Kernel Entropy Estimation for Linear Processes

Abstract

Let {X n :n∈N}be a linear process with bounded probability density function f(x). We study the estimation of the quadratic functional ∫ 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 2 false/ false( n false( n − 1 false) h n false) ∑ 1 ≤ i < j ≤ n K false( false( X i − X j false) false/ h n false) has similar asymptotical properties as the i.i.d. case studied in Giné and Nickl if the linear process {X n :n∈N}has the defined short range dependence. We also provide an application to L 2 2 divergence and the extension to multi‐variate linear processes. The simulation study for linear processes with Gaussian and α‐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.

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

“…extend the projection method applied in Sang et al (2018). In their Lemma 7.1, they applied the projection method to estimate the upper bound of the quantity 1≤i =j≤n E H(X i )(λ)H(X j )(λ) , where H(X j )(λ) is the conjugate of H(X j )(λ).…”

Shannon Entropy Estimation for Linear Processes

“…The estimation of Rényi entropy for the dependent case is challenging. A dependent case is treated by Sang et al (2018). They studied the estimation of the quadratic Entropy for the one-sided linear process.…”

Shannon entropy estimation for linear processes

“…The estimation of Rényi entropy for the dependent case is challenging. A dependent case is treated by Sang, Sang, and Xu [13]. They studied the estimation of the quadratic Entropy for the one-sided linear process.…”