This thesis addresses some problems that arise in signal processing when the noise is impulsive and follows a heavy tailed distribution. After reviewing several of the more well known heavy- tailed distributions the common problem of which of these hest models the observations is considered. To this end, a test is proposed for the symmetric alpha stable distribution. The test threshold is found using both asymptotic theory and parametric bootstrap resampling. In doing so, some modifications are proposed for Koutrouvelis' estimator of the symmetric alpha stable distributions parameters that improve performance. In electrical systems impulsive noise is generated externally to the receiver while thermal Gaussian noise is generated internally by the receiver electronics, the resultant noise is an additive combination of these two independent sources. A characteristic function domain estimator for the parameters of the resultant distribution is developed for the case when the impulsive noise is modeled by a symmetric alpha stable distribution. Having concentrated on validation and parameter estimation for the noise model, some problems in signal detection and estimation are considered. Detection of the number of sources impinging on an array is an important first. step in many array processing problems for which the development of optimal methods can be complicated even in the Gaussian case. Here, a multiple hypothesis test for the equality of the eigenvalues of the sample array covariance is proposed. / The nonparametric bootstrap is used to estimate the distributions of the test statistics removing the assumption of Gaussianity and offering improved performance for heavy tailed observations. Finally, some robust estimators are proposed for estimating parametric signals in additive noise. These are based on M-estimators but implicitly incorporate an estimate of the noise distribution. enabling the estimator to adapt to the unknown noise distribution. Two estimators are developed, one uses a nonparametric kernel density estimator while the other models the score function of the noise distribution with a linear combination of basis functions.

The consequences of substituting the denominator Q 3(p) - Q 1(p) by Q 2 - Q 1(p) in Groeneveld's class of quantile measures of kurtosis (γ 2(p)) for symmetric distributions, are explored using the symmetric influence function. The relationship between the measure γ 2(p) and the alternative class of kurtosis measures κ2(p) is derived together with the relationship between their influence functions. The Laplace, Logistic, symmetric Two-sided Power, Tukey and Beta distributions are considered in the examples in order to discuss the results obtained pertaining to unimodal, heavy tailed, bounded domain and U-shaped distributions.

Este trabalho tem como objetivo calcular intervalos de confiança para altos quantis oriundos de distribuições de caudas pesadas. Para isso, utilizamos os métodos da aproximação pela distribuição normal, razão de verossimilhanças, {\\it data tilting} e gama generalizada. Obtivemos, através de simulações, que os intervalos calculados a partir do método da gama generalizada apresentam probabilidades de cobertura bem próximas do nível de confiança, com amplitudes médias menores do que os outros três métodos, para dados gerados da distribuição Weibull. Todavia, para dados gerados da distribuição Fréchet, o método da razão de verossimilhanças fornece os melhores intervalos. Aplicamos os métodos utilizados neste trabalho a um conjunto de dados reais, referentes aos pagamentos de indenizações, em reais, de seguros de incêndio, de um determinado grupo de seguradoras no Brasil, no ano de 2003 / In this work, confidence intervals for high quantiles from heavy-tailed distributions were computed. More specifically, four methods, namely, normal approximation method, likelihood ratio method, data tilting method and generalised gamma method are used. A simulation study with data generated from Weibull distribution has shown that the generalised gamma method has better coverage probabilities with the smallest average length intervals. However, from data generated from Fréchet distribution, the likelihood ratio method gives the better intervals. Moreover, the methods used in this work are applied on a real data set from 1758 Brazilian fire claims

Este trabalho tem como objetivo calcular intervalos de confiança para altos quantis oriundos de distribuições de caudas pesadas. Para isso, utilizamos os métodos da aproximação pela distribuição normal, razão de verossimilhanças, {\\it data tilting} e gama generalizada. Obtivemos, através de simulações, que os intervalos calculados a partir do método da gama generalizada apresentam probabilidades de cobertura bem próximas do nível de confiança, com amplitudes médias menores do que os outros três métodos, para dados gerados da distribuição Weibull. Todavia, para dados gerados da distribuição Fréchet, o método da razão de verossimilhanças fornece os melhores intervalos. Aplicamos os métodos utilizados neste trabalho a um conjunto de dados reais, referentes aos pagamentos de indenizações, em reais, de seguros de incêndio, de um determinado grupo de seguradoras no Brasil, no ano de 2003 / In this work, confidence intervals for high quantiles from heavy-tailed distributions were computed. More specifically, four methods, namely, normal approximation method, likelihood ratio method, data tilting method and generalised gamma method are used. A simulation study with data generated from Weibull distribution has shown that the generalised gamma method has better coverage probabilities with the smallest average length intervals. However, from data generated from Fréchet distribution, the likelihood ratio method gives the better intervals. Moreover, the methods used in this work are applied on a real data set from 1758 Brazilian fire claims

This thesis is divided in four chapters. The two first chapters introduce a parametric quantile-based estimation method of univariate heavy-tailed distributions and elliptical distributions, respectively. If one is interested in estimating the tail index without imposing a parametric form for the entire distribution function, but only on the tail behaviour, we propose a multivariate Hill estimator for elliptical distributions in chapter three. In the first three chapters we assume an independent and identically distributed setting, and so as a first step to a dependent setting, using quantiles, we prove in the last chapter the asymptotic normality of marginal sample quantiles for stationary processes under the S-mixing condition.<p><p><p>The first chapter introduces a quantile- and simulation-based estimation method, which we call the Method of Simulated Quantiles, or simply MSQ. Since it is based on quantiles, it is a moment-free approach. And since it is based on simulations, we do not need closed form expressions of any function that represents the probability law of the process. Thus, it is useful in case the probability density functions has no closed form or/and moments do not exist. It is based on a vector of functions of quantiles. The principle consists in matching functions of theoretical quantiles, which depend on the parameters of the assumed probability law, with those of empirical quantiles, which depend on the data. Since the theoretical functions of quantiles may not have a closed form expression, we rely on simulations.<p><p><p>The second chapter deals with the estimation of the parameters of elliptical distributions by means of a multivariate extension of MSQ. In this chapter we propose inference for vast dimensional elliptical distributions. Estimation is based on quantiles, which always exist regardless of the thickness of the tails, and testing is based on the geometry of the elliptical family. The multivariate extension of MSQ faces the difficulty of constructing a function of quantiles that is informative about the covariation parameters. We show that the interquartile range of a projection of pairwise random variables onto the 45 degree line is very informative about the covariation.<p><p><p>The third chapter consists in constructing a multivariate tail index estimator. In the univariate case, the most popular estimator for the tail exponent is the Hill estimator introduced by Bruce Hill in 1975. The aim of this chapter is to propose an estimator of the tail index in a multivariate context; more precisely, in the case of regularly varying elliptical distributions. Since, for univariate random variables, our estimator boils down to the Hill estimator, we name it after Bruce Hill. Our estimator is based on the distance between an elliptical probability contour and the exceedance observations. <p><p><p>Finally, the fourth chapter investigates the asymptotic behaviour of the marginal sample quantiles for p-dimensional stationary processes and we obtain the asymptotic normality of the empirical quantile vector. We assume that the processes are S-mixing, a recently introduced and widely applicable notion of dependence. A remarkable property of S-mixing is the fact that it doesn't require any higher order moment assumptions to be verified. Since we are interested in quantiles and processes that are probably heavy-tailed, this is of particular interest.<p> / Doctorat en Sciences économiques et de gestion / info:eu-repo/semantics/nonPublished

The aim of this thesis is to show that the use of heavy-tailed distributions in finance is theoretically unfounded and may cause significant misunderstandings and fallacies in model interpretation. The main reason seems to be a wrong understanding of the concept of the distributional tail. Also in models based on real data it seems more reasonable to concentrate on the central part of the distribution not tails. Powered by TCPDF (www.tcpdf.org)

Cette thèse présente plusieurs aspects de la croissance stochastique des interfaces, par lebiais de son modèle le plus étudié, l'équation de Kardar-Parisi-Zhang (KPZ). Bien qued'expression très simple, cette équation recèle une grande richesse phénoménologiqueet est l'objet d'une recherche intensive depuis des dizaines d'années. Cela a conduit àl'émergence d'une nouvelle classe d'universalité, contenant des modèles de croissanceparmi les plus courants, tels que le Eden model ou encore le Polynuclear Growth Model.L'équation KPZ est également reliée à des problèmes d'optimisation en présence dedésordre (le Polymère Dirigé), ou encore à la turbulence des uides (l'équation de Burger), renforçant son intérêt. Cependant, les limites de cette classe d'universalitésont encore mal comprises. L'objet de cette thèse est, après avoir présenté les progrèsles plus récents dans le domaine, de tester les limites de cette classe d'universalité. Lathèse s'articule en quatre parties :i) Dans un premier temps, nous présentons des outils théoriques qui permettent decaractériser finement l'évolution de l'interface. Ces outils montrent une grande flexibilité, que nous illustrons en considérant le cas d'une géométrie confinée (une interfacecroissant le long d'une paroi).ii) Nous nous penchons ensuite sur l'influence du désordre, et plus particulièrementl'importance des évènements extrêmes dans la mécanique de croissance. Les largesfluctuations du désordre déforment l'interface et conduisent à une modification notabledes exposants de scaling. Nous portons une attention particulière aux conséquencesd'un tel désordre sur les stratégies d'optimisation en milieu désordonné.iii) La présence de corrélations dans le désordre est d'un intérêt expérimentalimmédiat. Bien qu'elles ne modifient pas la classe d'universalité, elles influent grandement sur la vitesse de croissance moyenne de l'interface. Cette partie est dédiée àl'étude de cette vitesse moyenne, souvent négligée car délicate à définir, et à l'existenced'un optimum de croissance intimement lié à la compétition entre exploration et exploitation.iv) Enfin, nous considérons un exemple expérimental de croissance stochastique (quin'appartient toutefois pas à la classe KPZ) et développons un formalisme phénoménologiquepour modéliser la propagation d'une interface chimique dans un milieu poreux désordonné.Tout au long du manuscrit, les conséquences des phénomènes observées dans desdomaines variés, tels que les stratégies d'optimisation, la dynamique des populations,la turbulence ou la finance, sont détaillées. / This Thesis presents several aspects of the stochastic growth, through its most paradig-matic model, the Kardar-Parisi-Zhang equation (KPZ). Albeit very simple, this equa-tion shows a rich behaviour and has been extensively studied for decades. The existenceof a new universality class is now well established, containing numerous growth modelslike the Eden model or the Polynuclear Growth Model. The KPZ equation is closelyrelated to optimisation problems (the Directed Polymer) or turbulence of uids (theBurgers equation), a feature that underlines its importance. Nonetheless, the bound-aries of this universality class are still vague. The focus of this Thesis is to probe thoselimits through various modifications of the models. It is divided in four chapters:i) First, we present theoretical tools, borrowed from integrable systems, that allowto characterize in great details the evolution of the interface. Those tools exhibitconsiderable exibility due to the large corpus of work on integrable systems, and weillustrate it by tackling the case of confined geometry (growth close to a hard wall).ii) We investigate the inuence of the disorder distribution, and more specificallythe importance of large events, with heavy-tailed distributions. Those extreme eventsstretch the interface and notably modify the main scaling exponents. The consequenceson optimization strategies in disorder landscapes are emphasized.iii) The presence of correlations in the disorder is of natural experimental interest.Although they do not impact the KPZ class, they greatly inuence the average speed ofgrowth. The latter quantity is often overlooked because it is non-universal and ratherill-defined. Nonetheless, we show that a generic optimal average speed exists in presenceof time correlations, due to a competition between exploration and exploitation.iv) Finally, we consider a set of experiments about chemical front growth in porousmedium. While this growth process is not related to KPZ in an immediate way, wepresent different tools that effciently reproduce the observations.Along that work, the consequences of each Chapter in various domains, like opti-misation strategies, turbulence, population dynamics or finance, are detailed.

Les modèles linéaires généralisés (GLMs) constituent l’une des classes de modèles les plus populaires en statistique. Cette classe contient une grande variété de modèles de régression fréquemment utilisés, tels que la régression linéaire normale, la régression logistique et les gamma GLMs. Dans les GLMs, la distribution de la variable de réponse définit une famille exponentielle. Un désavantage de ces modèles est qu’ils ne sont pas robustes par rapport aux valeurs aberrantes. Pour les modèles comme la régression linéaire normale et les gamma GLMs, la non-robustesse est une conséquence des ailes exponentielles des densités. La différence entre les tendances de l’ensemble des données et celles des valeurs aberrantes donne lieu à des inférences et des prédictions biaisées. A notre connaissance, il n’existe pas d’approche bayésienne robuste spécifique pour les GLMs. La méthode la plus populaire est fréquentiste ; c’est celle de Cantoni and Ronchetti (2001). Leur approche consiste à adapter les M-estimateurs robustes pour la régression linéaire au contexte des GLMs. Cependant, leur estimateur est dérivé d’une modification de la dérivée de la log-vraisemblance, au lieu d’une modification de la vraisemblance (comme avec les M-estimateurs robustes pour la régression linéaire). Par conséquent, il n’est pas possible d’établir une correspondance claire entre la fonction modifiée à optimiser et un modèle. Le fait de proposer un modèle robuste présente deux avantages. Premièrement, il permet de comprendre et d’interpréter la modélisation. Deuxièmement, il permet l’analyse fréquentiste et bayésienne. La méthode que nous proposons s’inspire des idées de la régression linéaire robuste bayésienne. Nous adaptons l’approche proposée par Gagnon et al. (2020), qui consiste à utiliser une distribution normale modifiée avec des ailes plus relevées pour le terme d’erreur. Dans notre contexte, la distribution de la variable de réponse est une version modifiée où la partie centrale de la densité est conservée telle quelle, tandis que les extrémités sont remplacées par des ailes log-Pareto, se comportant comme (1/|x|)(1/ log |x|)λ. Ce mémoire se concentre sur les gamma GLMs. La performance est mesurée à la fois théoriquement et empiriquement, avec une analyse des données sur les coûts hospitaliers. / Generalized linear models (GLMs) form one of the most popular classes of models in statistics. This class contains a large variety of commonly used regression models, such as normal linear regression, logistic regression and gamma GLMs. In GLMs, the response variable distribution defines an exponential family. A drawback of these models is that they are non-robust against outliers. For models like the normal linear regression and gamma GLMs, the non-robustness is a consequence of the exponential tails of the densities. The difference in trends in the bulk of the data and the outliers yields skewed inference and prediction. To our knowledge, there is no Bayesian robust approach specifically for GLMs. The most popular method is frequentist; it is that of Cantoni and Ronchetti (2001). Their approach is to adapt the robust M-estimators for linear regression to the context of GLMs. However, their estimator is derived from a modification of the derivative of the log-likelihood, instead of from a modification of the likelihood (as with robust M-estimators for linear regression). As a consequence, it is not possible to establish a clear correspondence between the modified function to optimize and a model. Having a robust model has two advantages. First, it allows for an understanding and an interpretation of the modelling. Second, it allows for both frequentist and Bayesian analysis. The method we propose is based on ideas from Bayesian robust linear regression. We adapt the approach proposed by Gagnon et al. (2020), which consists of using a modified normal distribution with heavier tails for the error term. In our context, the distribution of the response variable is a modified version where the central part of the density is kept as is, while the extremities are replaced by log-Pareto tails, behaving like (1/|x|)(1/ log |x|)λ. The focus of this thesis is on gamma GLMs. The performance is measured both theoretically and empirically, with an analysis of hospital costs data.

This thesis concentrates on different approaches of solving decision making problems with an aspect of randomness. The basic methodologies of converting stochastic optimization problems to deterministic optimization problems are described. The proximity of solution of a problem and its empirical counterpart is shown. The empirical counterpart is used when we don't know the distribution of the random elements of the former problem. The distribution with heavy tails, stable distribution and their relationship is described. The stochastic dominance and the possibility of defining problems with stochastic dominance is introduced. The proximity of solution of problem with second order stochastic dominance and the solution of its empirical counterpart is proven. A portfolio management problem with second order stochastic dominance is solved by solving the equivalent empirical problem. Powered by TCPDF (www.tcpdf.org)

Cette thèse s'inscrit dans le contexte de la statistique des valeurs extrêmes. Elle y apporte deux contributions principales. Dans la littérature récente en statistique des valeurs extrêmes, un modèle de queues de distributions a été introduit afin d'englober aussi bien les lois de type Pareto que les lois à queue de type Weibull. Les deux principaux types de décroissance de la fonction de survie sont ainsi modélisés. Un estimateur des quantiles extrêmes a été déduit de ce modèle mais il dépend de deux paramètres inconnus, le rendant inutile dans des situations pratiques. La première contribution de cette thèse est de proposer des estimateurs de ces paramètres. Insérer nos estimateurs dans l'estimateur des quantiles extrêmes précédent permet alors d'estimer des quantiles extrêmes pour des lois de type Pareto aussi bien que pour des lois à queue de type Weibull d'une façon unifiée. Les lois asymptotiques de nos trois nouveaux estimateurs sont établies et leur efficacité est illustrée sur des données simulées et sur un jeu de données réelles de débits de la rivière Nidd se situant dans le Yorkshire en Angleterre. La seconde contribution de cette thèse consiste à introduire et estimer une nouvelle mesure de risque appelé Conditional Tail Moment. Elle est définie comme le moment d'ordre a>0 de la loi des pertes au-delà du quantile d'ordre p appartenant à ]0,1[ de la fonction de survie. Estimer le Conditional Tail Moment permet d'estimer toutes les mesures de risque basées sur les moments conditionnels telles que la Value-at-Risk, la Conditional Tail Expectation, la Conditional Value-at-Risk, la Conditional Tail Variance ou la Conditional Tail Skewness. Ici, on s'intéresse à l'estimation de ces mesures de risque dans le cas de pertes extrêmes c'est-à-dire lorsque p tend vers 0 lorsque la taille de l'échantillon augmente. On suppose également que la loi des pertes est à queue lourde et qu'elle dépend d'une covariable. Les estimateurs proposés combinent des méthodes d'estimation non-paramétrique à noyau avec des méthodes issues de la statistique des valeurs extrêmes. Le comportement asymptotique de nos estimateurs est établi et illustré aussi bien sur des données simulées que sur des données réelles de pluviométrie provenant de la région Cévennes-Vivarais. / This thesis can be viewed within the context of extreme value statistics. It provides two main contributions to this subject area. In the recent literature on extreme value statistics, a model on tail distributions which encompasses Pareto-type distributions as well as Weibull tail-distributions has been introduced. The two main types of decreasing of the survival function are thus modeled. An estimator of extreme quantiles has been deduced from this model, but it depends on two unknown parameters, making it useless in practical situations. The first contribution of this thesis is to propose estimators of these parameters. Plugging our estimators in the previous extreme quantiles estimator allows us to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unified way. The asymptotic distributions of our three new estimators are established and their efficiency is illustrated on a simulation study and on a real data set of exceedances of the Nidd river in the Yorkshire (England). The second contribution of this thesis is the introduction and the estimation of a new risk measure, the so-called Conditional Tail Moment. It is defined as the moment of order a>0 of the loss distribution above the quantile of order p in (0,1) of the survival function. Estimating the Conditional Tail Moment permits to estimate all risk measures based on conditional moments such as the Value-at-Risk, the Conditional Tail Expectation, the Conditional Value-at-Risk, the Conditional Tail Variance or the Conditional Tail Skewness. Here, we focus on the estimation of these risk measures in case of extreme losses i.e. when p converges to 0 when the size of the sample increases. It is moreover assumed that the loss distribution is heavy-tailed and depends on a covariate. The estimation method thus combines nonparametric kernel methods with extreme-value statistics. The asymptotic distribution of the estimators is established and their finite sample behavior is illustrated both on simulated data and on a real data set of daily rainfalls in the Cévennes-Vivarais region (France).