Wavelet optimal estimations for a two-dimensional continuous-discrete density function over L^{p} risk

Lin Hu,Jinru Wang,Xiaochen Zeng

doi:10.1186/s13660-018-1868-7

Abstract

The mixed continuous-discrete density model plays an important role in reliability, finance, biostatistics, and economics. Using wavelets methods, Chesneau, Dewan, and Doosti provide upper bounds of wavelet estimations on L^{2} risk for a two-dimensional continuous-discrete density function over Besov spaces B^{s}_{r,q}. This paper deals with L^{p} (1leq p < infty) risk estimations over Besov space, which generalizes Chesneau–Dewan–Doosti’s theorems. In addition, we firstly provide a lower bound of L^{p} risk. It turns out that the linear wavelet estimator attains the optimal convergence rate for r geq p, and the nonlinear one offers optimal estimation up to a logarithmic factor.

Highlights

1 Introduction 1.1 Introduction The density estimation plays an important role in both statistics and econometrics
This paper considers a two-dimensional density estimation model defined over mixed continuous and discrete variables [2]
We are interested in estimating f (x, v) from (X1, Y1), (X2, Y2), . . . , (Xn, Yn). This continuous-discrete density model arises in survival analysis, economics, and social sciences

Summary

Introduction

1.1 Introduction The density estimation plays an important role in both statistics and econometrics. This paper considers a two-dimensional density estimation model defined over mixed continuous and discrete variables [2]. The conventional kernel method gives a nice estimation for the continuous-discrete density function [1, 10, 14]. Wavelet estimation attains optimality for densities in Besov spaces, which avoids the disadvantage of kernel methods. Chesneau et al [2] constructed linear and nonlinear wavelet estimators for a two-dimensional continuous-discrete density function and derived their mean integrated squared errors performance over Besov balls. Lemma 1.1 ([13]) Let φ be an m-regular orthonormal scaling function with the corresponding wavelets ψi (i = 1, 2, 3). Lemma 1.2 ([13]) Let φ ∈ L2(R2) be a scaling function or a wavelet with supk∈Z2 |φ(x – k)| < ∞. The construction of f∗ follows the idea proposed by Chesneau [2] but is different from [2]

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When θ