Kernel entropy estimation for long memory linear processes with infinite variance

TL;DR

This paper develops a kernel-based method to estimate the quadratic functional of the density for long memory linear processes with infinite variance, specifically those with stable law innovations, and supports it with simulation results.

Contribution

It introduces a kernel estimator for the density functional in long memory processes with infinite variance, extending existing methods to stable law innovations.

Findings

Estimator performs well under certain conditions

Simulation confirms theoretical properties

Applicable to processes with stable law innovations

Abstract

Let $X=\{X_n: n\in\mathbb{N}\}$ be a long memory linear process with innovations in the domain of attraction of an $\alpha$-stable law $(0<\alpha<2)$. Assume that the linear process $X$ has a bounded probability density function $f(x)$. Then, under certain conditions, we consider the estimation of the quadratic functional $\int_{\mathbb{R}} f^2(x) \,dx$ by using the kernel estimator \[ T_n(h_n)=\frac{2}{n(n-1)h_n}\sum_{1\leq j<i\leq n}K\left(\frac{X_i-X_j}{h_n}\right). \] The simulation study for long memory linear processes with symmetric $\alpha$-stable innovations is also given.