Kernel entropy estimation for linear processes

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Let {Xₙ: n∈ N} be a linear process with bounded probability density function f(x). We study the estimation of the quadratic functional ∫ᵣ f²(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 ∑₁≤ ᵢ<ⱼ≤ ₙK(Xᵢ-Xⱼ/hₙ) has similar asymptotical properties as the i.i.d. case studied in Giné and Nickl (2008) if the linear process {Xₙ: n∈ N} has the defined short range dependence. We also provide an application to L²₂ 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 L²₂ divergences among the density functions of average annual river flows for four rivers and obtain promising results.