KERNEL ENTROPY ESTIMATION FOR LINEAR PROCESSES

Let {X-n : n is an element of N} be a linear process with bounded probability density function f ( x). We study the estimation of the quadratic functional integral(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) hn) S 1= i< j= n K((Xi - Xj)/ h(n)) has similar asymptotical properties as the i. i. d. case studied in Gine and Nickl (2008) if the linear process {Xn : 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 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.