La détection de périodicités cachées

This thesis is a small part of the preparation of the launch of the Gaia mission, a satellite of the European Space Agency (ESA). One of the goals of the mission is to perform a classification among variable stars considering different attributes. Periodic behavior in the observed light curve is such an attribute. It is of importance to determine these hidden cycles with as high an accuracy as possible. A key difficulty is connected to the fact that observations are taken at irregularly distributed time points. Classical methods of frequency analysis do not work in this situation. In a first step, we made a catalogue of potential solutions and applied them to real and simulated data. The performances have been compared in order to select the best method. In a second step, we considered the asymptotic performance of estimators based on regression methods. We derived the asymptotic distribution under the hypothesis that the observation model contains a periodic signal with period P0 > 0, continuous and square integrable on [0; P0), with Gaussian correlated errors. The irregular sampling has to satisfy the asymptotic property that the whole interval [0, P0) can be observed. We have also considered the special case of a periodic signal with period P0 > 0, piecewise constant on [0; P0) and the asymptotic distribution of the estimator. If an irregular sampling scheme does not satisfy the asymptotic property that the signal can be fully observed, the period can still be estimated reliably. We determined the asymptotic distribution of an estimator in a particular situation and under the assumption that the observation model contains a periodic signal with period P0 > 0, continuous and square integrable on [0; P0), and is observed with additive errors that are independent and identically distributed.

Hybrid regularisation and the (in)admissibility of ridge regression in infinite dimensional Hilbert spaces

Anirvan Chakraborty, Victor Panaretos

We consider the problem of estimating the slope function in a functional regression with a scalar response and a functional covariate. This central problem of functional data analysis is well known to be ill-posed, thus requiring a regularised estimation procedure. The two most commonly used approaches are based on spectral truncation or Tikhonov regularisation of the empirical covariance operator. In principle, Tikhonov regularisation is the more canonical choice. Compared to spectral truncation, it is robust to eigenvalue ties, while it attains the optimal minimax rate of convergence in the mean squared sense, and not just in a concentration probability sense. In this paper, we show that, surprisingly, one can strictly improve upon the performance of the Tikhonov estimator in finite samples by means of a linear estimator, while retaining its stability and asymptotic properties by combining it with a form of spectral truncation. Specifically, we construct an estimator that additively decomposes the functional covariate by projecting it onto two orthogonal subspaces defined via functional PCA; it then applies Tikhonov regularisation to the one component, while leaving the other component unregularised. We prove that when the covariate is Gaussian, this hybrid estimator uniformly improves upon the MSE of the Tikhonov estimator in a non-asymptotic sense, effectively rendering it inadmissible. This domination is shown to also persist under discrete observation of the covariate function. The hybrid estimator is linear, straightforward to construct in practice, and with no computational overhead relative to the standard regularisation methods. By means of simulation, it is shown to furnish sizeable gains even for modest sample sizes.

2019

Robust parameter estimation for the Ornstein-Uhlenbeck process

Sonja Rieder

In this paper, we derive elementary M- and optimally robust asymptotic linear (AL)-estimates for the parameters of an Ornstein-Uhlenbeck process. Simulation and estimation of the process are already well-studied, see Iacus (Simulation and inference for stochastic differential equations. Springer, New York, 2008). However, in order to protect against outliers and deviations from the ideal law the formulation of suitable neighborhood models and a corresponding robustification of the estimators are necessary. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. In a first step, we extend the method of M-estimation from Huber (Robust statistics. Wiley, New York, 1981). In a second step, we apply the general theory based on local asymptotic normality, AL estimates, and shrinking neighborhoods due to Kohl et al. (Stat Methods Appl 19:333-354, 2010), Rieder (Robust asymptotic statistics. Springer, New York, 1994), Rieder (2003), and Staab (1984). This leads to optimally robust ICs whose graph exhibits surprising behavior. In the end, we discuss the estimator construction, i.e. the problem of constructing an estimator from the family of optimal ICs. Therefore we carry out in our context the One-Step construction dating back to LeCam (Asymptotic methods in statistical decision theory. Springer, New York, 1969) and compare it by means of simulations with MLE and M-estimator.

Springer-Verlag2012

Stratégies de compromis pour l'estimation des paramètres de régression et pour la classification floue

Nicolas Fournier

Model specification is an integral part of any statistical inference problem. Several model selection techniques have been developed in order to determine which model is the best one among a list of possible candidates. Another way to deal with this question is the so-called model averaging, and in particular the frequentist approach. An estimation of the parameters of interest is obtained by constructing a weighted average of the estimates of these quantities under each candidate model. We develop compromise frequentist strategies for the estimation of regression parameters, as well as for the probabilistic clustering problem. In the regression context, we construct compromise strategies based on the Pitman estimators associated with various underlying errors distributions. The weight given to each model is equal to its profile likelihood, which gives a measure of the goodness-of-fit. Asymptotic properties of both Pitman estimators and profile likelihood allow us to define a minimax strategy for choosing the distributions of the compromise, involving a notion of distance between distributions. Performances of such estimators are then compared to other usual and robust procedures. In the second part of the thesis, we develop compromise strategies in the probabilistic clustering context. Although this clustering method is based on mixtures of distributions, our compromise strategies are not applied directly to the estimates of the parameters, but on the posterior probabilities of membership. Two types of compromise are presented, and the performances of resulting classification rules are investigated.

EPFL2011