Contribution of Hermit polynomials to non-gaussian modeling and associated statistical tests.

Summary The objective of this thesis is to study the contributions of Hermit polynomials, when they have Gaussian random variables as arguments, to some fields of signal processing and statistics. A family of statistical tests of Gaussianity, called Hermit tests, has been introduced. The latter uses the orthonogonality of Hermit polynomials with respect to the Gaussian weight, through a sphericity statistic. We have conducted an asymptotic study of the Hermit test in the case of standard data, and a non-asymptotic study (with power comparison) in an invariant setting. The powers exhibited show that in addition to the advantage brought by the intrinsic modularity of Hermit tests, they exhibit good performances compared to the usual tests used. A class of non-linear / non-Gaussian processes, called h-arma, is studied. These consist of an arma-like linear filtering of a Gaussian input, followed by an instantaneous hermitian polynomial transformation. The use of hermits polynomials, and in particular the mehler and kibble-slepian formulas, allowed the writing of the temporal and spectral cumulants of these processes, as well as the non-asymptotic calculation of their empirical estimation variance. The identification of these models was first conducted in a supervised context, then in a blind context. The blind identification is confronted with the non-inversibility of these processes as soon as the polynomial non-linearity is no longer bi-univocal. After highlighting the limitations of traditional estimation methods (maximum likelihood, cumulant methods, etc.), we have employed stochastic mcmc algorithms, taking advantage of the augmentation of the model by hidden state variables. Implemented in the Bayesian paradigm, these methods provide a first solution to the identification of non-linear / non-invertible models.