Introductions The density estimation plays important roles in both statistics and econometrics
This paper considers the density model under bias and multiplicative censoring, which were introduced by Abbaszadeh et al [ ]
Zi = UiYi, i =, . . . , n, where U, U, . . . , Un are unobserved i.i.d. random variables with the common uniform distribution on [, ], Y, Y, . . . , Yn are unobserved i.i.d. random variables and the density function fY is given by fY

Summary

It should be pointed out that

Besov spaces contain Hölder spaces and Sobolev spaces with non-integer exponents for a particular choice of s, p, and r [ ]. ([ ]) Let φ be a scaling function or a wavelet with θ (φ) := supx∈R |φ(x – k)| < ∞. (A ) The two density functions fX and fY have the support [ , ] and fX belongs to the Besov ball Bsr,q(H) (H > ) defined as. To introduce the wavelet estimator, we define the operator T by h(x)ω(x) + xh (x)ω(x) – xh(x)ω (x). The linear estimator is given as follows: flin(x) := αj ,kφj ,k(x), k∈∧. We have the following approximation result, which extends Abbaszadeh et al.’s theorems [ ] from p = to p ∈ [ , +∞). [ ] taken with m = , ignoring the log factor In this case, the model reduces to the standard density estimation problem under multiplicative censoring.

To estimate

One chooses

This completes the proof of

When θ