The Choice of the Smoothing Parameter for Alpha Stable Signals

International audienceIn this work we consider the class of symmetricalpha stable processes which are a particular family ofprocesses with infinite energy. These processes used inmodeling the random signals with indefinitely growingvariance. The spectral density estimator of such signals isgiven in the literature by smoothing the periodogram by aspectral window. Thus, the estimator depends on the widthof the spectral window considered as a smoothing parameter.The choice of this parameter plays an important role sincethe rate of convergence of the estimator is a function of thisparameter. The objective of this paper is to propose amethod giving the optimal parameter based on the crossvalidation technique (minimization of MISE: MeanIntegrate Square of Error). We establish a criterionfunction and we prove that the mean of this criterionconverges to MISE. Thus, we show that the valueminimizing this criterion is the optimal smoothingparameter. The rate of convergence of the estimator hasbeen studied in order to prove that the smoothingparameter obtained by this method gives the fastestconvergence of the estimator towards the spectral density