The Choice of the Smoothing Parameter for Alpha Stable Signals

Abstract

In this work we consider the class of symmetric alpha stable processes which are a particular family of processes with infinite energy. These processes used in modeling the random signals with indefinitely growing variance. The spectral density estimator of such signals is given in the literature by smoothing the periodogram by a spectral window. Thus, the estimator depends on the width of the spectral window considered as a smoothing parameter. The choice of this parameter plays an important role since the rate of convergence of the estimator is a function of this parameter. The objective of this paper is to propose a method giving the optimal parameter based on the cross validation technique (minimization of MISE: Mean Integrate Square of Error). We establish a criterion function and we prove that the mean of this criterion converges to MISE. Thus, we show that the value minimizing this criterion is the optimal smoothing parameter. The rate of convergence of the estimator has been studied in order to prove that the smoothing parameter obtained by this method gives the fastest convergence of the estimator towards the spectral density.