Pair Constructions for High-Dimensional Dependence Models in Discrete and Continuous Time

Modeling high-dimensional dependence structures with parametric dependence models is a challenging and important part in statistics. The dependence models that we discuss here are based on bivariate building blocks and we illustrate how to assemble these building blocks to multivariate dependence structures. In the first part of the dissertation, we recall the pair-copula construction for random variables, present different extensions and apply the concept in an empirical study. In the second part, we introduce the new pair-Lévy copula construction to model the jump dependence of high-dimensional Lévy processes. Moreover, we present simulation and estimation algorithms, conduct a simulation study and discuss various applications of this concept

Modélisation de la dépendance et apprentissage automatique dans le contexte du provisionnement individuel et de la solvabilité en actuariat IARD

Les compagnies d'assurance jouent un rôle important dans l'économie des pays en s'impliquant de façon notable dans les marchés boursiers, obligataires et immobiliers, d'où la nécessité de préserver leur solvabilité. Le cycle spécifique de production en assurance amène des défis particuliers aux actuaires et aux gestionnaires de risque dans l'accomplissement de leurs tâches. Dans cette thèse, on a pour but de développer des approches et des algorithmes susceptibles d'aider à résoudre certaines problématiques liées aux opérations de provisionnement et de solvabilité d'une compagnie d'assurance. Les notions préliminaires pour ces contributions sont présentées dans l'introduction de cette thèse. Les modèles de provisionnement traditionnels sont fondés sur des informations agrégées. Ils ont connu un grand succès, comme en témoigne le nombre important d'articles et documents actuariels connexes. Cependant, en raison de la perte d'informations individuelles des sinistres, ces modèles représentent certaines limites pour fournir des estimations robustes et réalistes dans des contextes susceptibles d'évoluer. Dans ce sens, les modèles de réserve individuels représentent une alternative prometteuse. En s'inspirant des récentes recherches, on propose dans le Chapitre 1 un modèle de réserve individuel basé sur un réseau de neurones récurrent. Notre réseau a l'avantage d'être flexible pour plusieurs structures de base de données détaillés des sinistres et capable d'incorporer plusieurs informations statiques et dynamiques. À travers plusieurs études de cas avec des jeux de données simulés et réels, le réseau proposé est plus performant que le modèle agrégé chain-ladder. La détermination des exigences de capital pour un portefeuille repose sur une bonne connaissance des distributions marginales ainsi que les structures de dépendance liants les risques individuels. Dans les Chapitres 2 et 3 on s'intéresse à la modélisation de la dépendance et à l'estimation des mesures de risque. Le Chapitre 2 présente une analyse tenant compte des structures de dépendance extrême. Pour un portefeuille à deux risques, on considère en particulier à la dépendance négative extrême (antimonotonocité) qui a été moins étudiée dans la littérature contrairement à la dépendance positive extrême (comonotonocité). On développe des expressions explicites pour des mesures de risque de la somme d'une paire de variables antimontones pour trois familles de distributions. Les expressions explicites obtenues sont très utiles notamment pour quantifier le bénéfice de diversification pour des risques antimonotones. Face à une problématique avec plusieurs lignes d'affaires, plusieurs chercheurs et praticiens se sont intéressés à la modélisation en ayant recours à la théorie des copules au cours de la dernière décennie. Cette dernière fournit un outil flexible pour modéliser la structure de dépendance entre les variables aléatoires qui peuvent représenter, par exemple, des coûts de sinistres pour des contrats d'assurance. En s'inspirant des récentes recherches, dans le Chapitre 3, on définit une nouvelle famille de copules hiérarchiques. L'approche de construction proposée est basée sur une loi mélange exponentielle multivariée dont le vecteur commun est obtenu par une convolution descendante de variables aléatoires indépendantes. En se basant sur les mesures de corrélation des rangs, on propose un algorithme de détermination de la structure, tandis que l'estimation des paramètres est basée sur une vraisemblance composite. La flexibilité et l'utilité de cette famille de copules est démontrée à travers deux études de cas réelles.Insurance companies play an essential role in the countries economy by monopolizing a large part of the stock, bond, and estate markets, which implies the necessity to preserve their solvency and sustainability. However, the particular production cycle of the insurance industry may involve typical problems for actuaries and risk managers. This thesis project aims to develop approaches and algorithms that can help solve some of the reserving and solvency operations problems. The preliminary concepts for these contributions are presented in the introduction of this thesis. In current reserving practice, we use deterministic and stochastic aggregate methods. These traditional models based on aggregate information have been very successful, as evidenced by many related actuarial articles. However, due to the loss of individual claims information, these models represent some limitations in providing robust and realistic estimates, especially in variable settings. In this context, individual reserve models represent a promising alternative. Based on the recent researches, in Chapter 1, we propose an individual reserve model based on a recurrent neural network. Our network has the advantage of being flexible for several detailed claims datasets structures and incorporating several static and dynamic information. Furthermore, the proposed network outperforms the chain-ladder aggregate model through several case studies with simulated and real datasets. Determining the capital requirements for a portfolio relies on a good knowledge of the marginal distributions and the dependency structures linking the individual risks. In Chapters 2 and 3, we focus on the dependence modeling component as well as on risk measures. Chapter 2 presents an analysis taking into account extreme dependence structures. For a two-risk portfolio, we are particularly interested in extreme negative dependence (antimonotonicity), which has been less studied in the literature than extreme positive dependence (comonotonicity). We develop explicit expressions for risk measures of the sum of a pair of antimonotonic variables for three families of distributions. The explicit expressions obtained are very useful, e.g., to quantify the diversification benefit for antimonotonic risks. For a problem with several lines of business, over the last decade, several researchers and practitioners have been interested in modeling using copula theory. The latter provides a flexible tool for modeling the dependence structure between random variables that may represent, for example, claims costs for insurance contracts. Inspired by some recent researches, in Chapter 3, we define a new family of hierarchical copulas. The proposed construction approach is based on a multivariate exponential mixture distribution whose common vector is obtained by a top-down convolution of independent random variables. A structure determination algorithm is proposed based on rank correlation measures, while the parameter estimation is based on a composite likelihood. The flexibility and usefulness of this family of copulas are demonstrated through two real case studies

Stochastic processes for graphs, extreme values and their causality: inference, asymptotic theory and applications

This thesis provides some theoretical and practical statistical inference tools for multivariate stochastic processes to better understand the behaviours and properties present in the data. In particular, we focus on the modelling of graphs, that is a family of nodes linked together by a collection of edges, and extreme values, that are values above a high threshold to have their own dynamics compared to the typical behaviour of the process. We develop an ensemble of statistical models, statistical inference methods and their asymptotic study to ensure the good behaviour of estimation schemes in a wide variety of settings. We also devote a chapter to the formulation of a methodology based on pre-existing theory to unveil the causal dependency structure behind high-impact events.Open Acces

Development of an Agriculture Drought Risk model for the Iberian Peninsula

Among climate extremes, droughts are a major source of risk to agriculture and food security, which are expected to be increasingly affected considering the tendency towards a warmer climate. Within the context of climate change, the Iberian Peninsula (IP) is one of the regions recurrently highlighted as one of the areas expected to be particularly affected by drought episodes, due to the strong variations in the precipitation regime that make the region prone to drought events. In this way, this dissertation aimed the development of an agricultural drought risk model to contribute to more resilient systems in the IP. The skills of several drought indicators (SPEI, VCI, TCI and VHI) in predicting wheat and barley yields were firstly assessed based on neural networks and multiple linear regression models. Afterwards, copula-based models were designed to assess the joint probability of crop yields and droughts for a probabilistic risk assessment. The agricultural drought risk was then defined as the conditional probability of crop-loss under drought conditions and mapped at the province level of the IP. Ultimately, the additional risk associated with the occurrence of extreme temperatures during droughts was evaluated to characterize how the interaction between dry and hot conditions may exacerbate the impacts of the individual hazards in agriculture. The results showed the good performance of drought indicators in predicting the occurrence of crop failures. In general, barley exhibits greater agricultural drought risk in comparison to wheat. Overall, the risk of crop-loss increases with the severity of drought conditions, and drought-related risks increase with the interaction with extreme temperatures. Although compound dry and hot conditions lead to the larger damages in crop yield than the individual drought- or heat-stress, drought is still the dominant factor. From an operational point of view, this research intends contributing to the agricultural decision-making

Modèles de dépendance hiérarchique pour l'évaluation des passifs et la tarification en actuariat

Dans cette thèse on s’intéresse à la modélisation de la dépendance entre les risques en assurance non-vie, plus particulièrement dans le cadre des méthodes de provisionnement et en tarification. On expose le contexte actuel et les enjeux liés à la modélisation de la dépendance et l’importance d’une telle approche avec l’avènement des nouvelles normes et exigences des organismes réglementaires quant à la solvabilité des compagnies d’assurances générales. Récemment, Shi et Frees (2011) suggère d’incorporer la dépendance entre deux lignes d’affaires à travers une copule bivariée qui capture la dépendance entre deux cellules équivalentes de deux triangles de développement. Nous proposons deux approches différentes pour généraliser ce modèle. La première est basée sur les copules archimédiennes hiérarchiques, et la deuxième sur les effets aléatoires et la famille de distributions bivariées Sarmanov. Nous nous intéressons dans un premier temps, au Chapitre 2, à un modèle utilisant la classe des copules archimédiennes hiérarchiques, plus précisément la famille des copules partiellement imbriquées, afin d’inclure la dépendance à l’intérieur et entre deux lignes d’affaires à travers les effets calendaires. Par la suite, on considère un modèle alternatif, issu d’une autre classe de la famille des copules archimédiennes hiérarchiques, celle des copules totalement imbriquées, afin de modéliser la dépendance entre plus de deux lignes d’affaires. Une approche avec agrégation des risques basée sur un modèle formé d’une arborescence de copules bivariées y est également explorée. Une particularité importante de l’approche décrite au Chapitre 3 est que l’inférence au niveau de la dépendance se fait à travers les rangs des résidus, afin de pallier un éventuel risque de mauvaise spécification des lois marginales et de la copule régissant la dépendance. Comme deuxième approche, on s’intéresse également à la modélisation de la dépendance à travers des effets aléatoires. Pour ce faire, on considère la famille de distributions bivariées Sarmanov qui permet une modélisation flexible à l’intérieur et entre les lignes d’affaires, à travers les effets d’années de calendrier, années d’accident et périodes de développement. Des expressions fermées de la distribution jointe, ainsi qu’une illustration empirique avec des triangles de développement sont présentées au Chapitre 4. Aussi, nous proposons un modèle avec effets aléatoires dynamiques, où l’on donne plus de poids aux années les plus récentes, et utilisons l’information de la ligne corrélée afin d’effectuer une meilleure prédiction du risque. Cette dernière approche sera étudiée au Chapitre 5, à travers une application numérique sur les nombres de réclamations, illustrant l’utilité d’un tel modèle dans le cadre de la tarification. On conclut cette thèse par un rappel sur les contributions scientifiques de cette thèse, tout en proposant des angles d’ouvertures et des possibilités d’extension de ces travaux.The objective of this thesis is to propose innovative hierarchical approaches to model dependence within and between risks in non-life insurance in general, and in a loss reserving context in particular. One of the most critical problems in property/casualty insurance is to determine an appropriate reserve for incurred but unpaid losses. These provisions generally comprise most of the liabilities of a non-life insurance company. The global provisions are often determined under an assumption of independence between the lines of business. However, most risks are related to each other in practice, and this correlation needs to be taken into account. Recently, Shi and Frees (2011) proposed to include dependence between lines of business in a pairwise manner, through a copula that captures dependence between two equivalent cells of two different runoff triangles. In this thesis, we propose to generalize this model with two different approaches. Firstly, by using hierarchical Archimedean copulas to accommodate correlation within and between lines of business, and secondly by capturing this dependence through random effects. The first approach will be presented in chapters 2 and 3. In chapter 2, we use partially nested Archimedean copulas to capture dependence within and between two lines of business, through calendar year effects. In chapter 3, we use fully nested Archimedean copulas, to accommodate dependence between more than two lines of business. A copula-based risk aggregation model is also proposed to accommodate dependence. The inference for the dependence structure is performed with a rank-based methodology to bring more robustness to the estimation. In chapter 4, we introduce the Sarmanov family of bivariate distributions to a loss reserving context, and show that its flexibility proves to be very useful for modeling dependence between loss triangles. This dependence is captured by random effects, through calendar years, accident years or development periods. Closed-form expressions are given, and a real life illustration is shown again. In chapter 5, we use the Sarmanov family of bivariate distributions in a dynamic framework, where the random effects are considered evolutionary and evolve over time, to update the information and allow more weight to more recent claims. Hence, we propose an innovative way to jointly model the dependence between risks and over time with an illustration in a ratemaking context. Finally, a brief conclusion recalls the main contributions of this thesis and provides insights into future research and possible extensions to the proposed works

Estimation of Value at Risk and Conditional Value at Risk

【学位授与の要件】中央大学学位規則第4条第1項【論文審査委員主査】石村　直之(中央大学商学部教授)【論文審査委員副査】斎藤　正武(中央大学商学部教授),髙岡　浩一郎(中央大学商学部教授)博士(商学)中央⼤

Joint Composite Estimating Functions in Spatial and Spatio-Temporal Models.

Spatial or spatio-temporal data are frequently encountered in many scientific disciplines. One major challenge in modeling these processes is the high dimensionality of such data; that is, the number of observations is usually enormous. The first part of the dissertation proposes an efficient approach to analyzing spatio-temporal processes. We proposed a new method called joint composite estimating function (JCEF). It reduces the likelihood dimension by utilizing lower-dimensional marginal likelihoods in estimation and inference. This method allows us to account for high-order spatio-temporal dependences through Hansen's generalized method of moments. Simulation experiments show favorable improvement in estimation efficiency over the conventional composite likelihood methods when applied to estimating the spatio-temporal covariance functions. Large sample properties of the proposed JCEF estimator are derived under more realistic settings than what is available in the current literature. The second part of the thesis presents a much needed review of existing covariance estimation methods parallelly developed for massive spatial data sets. To thoroughly investigate their relative performances in spatio-temporal data analysis, we conduct extensive simulation experiments to compare estimation bias and efficiency among the most popularly used methods, including conventional pairwise composite likelihood, JCEF, Stein's conditional pseudo-likelihood, tapering, weighted least squares, and maximum likelihood, which is served as the golden standard. The third part of the thesis develops a new modeling and estimating framework for high-dimensional spatial-clustered data, termed as GeoCopula. Marginal distributions are assumed to be the generalized linear models, so that the new method can handle both discrete and continuous outcomes. The within-cluster and between-cluster spatial correlations are modeled by a multivariate Gaussian copula, which results in a fully parametric model for dependent data. This class of models generates population-level regression parameter estimates similar to GEE, while explicitly models the dependence structures separately from the mean model. Estimation and inference are achieved by applying the JCEF method. Through simulation experiments we show efficiency improvement over conventional pairwise composite likelihoods. The proposed model and method are illustrated by an analysis of the Gambia malaria data set.Ph.D.BiostatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/89679/1/yunbai_1.pd

Extensions to Gaussian copula models

A copula is the representation of a multivariate distribution. Copulas are used to model multivariate data in many fields. Recent developments include copula models for spatial data and for discrete marginals. We will present a new methodological approach for modeling discrete spatial processes and for predicting the process at unobserved locations. We employ Bayesian methodology for both estimation and prediction. Comparisons between the new method and Generalized Additive Model (GAM) are done to test the performance of the prediction. Although there exists a large variety of copula functions, only a few are practically manageable and in certain problems one would like to choose the Gaussian copula to model the dependence. Furthermore, most copulas are exchangeable, thus implying symmetric dependence. However, none of them is flexible enough to catch the tailed (upper tailed or lower tailed) distribution as well as elliptical distributions. An elliptical copula is the copula corresponding to an elliptical distribution by Sklar's theorem, so it can be used appropriately and effectively only to fit elliptical distributions. While in reality, data may be better described by a "fat-tailed" or "tailed" copula than by an elliptical copula. This dissertation proposes a novel pseudo-copula (the modified Gaussian pseudo-copula) based on the Gaussian copula to model dependencies in multivariate data. Our modified Gaussian pseudo-copula differs from the standard Gaussian copula in that it can model the tail dependence. The modified Gaussian pseudo-copula captures properties from both elliptical copulas and Archimedean copulas. The modified Gaussian pseudo-copula and its properties are described. We focus on issues related to the dependence of extreme values. We give our pseudo-copula characteristics in the bivariate case, which can be extended to multivariate cases easily. The proposed pseudo-copula is assessed by estimating the measure of association from two real data sets, one from finance and one from insurance. A simulation study is done to test the goodness-of-fit of this new model