Pointwise Optimality of Wavelet Density Estimation for Negatively Associated Biased Sample

Renyu Ye,Yuncai Yu,Xinsheng Liu

doi:10.3390/math8020176

Abstract

This paper focuses on the density estimation problem that occurs when the sample is negatively associated and biased. We constructed a block thresholding wavelet estimator to recover the density function from the negatively associated biased sample. The pointwise optimality of this wavelet density estimation is shown as L p ( 1 ≤ p < ∞ ) risks over Besov space. To validate the effectiveness of the block thresholding wavelet method, we provide some examples and implement the numerical simulations. The results indicate that our block thresholding wavelet density estimator is superior in terms of the mean squared error (MSE) when comparing with the nonlinear wavelet density estimator.

Highlights

Let X1, X2, · · ·, Xn be the unobserved realizations of random variable X with the density function g, and Y1, Y2, · · ·, Yn be the recorded observations of random variable Y with the density function: f (y) =
As the size-biased sample is dependent, many researchers modelled the dependence of the sample as being negatively associated (NA), the definition of which is as follows
The result indicated that the block thresholding wavelet density estimator is better than the nonlinear wavelet density estimator in terms of the mean squared error (MSE)

Summary

Introduction

Let X1 , X2 , · · · , Xn be the unobserved realizations of random variable X with the density function g, and Y1 , Y2 , · · · , Yn be the recorded observations of random variable Y with the density function:. As the linear wavelet method is not adaptive, Liu and Xu [14] and Guo and Kou [15] considered a nonlinear wavelet method to estimate the density function from a NA (stratified) size-biased sample. This nonlinear wavelet density estimation has been shown to be adaptive, and a pointwise convergence rate over L p risks was established. We structured an L p version of the block thresholding wavelet estimator for the density function g (y) in Model (1) This estimator is adaptive and simultaneously achieves the pointwise optimal convergence rate over Besov space. The result indicated that the block thresholding wavelet density estimator is better than the nonlinear wavelet density estimator in terms of the mean squared error (MSE)

Notations and Assumptions

Estimator and Main Result

Simulation Study

Proof of Theorem 1