Kernel entropy estimation for linear processes

12/01/2017

∙

by   Hailing Sang, et al.

∙

0

∙

share

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/n(n-1)h_n∑_1< i<j< nK(X_i-X_j/h_n) has similar asymptotical properties as the i.i.d. case studied in Giné and Nickl (2008) 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 multivariate 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.