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Kernel entropy estimation for linear processes II

Yudan Xiong, Fangjun Xu

Abstract

Let $X=\{X_n: n\in \mathbb{N}\}$ be a linear process with bounded probability density function $f(x)$. Under certain conditions, we use the kernel estimator \[ \frac{2}{n(n-1)h_n} \sum_{1\le i<j\le n}K\Big(\frac{X_i-X_j}{h_n}\Big) \] to estimate the quadratic functional of $\int_{\mathbb{R}}f^2(x)dx$ of the linear process $X=\{X_n: n\in \mathbb{N}\}$ and improve the corresponding results in [4].

Kernel entropy estimation for linear processes II

Abstract

Let be a linear process with bounded probability density function . Under certain conditions, we use the kernel estimator to estimate the quadratic functional of of the linear process and improve the corresponding results in [4].
Paper Structure (2 sections, 2 theorems, 61 equations)

This paper contains 2 sections, 2 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Proof of the main result

Key Result

Theorem 1.1

For some $\gamma\in(0,1]$, assume that Then there exist positive constants $c_1$ and $c_2$ such that where $Y_i=2(f(X_i)-{{\mathbb E}\,} f(X_i))$ for $1\leq i\leq n$, and, if additionally $nh_n\to\infty$ as $n\to\infty$, for some $\sigma^2\in(0,\infty)$.

Theorems & Definitions (2)

  • Theorem 1.1
  • Corollary 1.2