Kernel entropy estimation for linear processes

TL;DR

This paper develops a kernel-based method for estimating the quadratic functional of the density of linear processes, extending previous i.i.d. results to dependent data with short-range dependence, and demonstrates its effectiveness through simulations and real data applications.

Contribution

It introduces a kernel entropy estimator for linear processes, extending asymptotic properties known for i.i.d. data to dependent processes with short-range dependence.

Findings

Estimator has similar asymptotic properties as in the i.i.d. case.

Simulation confirms theoretical results for Gaussian and alpha-stable innovations.

Application to river flow data demonstrates practical utility.

Abstract

Let $\{X_n: n\in \mathbb{N}\}$ be a linear process with bounded probability density function $f(x)$. We study the estimation of the quadratic functional $\int_{\mathbb{R}} f^2(x)\, dx$. With a Fourier transform on the kernel function and the projection method, it is shown that, under certain mild conditions, the estimator \[ \frac{2}{n(n-1)h_n} \sum_{1\le i<j\le n}K\left(\frac{X_i-X_j}{h_n}\right) \] has similar asymptotical properties as the i.i.d. case studied in Gin\'{e} and Nickl (2008) if the linear process $\{X_n: n\in \mathbb{N}\}$ has the defined short range dependence. We also provide an application to $L^2_2$ divergence and the extension to multivariate linear processes. The simulation study for linear processes with Gaussian and $\alpha$-stable innovations confirms our theoretical results. As an illustration, we estimate the $L^2_2$ divergences among the density functions of average annual river flows for four rivers and obtain promising results.