Kernel Entropy Estimation for Linear Processes

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 (Formula presented.) 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 L₂² 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 L₂² divergences among the density functions of average annual river flows for four rivers and obtain promising results.