We compare two time series models: an ARMA(1,1)-ACD(1,1)-NIG model against an ARMA(1,1)-GARCH(1,1)-NIG model. Their out-of-sample performance is of interest rather than their in-sample properties. The models produce one-day ahead forecasts which are evaluated using three statistical tests: VaR-test, VaRdur-test and Berkowitz-test. All three tests are concerned with the the tail events, since our time series models are often used to estimate downside risk. When the two models are applied to data on Canadian stock market returns, our three statistical tests point in the direction that the ACD model and the GARCH model perform similarly. The difference between the models is small. We finish with comments on the model uncertainty inherit in the comparison.

Em uma quantidade substancial de problemas de astronomia, tem-se interesse na estimação do valor assumido, para diversas funções g, de alguma quantidade desconhecida z ∈ ℜ com base em covariáveis x ∈ ℜd. Isto é feito utilizando-se uma amostra (X1, Z1), ... (Xn, Zn). As duas abordagens usualmente utilizadas para resolver este problema consistem em (1) estimar a regressão de Z em x, e plugar esta na função g ou (2)estimar a densidade condicional f (z Ι x) e plugá-la em ∫ g(z) f (z Ι x)dz. Infelizmente, poucos estudos apresentam comparações quantitativas destas duas abordagens. Além disso, poucos métodos de estimação de densidade condicional tiveram seus desempenhos comparados nestes problemas. Em vista disso, o objetivo deste trabalho é apresentar diversas comparações de técnicas de estimação de funções de uma quantidade desconhecida. Em particular, damos destaque para métodos não paramétricos. Além dos estimadores (1) e (2), propomos também uma nova abordagem que consistem em estimar diretamente a função de regressão de g(Z) em x. Essas abordagens foram testadas em diferentes funções nos conjuntos de dados DEEP2 e Sheldon 2012. Para quase todas as funções testadas, o estimador (1) obteve os piores resultados, exceto quando utilizamos florestas aleatórias. Em diversos casos, a nova abordagem proposta apresentou melhores resultados, assim como o estimador (2). Em particular, verificamos que métodos via florestas aleatórias, em geral, levaram a bons resultados. / In a substantial a mount of astronomy problems, we are interested in estimating values assumed of some unknown quantity z ∈ ℜ, for many function g, based on covariates x ∈ ℜd. This is made using a sample (X1, Z1), ..., (Xn, Zn). Two approaches that are usually used to solve this problem consist in (1) estimating a regression function of Z in x and plugging it into the g or (2) estimating a conditional density f (z Ι x) and plugging it into ∫ g(z) f (z Ι x)dz. Unfortunately, few studies exhibit quantitative comparisons between these two approaches.Besides that, few conditional density estimation methods had their performance compared in these problems.In view of this, the objective of this work is to show several comparisons of techniques used to estimate functions of unknown quantity. In particular we highlight nonparametric methods. In addition to estimators (1) and (2), we also propose a new ap proach that consists in directly estimating the regression function from g(Z) on x. These approaches were tested in different functions in the DEEP 2 and Sheldon 2012 datasets. For almost all the functions tested, the estimator (1) obtained the worst results, except when we use the random forests methods. In several cases, the proposed new approach presented better results, as well as the estimator (2) .In particular, we verified that random forests methods generally present to good results.

Em uma quantidade substancial de problemas de astronomia, tem-se interesse na estimação do valor assumido, para diversas funções g, de alguma quantidade desconhecida z ∈ ℜ com base em covariáveis x ∈ ℜd. Isto é feito utilizando-se uma amostra (X1, Z1), ... (Xn, Zn). As duas abordagens usualmente utilizadas para resolver este problema consistem em (1) estimar a regressão de Z em x, e plugar esta na função g ou (2)estimar a densidade condicional f (z Ι x) e plugá-la em ∫ g(z) f (z Ι x)dz. Infelizmente, poucos estudos apresentam comparações quantitativas destas duas abordagens. Além disso, poucos métodos de estimação de densidade condicional tiveram seus desempenhos comparados nestes problemas. Em vista disso, o objetivo deste trabalho é apresentar diversas comparações de técnicas de estimação de funções de uma quantidade desconhecida. Em particular, damos destaque para métodos não paramétricos. Além dos estimadores (1) e (2), propomos também uma nova abordagem que consistem em estimar diretamente a função de regressão de g(Z) em x. Essas abordagens foram testadas em diferentes funções nos conjuntos de dados DEEP2 e Sheldon 2012. Para quase todas as funções testadas, o estimador (1) obteve os piores resultados, exceto quando utilizamos florestas aleatórias. Em diversos casos, a nova abordagem proposta apresentou melhores resultados, assim como o estimador (2). Em particular, verificamos que métodos via florestas aleatórias, em geral, levaram a bons resultados. / In a substantial a mount of astronomy problems, we are interested in estimating values assumed of some unknown quantity z ∈ ℜ, for many function g, based on covariates x ∈ ℜd. This is made using a sample (X1, Z1), ..., (Xn, Zn). Two approaches that are usually used to solve this problem consist in (1) estimating a regression function of Z in x and plugging it into the g or (2) estimating a conditional density f (z Ι x) and plugging it into ∫ g(z) f (z Ι x)dz. Unfortunately, few studies exhibit quantitative comparisons between these two approaches.Besides that, few conditional density estimation methods had their performance compared in these problems.In view of this, the objective of this work is to show several comparisons of techniques used to estimate functions of unknown quantity. In particular we highlight nonparametric methods. In addition to estimators (1) and (2), we also propose a new ap proach that consists in directly estimating the regression function from g(Z) on x. These approaches were tested in different functions in the DEEP 2 and Sheldon 2012 datasets. For almost all the functions tested, the estimator (1) obtained the worst results, except when we use the random forests methods. In several cases, the proposed new approach presented better results, as well as the estimator (2) .In particular, we verified that random forests methods generally present to good results.

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Dans cette thèse, nous nous intéressons à l'estimation non paramétrique de la densité conditionnelle d'une variable aléatoire réponse réelle conditionnée par une variable aléatoire explicative fonctionnelle de dimension éventuellement fi nie. Dans un premier temps, nous considérons l'estimation de ce modèle par la méthode du double noyaux. Nous proposons une méthode de sélection automatique du paramètre de lissage (global et puis local) intervenant dans l'estimateur à noyau, et puis nous montrons l'optimalité asymptotique du paramètre obtenu quand les observations sont indépendantes et identiquement distribuées. Le critère adopté est issu du principe de validation croisée. Dans cette partie nous avons procédé également à la comparaison de l'efficacité des deux types de choix (local et global). Dans la deuxième partie et dans le même contexte topologique, nous estimons la densité conditionnelle par la méthode des polynômes locaux. Sous certaines conditions, nous établissons des propriétés asymptotiques de cet estimateur telles que la convergence presque-complète et la convergence en moyenne quadratique dans le cas où les observations sont indépendantes et identiquement distribuées. Nous étendons aussi nos résultats au cas où les observations sont de type α- mélangeantes, dont on montre la convergence presque-complète (avec vitesse de convergence) de l'estimateur proposé. Enfi n, l'applicabilité rapide et facile de nos résultats théoriques, dans le cadre fonctionnel, est illustrée par des exemples (1) sur des données simulées, et (2) sur des données réelles. / In this thesis, we consider the problem of the nonparametric estimation of the conditional density when the response variable is real and the regressor is valued in a functional space. In the rst part, we use the double kernels method's as a estimation method where we focus on the choice of the smoothing parameters. We construct a data driven method permitting to select optimally and automatically bandwidths. As main results, we study the asymptotic optimality of this selection method in the case where observations are independent and identically distributed (i.i.d). Our selection rule is based on the classical cross-validation ideas and it deals with the both global and local choices. The performance of our approach is illustrated also by some simulation results on nite samples where we conduct a comparison between the two types of bandwidths choices (local and global). In the second part, we adopt a functional version of the local linear method, in the same topological context, to estimate some functional parameters. Under some general conditions, we establish the almost-complete convergence (with rates) of the proposed estimator in the both cases ( the i.i.d. case and the α-mixing case) . As application, we use the conditional density estimator to estimate the conditional mode estimation and to derive some asymptotic proprieties of the constructed estimator. Then, we establish the quadratic error of this estimator by giving its exact asymptotic expansion (involved in the leading in the bias and variance terms). Finally, the applicability of our results is then veri ed and validated for (1) simulated data, and (2) some real data.

Nous considérons le problème d’estimation de densités conditionnelles en modérément grandes dimensions. Beaucoup plus informatives que les fonctions de régression, les densités condi- tionnelles sont d’un intérêt majeur dans les méthodes récentes, notamment dans le cadre bayésien (étude de la distribution postérieure, recherche de ses modes...). Après avoir rappelé les problèmes liés à l’estimation en grande dimension dans l’introduction, les deux chapitres suivants développent deux méthodes qui s’attaquent au fléau de la dimension en demandant : d’être efficace computation- nellement grâce à une procédure itérative gloutonne, de détecter les variables pertinentes sous une hypothèse de parcimonie, et converger à vitesse minimax quasi-optimale. Plus précisément, les deux méthodes considèrent des estimateurs à noyau bien adaptés à l’estimation de densités conditionnelles et sélectionnent une fenêtre multivariée ponctuelle en revisitant l’algorithme glouton RODEO (Re- gularisation Of Derivative Expectation Operator). La première méthode ayant des problèmes d’ini- tialisation et des facteurs logarithmiques supplémentaires dans la vitesse de convergence, la seconde méthode résout ces problèmes, tout en ajoutant l’adaptation à la régularité. Dans l’avant-dernier cha- pitre, on traite de la calibration et des performances numériques de ces deux procédures, avant de donner quelques commentaires et perspectives dans le dernier chapitre. / We consider the problem of conditional density estimation in moderately large dimen- sions. Much more informative than regression functions, conditional densities are of main interest in recent methods, particularly in the Bayesian framework (studying the posterior distribution, find- ing its modes...). After recalling the estimation issues in high dimension in the introduction, the two following chapters develop on two methods which address the issues of the curse of dimensionality: being computationally efficient by a greedy iterative procedure, detecting under some suitably defined sparsity conditions the relevant variables, while converging at a quasi-optimal minimax rate. More precisely, the two methods consider kernel estimators well-adapted for conditional density estimation and select a pointwise multivariate bandwidth by revisiting the greedy algorithm RODEO (Regular- isation Of Derivative Expectation Operator). The first method having some initialization problems and extra logarithmic factors in its convergence rate, the second method solves these problems, while adding adaptation to the smoothness. In the penultimate chapter, we discuss the calibration and nu- merical performance of these two procedures, before giving some comments and perspectives in the last chapter.

<p> Generative models form powerful tools for learning data distributions and simulating new samples. Recent years have seen significant advances in the flexibility and applicability of such models, with Bayesian approaches like nonparametric Bayesian models and deep neural network models such as Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs) finding use in a wide range of domains. However, the black-box nature of these models means that they are often hard to interpret, and they often come with modeling implications that are inconsistent with side knowledge resulting from domain knowledge. This thesis studies situations where the modeler has side knowledge represented as probability distributions on functionals of the objects being modeled, and we study methods to incorporate this particular kind of side knowledge into flexible generative models. This dissertation covers three main parts. </p> <p><br></p> <p>The first part focuses on incorporating a special case of the aforementioned side knowledge into flexible nonparametric Bayesian models. Many times, practitioners have additional distributional information about a subset of the coordinates of the observations being modeled. The flexibility of nonparametric Bayesian models usually implies incompatibility with this side information. Such inconsistency triggers the necessity of developing methods to incorporate this side knowledge into flexible nonparametric Bayesian models. We design a specialized generative process to build in this side knowledge and propose a novel sigmoid Gaussian process conditional model. We also develop a corresponding posterior sampling method based on data augmentation to overcome a doubly intractable problem. We illustrate the efficacy of our proposed constrained nonparametric Bayesian model in a variety of real-world scenarios including modeling environmental and earthquake data. </p> <p><br></p> <p>The second part of the dissertation discusses neural network approaches to satisfying the said general side knowledge. Further, the generative models considered in this part broaden into black-box models. We formulate this side knowledge incorporation problem as a constrained divergence minimization problem and propose two scalable neural network approaches as its solution. We demonstrate their practicality using various synthetic and real examples. </p> <p><br></p> <p> The third part of the dissertation concentrates on a specific generative model of individual pixels of the fMRI data constructed from a latent group image. Usually there is two-fold side knowledge about the latent group image: spatial structure and partial activation zones. The former can be captured by modeling the prior for the group image with Markov random fields. The latter, which is often obtained from previous related studies, is left for future research. We propose a novel Bayesian model with Markov random fields and aim to estimate the maximum a posteriori for the group image. We also derive a variational Bayes algorithm to overcome local optima in the optimization.</p>

This thesis studies the mixed data types kernel estimation framework for the models of discrete dependent variables, which are known as kernel discrete conditional functions. The conventional parametric multinomial logit MNL model is compared with the mixed data types kernel conditional density estimator in Chapter (2). A new kernel estimator for discrete time single state hazard models is developed in Chapter (3), and named as the discrete time “external kernel hazard” estimator. The discrete time (mixed) proportional hazard estimators are then compared with the discrete time external kernel hazard estimator empirically in Chapter (4). The work in Chapter (2) attempts to estimate a labour force participation decision model using a cross-section data from the UK labour force survey in 2007. The work in Chapter (4) estimates a hazard rate for job-vacancies in weeks, using data from Lancashire Careers Service (LCS) between the period from March 1988 to June 1992. The evidences from the vast literature regarding female labour force participation and the job-market random matching theory are used to examine the empirical results of the estimators. The parametric estimator are tighten by the restrictive assumption regarding the link function of the discrete dependent variable and the dummy variables of the discrete covariates. Adding interaction terms improves the performance of the parametric models but encounters other risks like generating multicollinearity problem, increasing the singularity of the data matrix and complicates the computation of the ML function. On the other hand, the mixed data types kernel estimation framework shows an outstanding performance compared with the conventional parametric estimation methods. The kernel functions that are used for the discrete variables, including the dependent variable, in the mixed data types estimation framework, have substantially improved the performance of the kernel estimators. The kernel framework uses very few assumptions about the functional form of the variables in the model, and relay on the right choice of the kernel functions in the estimator. The outcomes of the kernel conditional density shows that female education level and fertility have high impact on females propensity to work and be in the labour force. The kernel conditional density estimator captures more heterogeneity among the females in the sample than the MNL model due to the restrictive parametric assumptions in the later. The (mixed) proportional hazard framework, on the other hand, missed to capture the effect of the job-market tightness in the job-vacancies hazard rate and produce inconsistent results when the assumptions regarding the distribution of the unobserved heterogeneity are changed. The external kernel hazard estimator overcomes those problems and produce results that consistent with the job market random matching theory. The results in this thesis are useful for nonparametric estimation research in econometrics and in labour economics research.