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No description available.

No description available.

No description available.

Bayesian consistency is an important issue in the context of non- parametric problems. The posterior consistency is a validation of a Bayesian approach and guarantees the posterior mass accumulates around the true density, which" is unknown in most circumstances, as the number of observations goes to infinity. This thesis mainly considers the consistency for nonparametric regression models over both random and non random covariates. The techniques to achieve consistency under random covariates are similar to that derived in Walker (2003, 2004) which is designed for the consistency of independent and identically distributed variables. We contribute a new idea to deal with the supremum metric over covariates when the regression model is with non random covariates. That is, if a regression density is away from the true density in the Hellinger sense, then there is a covariate, whose value is picked from a specific design, such that the density indexed by this value is also away from the true density. As a result, the posterior concentrates in the supremum Hellinger neighbourhood of the real model under conditions on the prior such as the Kullback-Leibler property and the summability of the square rooted prior mass on Hellinger covering balls. Furthermore, the predictive is also shown to be consistent and we illustrate our results on a normal mean regression function and demonstrate the usefulness of a model based on piecewise constant functions. We also investigate conditions under which a piecewise density model is consistent.

The thesis considers the Aalen additive regression model for recurrent event data. The model itself, estimation of the cumulative regression functions, testing procedures, checking goodness of fit and inclusion of dynamic covariates in the model are reviewed. A disadvantage of this model is that estimates of the conditional probabilities are not constrained to lie between zero and one, therefore a model with logistic intensity is considered. Results under the logistic model are shown to be qualitatively similar to those under the additive model. The additive model is extended to incorporate the possibility of spatial or spatio-temporal clustering, possibly caused by unobserved environmental factors or infectivity. Various tests for the presence of clustering are described and implemented. The issue of frailty modelling and its connection to dynamic modelling is presented and examined. We show that frailty and dynamic models are almost indistinguishable in terms of residual summary plots. A graphical procedure based on the property that the covariance between martingale residuals at time to and t > to is independent of t is proposed and supplemented by a formal test statistic to investigate the adequacy of the fitted models. The results can be used to compare models and to check the validity of the model being tested. Also we investigate properties under various types of model misspecification. All our works are illustrated using two sets of data measuring daily prevalence and incidence of infant diarrhoea in Salvador, Brazil. Significant clustering is identified in the data. We investigate risk factors for diarrhoea and there is strong evidence of dynamic effects being important, implying heterogeneity between individuals not explained by measured socio- economic and environmental factors.

Nonparametric regression estimation has become popular in the last 50 years. A commonly used nonparametric method for estimating the regression curve is the kernel estimator, exemplified by the Nadaraya- Watson estimator. The first part of thesis concentrates on the important issue of how to make a good choice of smoothing parameter for the Nadaraya- Watson estimator. In this study three types of smoothing parameter selectors are investigated: cross-validation, plug-in and bootstrap. In addition, two situations are examined: the same smoothing parameter and different smoothing parameters are employed for the estimates of the numerator and the denominator. We study the asymptotic bias and variance of the Nadaraya- Watson estimator when different smoothing parameters are used. We propose various plug-in methods for selecting smoothing parameter including a bootstrap smoothing parameter selector. The performances of the proposed selectors are investigated and also compared with cross-validation via a simulation study. Numerical results demonstrate that the proposed plug-in selectors outperform cross-validation when data is bivariate normal distributed. Numerical results also suggest that the proposed bootstrap selector with asymptotic pilot smoothing parameter compares favourably with cross-validation. We consider a circular-circular parametric regression model proposed by Taylor (2009), including parameter estimation and inference. In addition, we investigate diagnostic tools for circular regression which can be generally applied. A final thread is related to mixture models, in particular a mixture of linear regression models and a mixture of circular-circular regression models where there is unobserved group membership of the observation. We investigate methods for selecting starting values for EM algorithm which is used to fit mixture models and also the distributions of these values. Our experiments suggest that the proposed method compares favourably with the common method in mixture linear regression models.

Ce travail est une contribution à la sélection de modèles statistiques et plus précisément à la sélection de variables dans le cadre de régression linéaire sur les quantiles pénalisée lorsque la dimension est grande. On se focalise sur deux points lors de la procédure de sélection : la stabilité de sélection et la prise en compte de variables présentant un effet de groupe. Dans une première contribution, on propose une transition des moindres carrés pénalisés vers la régression sur les quantiles (QR). Une approche de type bootstrap fondée sur la fréquence de sélection de chaque variable est proposée pour la construction de modèles linéaires (LM). Dans la majorité des cas, l’approche QR fournit plus de coefficients significatifs. Une deuxième contribution consiste à adapter certains algorithmes de la famille « Random » LASSO (Least Absolute Solution and Shrinkage Operator) au cadre de la QR et à proposer des méthodes de stabilité de sélection. Des exemples provenant de la sécurité alimentaire illustrent les résultats obtenus. Dans le cadre de la QR pénalisée en grande dimension, on établit la propriété d’effet groupement sous des conditions plus faibles ainsi que les propriétés oracles. Deux exemples de données réelles et simulées illustrent les chemins de régularisation des algorithmes proposés. La dernière contribution traite la sélection de variables pour les modèles linéaires généralisés (GLM) via la vraisemblance nonconcave pénalisée. On propose un algorithme pour maximiser la vraisemblance pénalisée pour une large classe de fonctions de pénalité non convexes. La propriété de convergence de l’algorithme ainsi que la propriété oracle de l’estimateur obtenu après une itération ont été établies. Des simulations ainsi qu’une application sur données réelles sont également présentées. / This work is a contribution to the selection of statistical models and more specifically in the selection of variables in penalized linear quantile regression when the dimension is high. It focuses on two points in the selection process: the stability of selection and the inclusion of variables by grouping effect. As a first contribution, we propose a transition from the penalized least squares regression to quantiles regression (QR). A bootstrap approach based on frequency of selection of each variable is proposed for the construction of linear models (LM). In most cases, the QR approach provides more significant coefficients. A second contribution is to adapt some algorithms of "Random" LASSO (Least Absolute Shrinkage and Solution Operator) family in connection with the QR and to propose methods of selection stability. Examples from food security illustrate the obtained results. As part of the penalized QR in high dimension, the grouping effect property is established under weak conditions and the oracle ones. Two examples of real and simulated data illustrate the regularization paths of the proposed algorithms. The last contribution deals with variable selection for generalized linear models (GLM) using the nonconcave penalized likelihood. We propose an algorithm to maximize the penalized likelihood for a broad class of non-convex penalty functions. The convergence property of the algorithm and the oracle one of the estimator obtained after an iteration have been established. Simulations and an application to real data are also presented.

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Cette thèse porte sur la sélection des variables dans les modèles de régression linéaires multidimensionnels et les modèles de régression linéaires fonctionnels. Plus précisément, nous proposons trois nouvelles approches de sélection de variables qui généralisent des méthodes existantes dans la littérature. La première méthode permet de sélectionner des variables aléatoires continues dans un modèle linéaire multidimensionnel. Cette approche généralise celle de NKIET (2001) obtenue dans le cas d'un modèle linéaire unidimensionnel. Une étude comparative, par simulation, basée sur le calcul de la perte de prédiction montre que notre méthode est meilleure à celle de An et al. (2013). La deuxième approche propose une nouvelle méthode de sélection des variables mixtes (mélange de variables discrètes et de variables continues) en analyse discriminante pour plus de deux groupes. Cette méthode est basée sur la généralisation dans le cadre mixte de l'approche de NKIET (2012) obtenue dans le cas de l'analyse discriminante de plus de deux groupes. Une étude comparative par simulation montre, à partir du taux de bon classement que cette méthode a les mêmes performances que celle de MAHAT et al. (2007) dans le cas de deux groupes. Enfin, nous proposons dans la troisième approche une méthode de sélection de variables dans un modèle linéaire fonctionnel additif. Pour cela, nous considérons un modèle de régression d'une variable aléatoire réelle sur une somme de variables aléatoires fonctionnelles. En utilisant la distance de Hausdorff, qui mesure l'éloignement entre deux ensembles, nous montrons dans un exemple par simulation, une illustration de notre approche. / This thesis focuses on variables selection on linear models and additif functional linear model. More precisely we propose three variables selection methods. The first one is concerned with the selection continuous variables of multidimentional linear model. The comparative study based on prediction loss shows that our method is beter to method of An et al. (2013) Secondly, we propose a new selection method of mixed variables (mixing of discretes and continuous variables). This method is based on generalization in the mixed framwork of NKIET (2012) method, more precisely, is based on a generalization of linear canonical invariance criterion to the framework of discrimination with mixed variables. A comparative study based on the rate of good classification show that our method is equivalente to the method of MAHAT et al. (2007) in the case of two groups. In the third method, we propose an approach of variables selection on an additive functional linear model. A simulations study shows from Hausdorff distance an illustration of our approach.