退休金是退休人未來生活的依靠，確保在退休後能得到適足的退休給付，政府在退休金上實施保證收益制度，此制度為最低保證利率與投資報酬率連結。本文探討退休金給付標準為確定提撥制，當退休金的投資報酬率是根據其連結之股價指數的表現來計算時，股價指數報酬率的模型假設為馬可夫調控跳躍過程模型，考慮市場狀態與布朗運動項、跳躍項的跳躍頻率相關，即為Elliot et al. (2007) 的模型特例。使用1999年至2012年的道瓊工業指數與S&P 500指數的股價指數對數報酬率作為研究資料，採用EM演算法估計參數及SEM演算法估計參數共變異數矩陣。透過概似比檢定說明馬可夫調控跳躍過程模型比狀態轉換模型、跳躍風險下狀態轉換模型更適合描述股價指數報酬率變動情形，也驗證馬可夫調控跳躍過程模型具有描述報酬率不對稱、高狹峰及波動叢聚的特性。最後，假設最低保證利率為固定下，利用Esscher轉換法計算不同模型下型I保證之確定提撥制退休金的評價公式，從公式中可看出受雇人提領的退休金價值可分為政府補助與個人帳戶擁有之退休金兩部分。以執行敏感度分析探討估計參數對於馬可夫調控跳躍過程模型評價公式的影響，而型II保證之確定提撥制退休金的價值則以蒙地卡羅法模擬並探討其敏感度分析結果。 / Pension plan make people a guarantee life in their retirement. In order to ensure the appropriate amount of pension plan, government guarantees associated with pension plan which ties minimum rate of return guarantees and underlying asset rate of return. In this paper, we discussed the pension plan with defined contribution (DC). When the return of asset is based on the stock indices, the return model was set on the assumption that markov-modulated jump diffusion model (MMJDM) could the Brownian motion term and jump rate be both related to market states. This model is the specific case of Elliot et al. (2007) offering. The sample observations is Dow-Jones industrial average and S&P 500 index from 1999 to 2012 by logarithm return of the stock indices. We estimated the parameters by the Expectation-Maximization (EM) algorithm and calculated the covariance matrix of the estimates by supplemented EM (SEM) algorithm. Through the likelihood ratio test (LRT), the data fitted the MMJDM better than other models. The empirical evidence indicated that the MMJDM could describe the asset return for asymmetric, leptokurtic, volatility clustering particularly. Finally, we derived different model's valuation formula for DC pension plan with type-I guarantee by Esscher transformation under rate of return guarantees is constant. From the formula, the value of the pension plan could divide into two segment: government supplement and employees deposit made pension to their personal bank account. And then, we done sensitivity analysis through the MMJDM valuation formula. We used Monte Carlo simulations to evaluate the valuation of DC pension plan with type-II guarantee and discussed it from sensitivity analysis.

Nous y introduisons une nouvelle classe de distributions bivariées de type Marshall-Olkin, la distribution Erlang bivariée. La transformée de Laplace, les moments et les densités conditionnelles y sont obtenus. Les applications potentielles en assurance-vie et en finance sont prises en considération. Les estimateurs du maximum de vraisemblance des paramètres sont calculés par l'algorithme Espérance-Maximisation. Ensuite, notre projet de recherche est consacré à l'étude des processus de risque multivariés, qui peuvent être utiles dans l'étude des problèmes de la ruine des compagnies d'assurance avec des classes dépendantes. Nous appliquons les résultats de la théorie des processus de Markov déterministes par morceaux afin d'obtenir les martingales exponentielles, nécessaires pour établir des bornes supérieures calculables pour la probabilité de ruine, dont les expressions sont intraitables. / In this contribution, we introduce a new class of bivariate distributions of Marshall-Olkin type, called bivariate Erlang distributions. The Laplace transform, product moments and conditional densities are derived. Potential applications of bivariate Erlang distributions in life insurance and finance are considered. Further, our research project is devoted to the study of multivariate risk processes, which may be useful in analyzing ruin problems for insurance companies with a portfolio of dependent classes of business. We apply results from the theory of piecewise deterministic Markov processes in order to derive exponential martingales needed to establish computable upper bounds of the ruin probabilities, as their exact expressions are intractable.

This thesis is related to the Bit Error Rate (BER) estimation for any digital communication system. In many designs of communication systems, the BER is a Key Performance Indicator (KPI). The popular Monte-Carlo (MC) simulation technique is well suited to any system but at the expense of long time simulations when dealing with very low error rates. In this thesis, we propose to estimate the BER by using the Probability Density Function (PDF) estimation of the soft observations of the received bits. First, we have studied a non-parametric PDF estimation technique named the Kernel method. Simulation results in the context of several digital communication systems are proposed. Compared with the conventional MC method, the proposed Kernel-based estimator provides good precision even for high SNR with very limited number of data samples. Second, the Gaussian Mixture Model (GMM), which is a semi-parametric PDF estimation technique, is used to estimate the BER. Compared with the Kernel-based estimator, the GMM method provides better performance in the sense of minimum variance of the estimator. Finally, we have investigated the blind estimation of the BER, which is the estimation when the sent data are unknown. We denote this case as unsupervised BER estimation. The Stochastic Expectation-Maximization (SEM) algorithm combined with the Kernel or GMM PDF estimation methods has been used to solve this issue. By analyzing the simulation results, we show that the obtained BER estimate can be very close to the real values. This is quite promising since it could enable real-time BER estimation on the receiver side without decreasing the user bit rate with pilot symbols for example.

獨立因子模型廣泛的應用在信用風險領域，此模型可用來估計經濟資本與投資組合的損失率分配。然而獨立因子模型假設因子獨立地服從同分配，因而可能會得到估計不精確的違約機率與資產相關係數。因此我們在本論文中提出序列相關因子模型來改進獨立因子模型的缺失，同時可以捕捉違約率的動態行為與授信戶間相關性。我們也分別從古典與貝氏的角度下估計序列相關因子模型。首先，我們在序列相關因子模型下利用貝氏的方法應用馬可夫鍊蒙地卡羅技巧估計違約機率與資產相關係數，使用標準普爾違約資料進行外樣本資料預測，能夠證明序列相關因子模型是比獨立因子模型合理。第二，蒙地卡羅期望最大法與蒙地卡羅最大概似法這兩種估計方法也使用在本篇論文。從模擬結果發現，若違約資料具有較大的序列相關與資產相關特性，蒙地卡羅最大概似法能夠配適的比蒙地卡羅期望最大法好。 / The independent factor model has been widely used in the credit risk field, and has been applied in estimating the economic capital allocations and loss rate distribution on a credit portfolio. However, this model assumes independent and identically distributed common factor which may produce inaccurate estimates of default probabilities and asset correlation. In this thesis, we address a serially dependent factor model (SDFM) to improve this phenomenon. This model can capture both dynamic behavior of default risk and dependence among individual obligors. We also address the estimation of the SDFM from both frequentist and Bayesian point of view. Firstly, we consider the Bayesian approach by applying Markov chain Monte Carlo (MCMC) techniques in estimating default probability and asset correlation under SDFM. The out-of-sample forecasting for S&P default data provide strong evidence to support that the SDFM is more reliable than the independent factor model. Secondly, we use two frequentist estimation methods to estimate the default probability and asset correlation under SDFM. One is Monte Carlo Expectation Maximization (MCEM) estimation method along with a Gibbs sampler and an acceptance method and the other is Monte Carlo maximum likelihood (MCML) estimation method with importance sampling techniques.

Nous proposons deux nouvelles approches pour les systèmes de recommandation et les réseaux. Dans la première partie, nous donnons d’abord un aperçu sur les systèmes de recommandation avant de nous concentrer sur les approches de rang faible pour la complétion de matrice. En nous appuyant sur une approche probabiliste, nous proposons de nouvelles fonctions de pénalité sur les valeurs singulières de la matrice de rang faible. En exploitant une représentation de modèle de mélange de cette pénalité, nous montrons qu’un ensemble de variables latentes convenablement choisi permet de développer un algorithme espérance-maximisation afin d’obtenir un maximum a posteriori de la matrice de rang faible complétée. L’algorithme résultant est un algorithme à seuillage doux itératif qui adapte de manière itérative les coefficients de réduction associés aux valeurs singulières. L’algorithme est simple à mettre en œuvre et peut s’adapter à de grandes matrices. Nous fournissons des comparaisons numériques entre notre approche et de récentes alternatives montrant l’intérêt de l’approche proposée pour la complétion de matrice à rang faible. Dans la deuxième partie, nous présentons d’abord quelques prérequis sur l’approche bayésienne non paramétrique et en particulier sur les mesures complètement aléatoires et leur extension multivariée, les mesures complètement aléatoires composées. Nous proposons ensuite un nouveau modèle statistique pour les réseaux creux qui se structurent en communautés avec chevauchement. Le modèle est basé sur la représentation du graphe comme un processus ponctuel échangeable, et généralise naturellement des modèles probabilistes existants à structure en blocs avec chevauchement au régime creux. Notre construction s’appuie sur des vecteurs de mesures complètement aléatoires, et possède des paramètres interprétables, chaque nœud étant associé un vecteur représentant son niveau d’affiliation à certaines communautés latentes. Nous développons des méthodes pour simuler cette classe de graphes aléatoires, ainsi que pour effectuer l’inférence a posteriori. Nous montrons que l’approche proposée peut récupérer une structure interprétable à partir de deux réseaux du monde réel et peut gérer des graphes avec des milliers de nœuds et des dizaines de milliers de connections. / We propose two novel approaches for recommender systems and networks. In the first part, we first give an overview of recommender systems and concentrate on the low-rank approaches for matrix completion. Building on a probabilistic approach, we propose novel penalty functions on the singular values of the low-rank matrix. By exploiting a mixture model representation of this penalty, we show that a suitably chosen set of latent variables enables to derive an expectation-maximization algorithm to obtain a maximum a posteriori estimate of the completed low-rank matrix. The resulting algorithm is an iterative soft-thresholded algorithm which iteratively adapts the shrinkage coefficients associated to the singular values. The algorithm is simple to implement and can scale to large matrices. We provide numerical comparisons between our approach and recent alternatives showing the interest of the proposed approach for low-rank matrix completion. In the second part, we first introduce some background on Bayesian nonparametrics and in particular on completely random measures (CRMs) and their multivariate extension, the compound CRMs. We then propose a novel statistical model for sparse networks with overlapping community structure. The model is based on representing the graph as an exchangeable point process, and naturally generalizes existing probabilistic models with overlapping block-structure to the sparse regime. Our construction builds on vectors of CRMs, and has interpretable parameters, each node being assigned a vector representing its level of affiliation to some latent communities. We develop methods for simulating this class of random graphs, as well as to perform posterior inference. We show that the proposed approach can recover interpretable structure from two real-world networks and can handle graphs with thousands of nodes and tens of thousands of edges.

Introduction: No Fixed-Dose Combination (FDC) formulations currently exist for pediatric tuberculosis (TB) treatment. Earlier work implemented, in the software NONMEM, a rational method for optimizing design and individualization of pediatric anti-TB FDC formulations based on patient body weight, but issues with parameter estimation, dosage strata heterogeneity and representative pharmacokinetics remained. Aim: To further develop the rational model-based methodology aiding the selection of appropriate FDC formulation designs and dosage regimens, in pediatric TB treatment. Materials and Methods: Optimization of the method with respect to the estimation of body weight breakpoints was sought. Heterogeneity of dosage groups with respect to treatment efficiency was sought to be improved. Recently published pediatric pharmacokinetic parameters were implemented and the model translated to MATLAB, where also the performance was evaluated by stochastic estimation and graphical visualization. Results: A logistic function was found better suited as an approximation of breakpoints. None of the estimation methods implemented in NONMEM were more suitable than the originally used FO method. Homogenization of dosage group treatment efficiency could not be solved. MATLAB translation was successful but required stochastic estimations and highlighted high densities of local minima. Representative pharmacokinetics were successfully implemented. Conclusions: NONMEM was found suboptimal for the task due to problems with discontinuities and heterogeneity, but a stepwise method with representative pharmacokinetics were successfully implemented. MATLAB showed more promise in the search for a method also addressing the heterogeneity issue.

En écologie des populations, les distributions spatiales d'espèces sont étudiées afin d'inférer l'existence de processus sous-jacents, tels que les interactions intra- et interspécifiques et les réponses des espèces à l'hétérogénéité de l'environnement. Nous proposons d'analyser les données spatiales multi-spécifiques sous l'angle des assemblages d'espèces, que nous considérons en termes d'abondances absolues et non de diversité des espèces. Les assemblages d'espèces sont une des signatures des interactions spatiales locales des espèces entre elles et avec leur environnement. L'étude des assemblages d'espèces peut permettre de détecter plusieurs types d'équilibres spatialisés et de les associer à l'effet de variables environnementales. Les assemblages d'espèces sont définis ici par classification non spatiale des observations multivariées d'abondances d'espèces. Les méthodes de classification basées sur les modèles de mélange ont été choisies afin d'avoir une mesure de l'incertitude de la classification et de modéliser un assemblage par une loi de probabilité multivariée. Dans ce cadre, nous proposons : 1. une méthode d'analyse exploratoire de données spatiales multivariées d'abondances d'espèces, qui permet de détecter des assemblages d'espèces par classification, de les cartographier et d'analyser leur structure spatiale. Des lois usuelles, telle que la Gaussienne multivariée, sont utilisées pour modéliser les assemblages, 2. un modèle hiérarchique pour les assemblages d'abondances lorsque les lois usuelles ne suffisent pas. Ce modèle peut facilement s'adapter à des données contenant des variables de types différents, qui sont fréquemment rencontrées en écologie, 3. une méthode de classification de données contenant des variables de types différents basée sur des mélanges de lois à structure hiérarchique (définies en 2.). Deux applications en écologie ont guidé et illustré ce travail : l'étude à petite échelle des assemblages de deux espèces de pucerons sur des feuilles de clémentinier et l'étude à large échelle des assemblages d'une plante hôte, le plantain lancéolé, et de son pathogène, l'oïdium, sur les îles Aland en Finlande / In population ecology, species spatial patterns are studied in order to infer the existence of underlying processes, such as interactions within and between species, and species response to environmental heterogeneity. We propose to analyze spatial multi-species data by defining species abundance assemblages. Species assemblages are one of the signatures of the local spatial interactions between species and with their environment. Species assemblages are defined here by a non spatial classification of the multivariate observations of species abundances. Model-based clustering procedures using mixture models were chosen in order to have an estimation of the classification uncertainty and to model an assemblage by a multivariate probability distribution. We propose : 1. An exploratory tool for the study of spatial multivariate observations of species abundances, which defines species assemblages by a model-based clustering procedure, and then maps and analyzes the spatial structure of the assemblages. Common distributions, such as the multivariate Gaussian, are used to model the assemblages. 2. A hierarchical model for abundance assemblages which cannot be modeled with common distributions. This model can be easily adapted to mixed mode data, which are frequent in ecology. 3. A clustering procedure for mixed-mode data based on mixtures of hierarchical models. Two ecological case-studies guided and illustrated this work: the small-scale study of the assemblages of two aphid species on leaves of Citrus trees, and the large-scale study of the assemblages of a host plant, Plantago lanceolata, and its pathogen, the powdery mildew, on the Aland islands in south-west Finland

Indiana University-Purdue University Indianapolis (IUPUI) / Mammographic density is an important risk factor for breast cancer, detecting and screening at an early stage could help save lives. To analyze breast density distribution, a good segmentation algorithm is needed. In this thesis, we compared two popularly used segmentation algorithms, EM-MPM and K-means Clustering. We applied them on twenty cases of synthetic phantom ultrasound tomography (UST), and nine cases of clinical mammogram and UST images. From the synthetic phantom segmentation comparison we found that EM-MPM performs better than K-means Clustering on segmentation accuracy, because the segmentation result fits the ground truth data very well (with superior Tanimoto Coefficient and Parenchyma Percentage). The EM-MPM is able to use a Bayesian prior assumption, which takes advantage of the 3D structure and finds a better localized segmentation. EM-MPM performs significantly better for the highly dense tissue scattered within low density tissue and for volumes with low contrast between high and low density tissues. For the clinical mammogram, image segmentation comparison shows again that EM-MPM outperforms K-means Clustering since it identifies the dense tissue more clearly and accurately than K-means. The superior EM-MPM results shown in this study presents a promising future application to the density proportion and potential cancer risk evaluation.

Indiana University-Purdue University Indianapolis (IUPUI) / Joint models of longitudinal and survival outcomes have been used with increasing frequency in clinical investigations. Correct specification of fixed and random effects, as well as their functional forms is essential for practical data analysis. However, no existing methods have been developed to meet this need in a joint model setting. In this dissertation, I describe a penalized likelihood-based method with adaptive least absolute shrinkage and selection operator (ALASSO) penalty functions for model selection. By reparameterizing variance components through a Cholesky decomposition, I introduce a penalty function of group shrinkage; the penalized likelihood is approximated by Gaussian quadrature and optimized by an EM algorithm. The functional forms of the independent effects are determined through a procedure for structural discovery. Specifically, I first construct the model by penalized cubic B-spline and then decompose the B-spline to linear and nonlinear elements by spectral decomposition. The decomposition represents the model in a mixed-effects model format, and I then use the mixed-effects variable selection method to perform structural discovery. Simulation studies show excellent performance. A clinical application is described to illustrate the use of the proposed methods, and the analytical results demonstrate the usefulness of the methods.

Multiple-outcome longitudinal data are abundant in clinical investigations. For example, infections with different pathogenic organisms are often tested concurrently, and assessments are usually taken repeatedly over time. It is therefore natural to consider a multivariate modeling approach to accommodate the underlying interrelationship among the multiple longitudinally measured outcomes. This dissertation proposes a multivariate semiparametric modeling framework for such data. Relevant estimation and inference procedures as well as model selection tools are discussed within this modeling framework. The first part of this research focuses on the analytical issues concerning binary data. The second part extends the binary model to a more general situation for data from the exponential family of distributions. The proposed model accounts for the correlations across the outcomes as well as the temporal dependency among the repeated measures of each outcome within an individual. An important feature of the proposed model is the addition of a bivariate smooth function for the depiction of concurrent nonlinear and possibly interacting influences of two independent variables on each outcome. For model implementation, a general approach for parameter estimation is developed by using the maximum penalized likelihood method. For statistical inference, a likelihood-based resampling procedure is proposed to compare the bivariate nonlinear effect surfaces across the outcomes. The final part of the dissertation presents a variable selection tool to facilitate model development in practical data analysis. Using the adaptive least absolute shrinkage and selection operator (LASSO) penalty, the variable selection tool simultaneously identifies important fixed effects and random effects, determines the correlation structure of the outcomes, and selects the interaction effects in the bivariate smooth functions. Model selection and estimation are performed through a two-stage procedure based on an expectation-maximization (EM) algorithm. Simulation studies are conducted to evaluate the performance of the proposed methods. The utility of the methods is demonstrated through several clinical applications.