Responding to the changes in the insurance environment of the past decade, insurance regulators globally have been revamping the valuation and capital regulations. This thesis is concerned with the design and analysis of statistical inference procedures that are used to implement these new and upcoming insurance regulations, and their analysis in a more general setting toward lending further insights into their performance in practical situations. The quantitative measure of risk that is used in these new and upcoming regulations is the risk measure known as the Tail Value-at-Risk (T-VaR). In implementing these regulations, insurance companies often have to estimate the T-VaR of product portfolios from the output of a simulation of its cash flows. The distributions for the underlying economic variables are either estimated or prescribed by regulations. In this situation the computational complexity of estimating the T-VaR arises due to the complexity in determining the portfolio cash flows for a given realization of economic variables. A technique that has proved promising in such settings is that of importance sampling. While the asymptotic behavior of the natural non-parametric estimator of T-VaR under importance sampling has been conjectured, the literature has lacked an honest result. The main goal of the first part of the thesis is to give a precise weak convergence result describing the asymptotic behavior of this estimator under importance sampling. Our method also establishes such a result for the natural non-parametric estimator for the Value-at-Risk, another popular risk measure, under weaker assumptions than those used in the literature. We also report on a simulation study conducted to examine the quality of these asymptotic approximations in small samples. The Haezendonck-Goovaerts class of risk measures corresponds to a premium principle that is a multiplicative analog of the zero utility principle, and is thus of significant academic interest. From a practical point of view our interest in this class of risk measures arose primarily from the fact that the T-VaR is, in a sense, a minimal member of the class. Hence, a study of the natural non-parametric estimator for these risk measures will lend further insights into the statistical inference for the T-VaR. Analysis of the asymptotic behavior of the generalized estimator has proved elusive, largely due to the fact that, unlike the T-VaR, it lacks a closed form expression. Our main goal in the second part of this thesis is to study the asymptotic behavior of this estimator. In order to conduct a simulation study, we needed an efficient algorithm to compute the Haezendonck-Goovaerts risk measure with precise error bounds. The lack of such an algorithm has clearly been noticed in the literature, and has impeded the quality of simulation results. In this part we also design and analyze an algorithm for computing these risk measures. In the process of doing we also derive some fundamental bounds on the solutions to the optimization problem underlying these risk measures. We also have implemented our algorithm on the R software environment, and included its source code in the Appendix.

Cette thèse se propose de traiter de trois problèmes de gestion des risques financiers en utilisant différentes approches asymptotiques. La première partie présente un algorithme Monte Carlo d’échantillonnage d’importance pour la valorisation d’options asiatiques dans des modèles exponentiels de Lévy. La mesure optimale d’échantillonnage d’importance est obtenue grâce à la théorie des grandes déviations. La seconde partie présente l’étude du comportement asymptotique de la somme de n variables aléatoires positives et dépendantes dont la distribution est un mélange log-normal ainsi que des applications en gestion des risque de portefeuille d’actifs. Enfin, la dernière partie, présente une application de la notion de variations régulières pour l’analyse du comportement des queues de distribution d’un vecteur aléatoire dont les composantes suivent des distributions à queues épaisses et dont la structure de dépendance est modélisée par une copule Gaussienne. Ces résultats sont ensuite appliqués au comportement asymptotique d’un portefeuille d’options dans le modèle de Black-Scholes / This thesis focuses on three problems from the area of financial risk management, using various asymptotic approaches. The first part presents an importance sampling algorithm for Monte Carlo pricing of exotic options in exponential Lévy models. The optimal importance sampling measure is computed using techniques from the theory of large deviations. The second part uses the Laplace method to study the tail behavior of the sum of n dependent positive random variables, following a log-normal mixture distribution, with applications to portfolio risk management. Finally, the last part employs the notion of multivariate regular variation to analyze the tail behavior of a random vector with heavy-tailed components, whose dependence structure is modeled by a Gaussian copula. As application, we consider the tail behavior of a portfolio of options in the Black-Scholes model

In the computer graphics production environment, artists often must tweak specific lighting and material parameters to match a mind's eye vision of the appearance of a 3D scene. However, the interaction between a material and a lighting environment is often too complex to cognitively predict without visualization. Therefore, artists operate in a design cycle, where they tweak the parameters, wait for a visualization, and repeat, seeking to obtain a desired look. We propose the use of appearance-driven material design. Here, artists directly design the appearance of reflected light for a specific view, surface point, and time. In this thesis, we discuss several methods for appearance-driven design with homogeneous materials, spatially-varying materials, and appearance-matching materials, where each uses a unique modeling and optimization paradigm. Moreover, we present a novel treatment of the illumination integral using sampling theory that can utilize the computational power of the graphics processing unit (GPU) to provide real-time visualization of the appearance of various materials illuminated by complex environment lighting. As a system, the modeling, optimization and rendering steps all operate on arbitrary geometry and in detailed lighting environments, while still providing instant feedback to the designer. Thus, our approach allows materials to play an active role in the process of set design and story-telling, a capability that was, until now, difficult to achieve due to the unavailability of interactive tools appropriate for artists.

The majority of aircraft structures are designed to be damage-tolerant such that safe operation can continue in the presence of minor damage. It is necessary to schedule inspections so that minor damage can be found and repaired. It is generally not possible to perform structural inspections prior to every flight. The scheduling is traditionally accomplished through a deterministic set of methods referred to as Damage Tolerance Analysis (DTA). DTA has proven to produce safe aircraft but does not provide estimates of the probability of failure of future flights or the probability of repair of future inspections. Without these estimates maintenance costs cannot be accurately predicted. Also, estimation of failure probabilities is now a regulatory requirement for some aircraft. The set of methods concerned with the probabilistic formulation of this problem are collectively referred to as Probabilistic Damage Tolerance Analysis (PDTA). The goal of PDTA is to control the failure probability while holding maintenance costs to a reasonable level. This work focuses specifically on PDTA for fatigue cracking of metallic aircraft structures. The growth of a crack (or cracks) must be modeled using all available data and engineering knowledge. The length of a crack can be assessed only indirectly through evidence such as non-destructive inspection results, failures or lack of failures, and the observed severity of usage of the structure. The current set of industry PDTA tools are lacking in several ways: they may in some cases yield poor estimates of failure probabilities, they cannot realistically represent the variety of possible failure and maintenance scenarios, and they do not allow for model updates which incorporate observed evidence. A PDTA modeling methodology must be flexible enough to estimate accurately the failure and repair probabilities under a variety of maintenance scenarios, and be capable of incorporating observed evidence as it becomes available. This dissertation describes and develops new PDTA methodologies that directly address the deficiencies of the currently used tools. The new methods are implemented as a free, publicly licensed and open source R software package that can be downloaded from the Comprehensive R Archive Network. The tools consist of two main components. First, an explicit (and expensive) Monte Carlo approach is presented which simulates the life of an aircraft structural component flight-by-flight. This straightforward MC routine can be used to provide defensible estimates of the failure probabilities for future flights and repair probabilities for future inspections under a variety of failure and maintenance scenarios. This routine is intended to provide baseline estimates against which to compare the results of other, more efficient approaches. Second, an original approach is described which models the fatigue process and future scheduled inspections as a hidden Markov model. This model is solved using a particle-based approximation and the sequential importance sampling algorithm, which provides an efficient solution to the PDTA problem. Sequential importance sampling is an extension of importance sampling to a Markov process, allowing for efficient Bayesian updating of model parameters. This model updating capability, the benefit of which is demonstrated, is lacking in other PDTA approaches. The results of this approach are shown to agree with the results of the explicit Monte Carlo routine for a number of PDTA problems. Extensions to the typical PDTA problem, which cannot be solved using currently available tools, are presented and solved in this work. These extensions include incorporating observed evidence (such as non-destructive inspection results), more realistic treatment of possible future repairs, and the modeling of failure involving more than one crack (the so-called continuing damage problem). The described hidden Markov model / sequential importance sampling approach to PDTA has the potential to improve aerospace structural safety and reduce maintenance costs by providing a more accurate assessment of the risk of failure and the likelihood of repairs throughout the life of an aircraft. / Statistics

The increasing size and complexity of datasets have accelerated the development of machine learning models and exposed the need for more scalable solutions. This thesis explores challenges associated with large-scale machine learning under data privacy constraints. With the growth of machine learning models, traditional privacy methods such as data anonymization are becoming insufficient. Thus, we delve into alternative approaches, such as differential privacy. Our research addresses the following core areas in the context of scalable privacy-preserving machine learning: First, we examine the implications of data dimensionality on privacy for the application of medical image analysis. We extend the classification algorithm Private Aggregation of Teacher Ensembles (PATE) to deal with high-dimensional labels, and demonstrate that dimensionality reduction can be used to improve privacy. Second, we consider the impact of hyperparameter selection on privacy. Here, we propose a novel adaptive technique for hyperparameter selection in differentially gradient-based optimization. Third, we investigate sampling-based solutions to scale differentially private machine learning to dataset with a large number of records. We study the privacy-enhancing properties of importance sampling, highlighting that it can outperform uniform sub-sampling not only in terms of sample efficiency but also in terms of privacy. The three techniques developed in this thesis improve the scalability of machine learning while ensuring robust privacy protection, and aim to offer solutions for the effective and safe application of machine learning in large datasets. / Den ständigt ökande storleken och komplexiteten hos datamängder har accelererat utvecklingen av maskininlärningsmodeller och gjort behovet av mer skalbara lösningar alltmer uppenbart. Den här avhandlingen utforskar tre utmaningar förknippade med storskalig maskininlärning under dataskyddskrav. För stora och komplexa maskininlärningsmodeller blir traditionella metoder för integritet, såsom datananonymisering, otillräckliga. Vi undersöker därför alternativa tillvägagångssätt, såsom differentiell integritet. Vår forskning behandlar följande utmaningar inom skalbar och integitetsmedveten maskininlärning: För det första undersöker vi hur hög data-dimensionalitet påverkar integriteten för medicinsk bildanalys. Vi utvidgar klassificeringsalgoritmen Private Aggregation of Teacher Ensembles (PATE) för att hantera högdimensionella etiketter och visar att dimensionsreducering kan användas för att förbättra integriteten. För det andra studerar vi hur valet av hyperparametrar påverkar integriteten. Här föreslår vi en ny adaptiv teknik för val av hyperparametrar i gradient-baserad optimering med garantier på differentiell integritet. För det tredje granskar vi urvalsbaserade lösningar för att skala differentiellt privat maskininlärning till stora datamängder. Vi studerar de integritetsförstärkande egenskaperna hos importance sampling och visar att det kan överträffa ett likformigt urval av sampel, inte bara när det gäller effektivitet utan även för integritet. De tre teknikerna som utvecklats i denna avhandling förbättrar skalbarheten för integritetsskyddad maskininlärning och syftar till att erbjuda lösningar för effektiv och säker tillämpning av maskininlärning på stora datamängder. / <p>QC 20231101</p>