On the integrated mean squared error of wavelet density estimation for linear processes

TL;DR

This paper analyzes the integrated mean squared error of wavelet density estimators for linear processes, establishing minimax optimal convergence rates under certain conditions, with simulations demonstrating the theoretical findings.

Contribution

It provides the first derivation of minimax optimal convergence rates for wavelet density estimation of linear processes with specific wavelet properties.

Findings

Achieves minimax optimal convergence rate for density estimation

Wavelet properties depend on the number of nonzero coefficients

Simulation studies confirm theoretical results with various innovation distributions

Abstract

Let $\{X_n: n\in \N\}$ be a linear process with density function $f(x)\in L^2(\R)$. We study wavelet density estimation of $f(x)$. Under some regular conditions on the characteristic function of innovations, we achieve, based on the number of nonzero coefficients in the linear process, the minimax optimal convergence rate of the integrated mean squared error of density estimation. Considered wavelets have compact support and are twice continuously differentiable. The number of vanishing moments of mother wavelet is proportional to the number of nonzero coefficients in the linear process and to the rate of decay of characteristic function of innovations. Theoretical results are illustrated by simulation studies with innovations following Gaussian, Cauchy and chi-squared distributions.