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In the master thesis the issue of bayesian approach to stochastic reserving is solved. Reserving problem is very discussed in insurance industry. The text introduces the basic actuarial notation and terminology and explains the bayesian inference in statistics and estimation. The main part of the thesis is framed by the description of the particular bayesian models. It is focused on the derivation of estimators for the reserves and ultimate claims. The aim of the thesis is to show the practical uses of the models and the relations between them. For this purpose the methods are applied on a real data set. Obtained results are summarized in tables and the comparison of the methods is provided. Finally the impact of a prior distribution on the resulting reserves is showed. Powered by TCPDF (www.tcpdf.org)

Estimation of location and scale parameters from a random sample of size n is of paramount importance in Statistics. An estimator is called fully efficient if it attains the Cramer-Rao minimum variance bound besides being unbiased. The method that yields such estimators, at any rate for large n, is the method of modified maximum likelihood estimation. Apparently, such estimators cannot be made more efficient by using sample based classical methods. That makes room for Bayesian method of estimation which engages prior distributions and likelihood functions. A formal combination of the prior knowledge and the sample information is called posterior distribution. The posterior distribution is maximized with respect to the unknown parameter(s). That gives HPD (highest probability density) estimator(s). Locating the maximum of the posterior distribution is, however, enormously difficult (computationally and analytically) in most situations. To alleviate these difficulties, we use modified likelihood function in the posterior distribution instead of the likelihood function. We derived the HPD estimators of location and scale parameters of distributions in the family of Generalized Logistic. We have extended the work to experimental design, one way ANOVA. We have obtained the HPD estimators of the block effects and the scale parameter (in the distribution of errors) / they have beautiful algebraic forms. We have shown that they are highly efficient. We have given real life examples to illustrate the usefulness of our results. Thus, the enormous computational and analytical difficulties with the traditional Bayesian method of estimation are circumvented at any rate in the context of experimental design.

Processos estocásticos complexos são muitas vezes utilizados em modelagem, com o intuito de capturar uma maior proporção das principais características dos sistemas biológicos. A descrição do comportamento desses sistemas tem sido realizada por muitos amostradores baseados na distribuição a posteriori de Monte Carlo. Modelos probabilísticos que descrevem esses processos podem levar a funções de verossimilhança computacionalmente intratáveis, impossibilitando a utilização de métodos de inferência estatística clássicos e os baseados em amostragem por meio de MCMC. A Computação Bayesiana Aproximada (ABC) é considerada um novo método de inferência com base em estatísticas de resumo, ou seja, valores calculados a partir do conjunto de dados (média, moda, variância, etc.). Essa metodologia combina muitas das vantagens da eficiência computacional de processos baseados em estatísticas de resumo com inferência estatística bayesiana uma vez que, funciona bem para pequenas amostras e possibilita incorporar informações passadas em um parâmetro e formar uma priori para análise futura. Nesse trabalho foi realizada uma comparação entre os métodos de estimação, clássico, bayesiano e ABC, para estudos de simulação de modelos simples e para análise de dados de dinâmica populacional. Foram implementadas no software R as distâncias modular e do máximo como alternativas de função distância a serem utilizadas no ABC, além do algoritmo ABC de rejeição para equações diferenciais estocásticas. Foi proposto sua utilização para a resolução de problemas envolvendo modelos de interação populacional. Os estudos de simulação mostraram melhores resultados quando utilizadas as distâncias euclidianas e do máximo juntamente com distribuições a priori informativas. Para os sistemas dinâmicos, a estimação por meio do ABC apresentou resultados mais próximos dos verdadeiros bem como menores discrepâncias, podendo assim ser utilizado como um método alternativo de estimação. / Complex stochastic processes are often used in modeling in order to capture a greater proportion of the main features of natural systems. The description of the behavior of these systems has been made by many Monte Carlo based samplers of the posterior distribution. Probabilistic models describing these processes can lead to computationally intractable likelihood functions, precluding the use of classical statistical inference methods and those based on sampling by MCMC. The Approxi- mate Bayesian Computation (ABC) is considered a new method for inference based on summary statistics, that is, calculated values from the data set (mean, mode, variance, etc.). This methodology combines many of the advantages of computatio- nal efficiency of processes based on summary statistics with the Bayesian statistical inference since, it works well for small samples and it makes possible to incorporate past information in a parameter and form a prior distribution for future analysis. In this work a comparison between, classical, Bayesian and ABC, estimation methods was made for simulation studies considering simple models and for data analysis of population dynamics. It was implemented in the R software the modular and maxi- mum as alternative distances function to be used in the ABC, besides the rejection ABC algorithm for stochastic differential equations. It was proposed to use it to solve problems involving models of population interaction. The simulation studies showed better results when using the Euclidean and maximum distances together with informative prior distributions. For the dynamic systems, the ABC estimation presented results closer to the real ones as well as smaller discrepancies and could thus be used as an alternative estimation method.

Processos estocásticos complexos são muitas vezes utilizados em modelagem, com o intuito de capturar uma maior proporção das principais características dos sistemas biológicos. A descrição do comportamento desses sistemas tem sido realizada por muitos amostradores baseados na distribuição a posteriori de Monte Carlo. Modelos probabilísticos que descrevem esses processos podem levar a funções de verossimilhança computacionalmente intratáveis, impossibilitando a utilização de métodos de inferência estatística clássicos e os baseados em amostragem por meio de MCMC. A Computação Bayesiana Aproximada (ABC) é considerada um novo método de inferência com base em estatísticas de resumo, ou seja, valores calculados a partir do conjunto de dados (média, moda, variância, etc.). Essa metodologia combina muitas das vantagens da eficiência computacional de processos baseados em estatísticas de resumo com inferência estatística bayesiana uma vez que, funciona bem para pequenas amostras e possibilita incorporar informações passadas em um parâmetro e formar uma priori para análise futura. Nesse trabalho foi realizada uma comparação entre os métodos de estimação, clássico, bayesiano e ABC, para estudos de simulação de modelos simples e para análise de dados de dinâmica populacional. Foram implementadas no software R as distâncias modular e do máximo como alternativas de função distância a serem utilizadas no ABC, além do algoritmo ABC de rejeição para equações diferenciais estocásticas. Foi proposto sua utilização para a resolução de problemas envolvendo modelos de interação populacional. Os estudos de simulação mostraram melhores resultados quando utilizadas as distâncias euclidianas e do máximo juntamente com distribuições a priori informativas. Para os sistemas dinâmicos, a estimação por meio do ABC apresentou resultados mais próximos dos verdadeiros bem como menores discrepâncias, podendo assim ser utilizado como um método alternativo de estimação. / Complex stochastic processes are often used in modeling in order to capture a greater proportion of the main features of natural systems. The description of the behavior of these systems has been made by many Monte Carlo based samplers of the posterior distribution. Probabilistic models describing these processes can lead to computationally intractable likelihood functions, precluding the use of classical statistical inference methods and those based on sampling by MCMC. The Approxi- mate Bayesian Computation (ABC) is considered a new method for inference based on summary statistics, that is, calculated values from the data set (mean, mode, variance, etc.). This methodology combines many of the advantages of computatio- nal efficiency of processes based on summary statistics with the Bayesian statistical inference since, it works well for small samples and it makes possible to incorporate past information in a parameter and form a prior distribution for future analysis. In this work a comparison between, classical, Bayesian and ABC, estimation methods was made for simulation studies considering simple models and for data analysis of population dynamics. It was implemented in the R software the modular and maxi- mum as alternative distances function to be used in the ABC, besides the rejection ABC algorithm for stochastic differential equations. It was proposed to use it to solve problems involving models of population interaction. The simulation studies showed better results when using the Euclidean and maximum distances together with informative prior distributions. For the dynamic systems, the ABC estimation presented results closer to the real ones as well as smaller discrepancies and could thus be used as an alternative estimation method.

In the paper at hand we apply it to Bayesian statistics to obtain "Fuzzy Bayesian Inference". In the subsequent sections we will discuss a fuzzy valued likelihood function, Bayes' theorem for both fuzzy data and fuzzy priors, a fuzzy Bayes' estimator, fuzzy predictive densities and distributions, and fuzzy H.P.D .-Regions. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

本論文主要討論貝氏漸近的比較，推導出參數的聯合後驗分配與利用圖形來診斷馬可夫鏈蒙地卡羅的收斂。Johnson (1970)利用泰勒展開式得到個別後驗分配的展開式，此展開式是根據概似函數與先驗分配。 Weng (2010b) 和 Weng and Hsu (2011) 利用 Stein’s 等式且由概似函數與先驗分配估計後驗動差；將這些後驗動差代入Edgeworth 展開式得到近似後驗分配，此近似分配的誤差可精確到大O的負3/2次方與Johnson’s 相同。另外Weng and Hsu (2011)發現Weng (2010b) 和Johnson (1970)的近似展開式各別項誤差到大O的負1次方不一致，由模擬結果得到Weng’s 在此項表現比Johnson’s 好。另外由Weng (2010b)得到一維參數 的Edgeworth 近似後驗分配延伸到二維參數的聯合後驗分配；並應用二維參數的聯合後驗分配於多階段資料。本論文我們提出利用圖形來診斷馬可夫鏈蒙地卡羅收斂的方法，並且應用一般化線性模型與混合常態模型做為模擬。 關鍵字: Edgeworth 展開式；馬可夫鏈蒙地卡羅；個別後驗分配；Stein’s 等式 / Johnson (1970) obtained expansions for marginal posterior distributions through Taylor expansions. The expansion in Johnson (1970) is expressed in terms of the likelihood and the prior. Weng (2010b) and Weng and Hsu (2011) showed that by using Stein's identity we can approximate the posterior moments in terms of the likelihood and the prior; then substituting these approximations into an Edgeworth series one can obtain an expansion which is correct to O(t{-3/2}), similar to Johnson's. Weng and Hsu (2011) found that the O(t{-1}) terms in Weng (2010b) and Johnson (1970) do not agree and further compared these two expansions by simulation study. The simulations confirmed this finding and revealed that our O(t{-1}) term gives better performance than Johnson's. In addition to the comparison of Bayesian asymptotics, we try to extend Weng (2010a)'s Edgeworth series for the distribution of a single parameter to the joint distribution of all parameters. Since the calculation is quite complicated, we only derive expansions for the two-parameter case and apply it to the experiment of multi-stage data. Markov Chain Monte Carlo (MCMC) is a popular method for making Bayesian inference. However, convergence of the chain is always an issue. Most of convergence diagnosis in the literature is based solely on the simulation output. In this dissertation, we proposed a graphical method for convergence diagnosis of the MCMC sequence. We used some generalized linear models and mixture normal models for simulation study. In summary, the goals of this dissertation are threefold: to compare some results in Bayesian asymptotics, to study the expansion for the joint distribution of the parameters and its applications, and to propose a method for convergence diagnosis of the MCMC sequence. Key words: Edgeworth expansion; Markov Chain Monte Carlo; marginal posterior distribution; Stein's identity.

L’objectif de cette thèse est de proposer un Alphabet Structural (AS) permettant une caractérisation fine et précise des structures tridimensionnelles (3D) des protéines, à l’aide des chaînes de Markov cachées (HMM) qui permettent de prendre en compte la logique issue de l’enchaînement des fragments structuraux en intégrant l’augmentation des conformations 3D des structures protéiques désormais disponibles dans la banque de données de la Protein Data Bank (PDB). Nous proposons dans cette thèse un nouvel alphabet, améliorant l’alphabet structural HMM-SA27,appelé SAFlex (Structural Alphabet Flexibility), dans le but de prendre en compte l’incertitude des données (données manquantes dans les fichiers PDB) et la redondance des structures protéiques. Le nouvel alphabet structural SAFlex obtenu propose donc un nouveau modèle d’encodage rigoureux et robuste. Cet encodage permet de prendre en compte l’incertitude des données en proposant trois options d’encodages : le Maximum a posteriori (MAP), la distribution marginale a posteriori (POST)et le nombre effectif de lettres à chaque position donnée (NEFF). SAFlex fournit également un encodage consensus à partir de différentes réplications (chaînes multiples, monomères et homomères) d’une même protéine. Il permet ainsi la détection de la variabilité structurale entre celles-ci. Les avancées méthodologiques ainsi que l’obtention de l’alphabet SAFlex constituent les contributions principales de ce travail de thèse. Nous présentons aussi le nouveau parser de la PDB (SAFlex-PDB) et nous démontrons que notre parser a un intérêt aussi bien sur le plan qualitatif (détection de diverses erreurs)que quantitatif (rapidité et parallélisation) en le comparant avec deux autres parsers très connus dans le domaine (Biopython et BioJava). Nous proposons également à la communauté scientifique un site web mettant en ligne ce nouvel alphabet structural SAFlex. Ce site web représente la contribution concrète de cette thèse alors que le parser SAFlex-PDB représente une contribution importante pour le fonctionnement du site web proposé. Cette caractérisation précise des conformations 3D et la prise en compte de la redondance des informations 3D disponibles, fournies par SAFlex, a en effet un impact très important pour la modélisation de la conformation et de la variabilité des structures 3D, des boucles protéiques et des régions d’interface avec différents partenaires, impliqués dans la fonction des protéines / The purpose of this PhD is to provide a Structural Alphabet (SA) for more accurate characterization of protein three-dimensional (3D) structures as well as integrating the increasing protein 3D structure information currently available in the Protein Data Bank (PDB). The SA also takes into consideration the logic behind the structural fragments sequence by using the hidden Markov Model (HMM). In this PhD, we describe a new structural alphabet, improving the existing HMM-SA27 structural alphabet, called SAFlex (Structural Alphabet Flexibility), in order to take into account the uncertainty of data (missing data in PDB files) and the redundancy of protein structures. The new SAFlex structural alphabet obtained therefore offers a new, rigorous and robust encoding model. This encoding takes into account the encoding uncertainty by providing three encoding options: the maximum a posteriori (MAP), the marginal posterior distribution (POST), and the effective number of letters at each given position (NEFF). SAFlex also provides and builds a consensus encoding from different replicates (multiple chains, monomers and several homomers) of a single protein. It thus allows the detection of structural variability between different chains. The methodological advances and the achievement of the SAFlex alphabet are the main contributions of this PhD. We also present the new PDB parser(SAFlex-PDB) and we demonstrate that our parser is therefore interesting both qualitative (detection of various errors) and quantitative terms (program optimization and parallelization) by comparing it with two other parsers well-known in the area of Bioinformatics (Biopython and BioJava). The SAFlex structural alphabet is being made available to the scientific community by providing a website. The SAFlex web server represents the concrete contribution of this PhD while the SAFlex-PDB parser represents an important contribution to the proper function of the proposed website. Here, we describe the functions and the interfaces of the SAFlex web server. The SAFlex can be used in various fashions for a protein tertiary structure of a given PDB format file; it can be used for encoding the 3D structure, identifying and predicting missing data. Hence, it is the only alphabet able to encode and predict the missing data in a 3D protein structure to date. Finally, these improvements; are promising to explore increasing protein redundancy data and obtain useful quantification of their flexibility