random-projection based procedure to test if a stationary process is
Universidad de Cantabria, Spain
The procedures known until now to test if a given stationary process is
Gaussian determine when the one-dimensional marginals of the stationary
process are Gaussian. Obviously, these provide tests at the right level
for the intended problem; but these tests could be at the nominal power
against some non-gaussian alternatives as those stationary non-Gaussian
process with Gaussian marginals.
The procedure we propose to test if a given stationary process of
real-valued random variables, X=(X_n)
is Gaussian is a combination of
the random projection method joined to some available procedures which
allow to determine when the one-dimensional marginals of a stationary
process are Gaussian. Here, we have chosen the proposal by Epps.
So, we are interested in constructing
a test for the null hypothesis H_0: "X
Notice that H_0 holds if and only if (X_1,..., X_n) is Gaussian and,
using the random projection method, this is, roughly speaking,
equivalent to that the projection of (X_1,..., X_n) in random
Hilbert-valued element is Gaussian. Now, is the turn of using Epps's
procedure which checks if the characteristic function of the
one-dimensional marginal of a stationary process
coincides with the characteristic function of a Gaussian distribution
a finite set of points. The difference here is that the points employed
in Epps's procedure are also chosen at random what guarantees the
consistency of the whole test.
This is a joint work with Juan Antonio Cuesta-Albertos, Universidad de
Cantabria, and Fabrice Gamboa, Université de Toulouse.