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.
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