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In this thesis, we use logarithmic methods to study motivic objects. Let R be a complete discrete valuation ring with perfect residue field k, and denote by K its fraction field. We give in chapter 2 a new construction of the motivic Serre invariant of a smooth K-variety and extend it additively to arbitrary K-varieties. The main advantage of this construction is to rely only on resolution of singularities and not on a characteristicnbsp;assumption, asnbsp;previous results. As an application, we give a conditional positive answer to Serre's question on the existence of rational fixed points of a G-action on the affine space, for G a finite l-group. We end the chapter by showing how the logarithmic point of view that we use in our construction leads to a newnbsp;of the motivic nearby cycles with support of Guibert, Loesernbsp;Merle as a motivic volume. In chapter 4 we use the theory of logarithmic geometry to derive a new formula for the motivic zeta function via the volume Poincaré series. More precisely, we show how to compute the volume Poincaré series associated to a generically smooth log smooth R-scheme in terms of its log geometry, more specifically in terms of its associated fan in the sense of Kato. This formula yields a much smaller set of candidate poles for the motivic zeta function and seems especially well suited to tacklenbsp;monodromy conjecture of Halle and Nicaise for Calabi-Yau K-varieties, for which log smooth models appear naturally through the Gross-Siebert programme on mirror symmetry. We end the chapter by showingnbsp;this formula sheds new light on previous results regarding the motivic zeta function of a polynomial nondegenerate with respect tonbsp;Newton polyhedron, and of a polynomial in two variables.

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In this thesis we study robust variable selection in (heteroscedastic) linear regression using the nonnegative garrote method. Many variable selection methods are available, but very few methods are designed to avoid sensitivity to outliers in the response and in the covariates. The nonnegative garrote method is a powerful variable selection method, developed originally for linear regression (see Chapter 1). This method starts from an initial estimator, the ordinary least-squares regression estimator, and then it shrinks or puts some coefficients of this initial estimator to zero using the nonnegative garrote shrinkage factors. Since this method is based on the ordinary least-squares regression estimator, it is not robust to outliers. In Chapter 2 we robustify the nonnegative garrote method for outliers in the response and in the covariates by using robust alternatives to the ordinary least-squares regression. We present three robust versions of the nonnegative garrote, namely the M-, S-, and LTS-nonnegative garrote and propose an extra reweighting step to improve the results of the S-nonnegative garrote. The performances of the methods are investigated via simulations and their use is illustrated on a real data example. In Chapter 3 some theoretical properties of the S-nonnegative garrote method are proved. We show that this method is consistent in variable selection and in estimation. We also provide a lower bound for its breakdown point and derive its influence function. Some illustrations regarding the influence function are given. The original nonnegative garrote method tends to select too many variables, when the error terms do not have a constant variance. The aim in Chapter 4 is to select and estimate the variables that explain the mean response and/or the variance in a heteroscedastic linear regression model. First, different (robust) heteroscedastic initial estimators are introduced and the original nonnegative garrote method for variable selection in a linear regression model is extended to a variable selection method in a heteroscedastic linear regression model. Then, three approaches to robustify this heteroscedastic nonnegative garrote method for outliers in the response and in the covariates are discussed, namely the heteroscedastic M-, S-, and T-nonnegative garrote. We further compare the performances of the different heteroscedastic nonnegative garrote methods in a simulation study. In Chapter 5 we provide the influence functions of the heteroscedastic nonnegative garrote method and the heteroscedastic S-nonnegative garrote method. Since the influence functions of initial estimators are needed in the influence functions of the heteroscedastic (S-)nonnegative garrote methods, we also derive the influence functions of the maximum likelihood estimator and the heteroscedastic S-estimator. With these influence functions we can see that the heteroscedastic nonnegative garrote method is not robust for outliers and that the influence function of the heteroscedastic S-nonnegative garrote method is bounded for outliers. Illustrations regarding these influence functions are provided and we illustrate the use of the different heteroscedastic nonnegative garrote methods of Chapter 4 in a real data example. Finally, in Chapter 6 we draw some conclusions and discuss some possible topics for further research.

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