Modelling of complex corroding industrial systems is ritical to effective inspection and maintenance for ssurance of system integrity. Wall thickness and corrosion rate are modelled for multiple dependent corroding omponents, given observations of minimum wall thickness per component. At each inspection, partial observations of the system are considered. A Bayes Linear approach is adopted simplifying parameter estimation and avoiding often unrealistic distributional assumptions. Key system variances are modelled, making exchangeability assumptions to facilitate analysis for sparse inspection time-series. A utility based criterion is used to assess quality of inspection design and aid decision making. The model is applied to inspection data from pipework networks on a full-scale offshore platform.

Dans le chapitre introductif, nous dressons une étude comparée des approches en économétrie et en apprentissage statistique sur les questions de l'estimation et de l'inférence en statistique.Dans le deuxième chapitre, nous nous intéressons à une classe générale de modèles de variables instrumentales nonparamétriques. Nous généralisons la procédure d'estimation de Otsu (2011) en y ajoutant un terme de régularisation. Nous prouvons la convergence de notre estimateur pour la norme L2 de Lebesgue.Dans le troisième chapitre, nous montrons que lorsque les données ne sont pas indépendantes et identiquement distribuées (i.i.d) mais simplement jointement échangeables, une version modifiée du processus empirique converge faiblement vers un processus gaussien sous les mêmes conditions que dans le cas i.i.d. Nous obtenons un résultat similaire pour une version adaptée du processus empirique bootstrap. Nous déduisons de nos résultats la normalité asymptotique de plusieurs estimateurs non-linéaires ainsi que la validité de l'inférence basée sur le bootstrap. Nous revisitons enfin l'article empirique de Santos Silva et Tenreyro (2006).Dans le quatrième chapitre, nous abordons la question de l'inférence pour des ratios d'espérances. Nous trouvons que lorsque le dénominateur ne tend pas trop vite vers zéro quand le nombre d'observations n augmente, le bootstrap nonparamétrique est valide pour faire de l'inférence asymptotique. Dans un second temps, nous complétons un résultat d'impossibilité de Dufour (1997) en montrant que quand n est fini, il est possible de construire des intervalles de confiance qui ne sont pas pathologiques sont certaines conditions sur le dénominateur.Dans le cinquième chapitre, nous présentons une commande Stata qui implémente les estimateurs proposés par de Chaisemartin et d'Haultfoeuille (2018) pour mesurer plusieurs types d'effets de traitement très étudiés en pratique. / In the introductory chapter, we compare views on estimation and inference in the econometric and statistical learning disciplines.In the second chapter, our interest lies in a generic class of nonparametric instrumental models. We extend the estimation procedure in Otsu (2011) by adding a regularisation term to it. We prove the consistency of our estimator under Lebesgue's L2 norm.In the third chapter, we show that when observations are jointly exchangeable rather than independent and identically distributed (i.i.d), a modified version of the empirical process converges weakly towards a Gaussian process under the same conditions as in the i.i.d case. We obtain a similar result for a modified version of the bootstrapped empirical process. We apply our results to get the asymptotic normality of several nonlinear estimators and the validity of bootstrap-based inference. Finally, we revisit the empirical work of Santos Silva and Tenreyro (2006).In the fourth chapter, we address the issue of conducting inference on ratios of expectations. We find that when the denominator tends to zero slowly enough when the number of observations n increases, bootstrap-based inference is asymptotically valid. Secondly, we complement an impossibility result of Dufour (1997) by showing that whenever n is finite it is possible to construct confidence intervals which are not pathological under some conditions on the denominator.In the fifth chapter, we present a Stata command which implements estimators proposed in de Chaisemartin et d'Haultfoeuille (2018) to measure several types of treatment effects widely studied in practice.

We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this thesis, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour. We further extend the underlying exchangeable fragmentation processes of such trees into partly exchangeable fragmentation processes by weakening the exchangeability. We obtain the integral representations for the measures associated with partly exchangeable fragmentation processes and subordinator of the tagged fragments. We also embed the trees associated with such processes into continuum random trees and study their limiting behaviour. In the end, we generate a three-parameter family of partly exchangeable trees which contains the family of the alpha-gamma trees and another important two-parameter family based on Poisson-Dirichlet distributions.

O elemento central deste estudo é o problema de predição em seqüências de variáveis aleatórias binárias (0-1). Modelos são estudados para esse tipo de situação e então relacionados com a Falácia do Apostador - um famoso caso de estudo da Psicologia (também conhecida como Lei da Maturidade). Estudos estatísticos anteriores propõem tal modelagem sob a perspectiva bayesiana. Neles, tem-se a suposição de permutabilidade infinita e, como conseqüência, a maturidade é um comportamento inadmissível. Neste estudo, um novo modelo é apresentado, no qual a crença do apostador não é necessariamente uma falácia. Este é o modelo preditivista usual de população finita e, portanto, somente quantidades com significado operacional (parâmetros operacionais) são envolvidas. Uma classe de prioris para o parâmetro operacional que resulta em modelos não estendíveis é apresentada. Trata-se de uma classe de distribuições que definimos como mais estreitas que a Binomial. Maturidade é uma conseqüência da crença em prioris dessa classe. Apresenta-se ainda uma subclasse referente às distribuições mais estreitas de segunda ordem que a Binomial. Para prioris dessa subclasse tem-se taxa de falha preditiva crescente, que pode ser interpretado como o resultado mais extremo de maturidade. Os resultados deste estudo podem contribuir para o julgamento de quão razoável é a suposição de permutabilidade infinita em relação ao típico comportamento humano. Outra principal contribuição está associada ao estudo de condições de estendibilidade em processos binários. / We study the problem of prediction in sequences of binary random variables. Models are studied for this kind of situation and then considered vis-à-vis the Gambler\'s Fallacy - a famous case study in Psychology (also known as Law of Maturity). Previous statistical studies proposed such modeling under the bayesian perspective. In them there is the assumption of exchangeability and, as a result, maturity is a inadmissible behavior. In this study, a new model in which the Gambler\'s belief need not be a fallacy is presented. This one is the usual finite population model and, therefore, only operationally meaningful quantities (operational parameters) are involved. A class of prior distributions for the operational parameter which yield non-extendable models is presented. It is a class of distributions which we defined as tighter than the Binomial. Maturity is a consequence of the belief in the prior distributions of this class. Furthermore, a subclass which refers to the distributions that are second-order tighter than the Binomial is presented. For prior distributions of this subclass the predictive failure rate is increasing, which can be interpreted as the most extreme case of maturity. The results of this study may contribute on the judgment of how reasonable the assumption of infinite exchangeability is relative to typical human perception. Another major contribution is related to the study on extendibility conditions in binary processes.

Esta tese trata de dois assuntos importantes na teoria de risco: o fenômeno da dependência local e a identificação e mensuração de assimetrias apresentadas pelos dados. A primeira parte trata de dependência local, sendo abordadas algumas medidas já analisadas na literatura. Versões locais dos coeficientes de Kendall e Spearman , baseadas na distribuição condicional dos dados, são propostas. São apresentadas algumas propriedades dessas medidas e a aplicação das mesmas a algumas cópulas. Na segunda parte são apresentados resultados sobre cópulas bivariadas que são as menos associativas e menos bi-simétricas segundo o critério de máxima distância modular. A última parte trata da não-permutabilidade e assimetria radial dos dados. Uma medida de não-permutabilidade baseada nos coeficientes de correlação condicional é proposta e aplicada a algumas distribuições. No final, o conceito de quantil bivariado é aplicado nas definições de medidas para avaliar o grau de permutabilidade e de simetria radial presentes na estrutura de dependência dos dados e de testes de hipóteses para verificar se a cópula subjacente aos dados é permutável ou radialmente simétrica. / In this thesis two important fields in risk theory are studied: the local dependence phenomenon and the identification and measuring of asymmetries contained in data. The first part deals with local dependence: some measures already studied in the literature are presented and discussed, and local versions of the coefficients Kendall and Spearman , based on the conditional distribution of data, are proposed. Properties of these measures and some examples concerning its application are treated. In the second part are presented some results about bivariate copulas which are the least associative and the least bi-symmetric according to the maximum modular distance. The last part analyses the nonexchangeability and the radial asymmetry of data. A measure of nonexchangeability based on the conditional correlation coefficient is proposed and applied to some distribution functions. At the end, the concept of bivariate quantile is applied in the definitions of measures for evaluating the degree of exchangeability and radial symmetry present in data and of hypothesis tests proposed for verifying whether the underlying copula is exchangeable or radially symmetric.

O elemento central deste estudo é o problema de predição em seqüências de variáveis aleatórias binárias (0-1). Modelos são estudados para esse tipo de situação e então relacionados com a Falácia do Apostador - um famoso caso de estudo da Psicologia (também conhecida como Lei da Maturidade). Estudos estatísticos anteriores propõem tal modelagem sob a perspectiva bayesiana. Neles, tem-se a suposição de permutabilidade infinita e, como conseqüência, a maturidade é um comportamento inadmissível. Neste estudo, um novo modelo é apresentado, no qual a crença do apostador não é necessariamente uma falácia. Este é o modelo preditivista usual de população finita e, portanto, somente quantidades com significado operacional (parâmetros operacionais) são envolvidas. Uma classe de prioris para o parâmetro operacional que resulta em modelos não estendíveis é apresentada. Trata-se de uma classe de distribuições que definimos como mais estreitas que a Binomial. Maturidade é uma conseqüência da crença em prioris dessa classe. Apresenta-se ainda uma subclasse referente às distribuições mais estreitas de segunda ordem que a Binomial. Para prioris dessa subclasse tem-se taxa de falha preditiva crescente, que pode ser interpretado como o resultado mais extremo de maturidade. Os resultados deste estudo podem contribuir para o julgamento de quão razoável é a suposição de permutabilidade infinita em relação ao típico comportamento humano. Outra principal contribuição está associada ao estudo de condições de estendibilidade em processos binários. / We study the problem of prediction in sequences of binary random variables. Models are studied for this kind of situation and then considered vis-à-vis the Gambler\'s Fallacy - a famous case study in Psychology (also known as Law of Maturity). Previous statistical studies proposed such modeling under the bayesian perspective. In them there is the assumption of exchangeability and, as a result, maturity is a inadmissible behavior. In this study, a new model in which the Gambler\'s belief need not be a fallacy is presented. This one is the usual finite population model and, therefore, only operationally meaningful quantities (operational parameters) are involved. A class of prior distributions for the operational parameter which yield non-extendable models is presented. It is a class of distributions which we defined as tighter than the Binomial. Maturity is a consequence of the belief in the prior distributions of this class. Furthermore, a subclass which refers to the distributions that are second-order tighter than the Binomial is presented. For prior distributions of this subclass the predictive failure rate is increasing, which can be interpreted as the most extreme case of maturity. The results of this study may contribute on the judgment of how reasonable the assumption of infinite exchangeability is relative to typical human perception. Another major contribution is related to the study on extendibility conditions in binary processes.

This thesis is divided into two parts. Part I deals with demand management. For goods with no substitutes, under supply constraints, fairness considerations introduce negative externalities and lead to a market failure. One example of such a good with no substitutes is water. In case of a market failure, it is necessary to design coordination mechanisms called contracts which provide the right incentives for coordination. As “repetition can yield coordination”, the aim in this part is to design price based dynamic demand management contracts which, under supply constraints, mitigate the market failure. In these contracts, we consider complete information settings; and use the status quo proposition as a fairness criterion for designing them. The contracts are designed as almost noncooperative dynamic games, within the agency theory framework, where the agent (the consumer) is induced to consume at a specified consumption level based on the incentive mechanism offered by the principal (the producer). These contracts use the solution concept of sub-game perfect Nash equilibrium (SPNE) to compute the price (mal-incentive) that acts as a credible threat for deviation from the specified consumption level. In these contracts, unlike the dynamic contracts with asymmetric information, the penalty for deviation is proportional to the amount of deviation. First, we consider a two-period demand management contract for a single consumer satisfying the status quo proposition. Under the assumption that the gain to the consumer and the loss to the producer by deviation is small, the contract is shown to be economically efficient. It is shown that, in the finite horizon, a fair demand management contract cannot be efficient. The demand management contract is homeomorphic to finite horizon alternating bargaining model. In the finite horizon alternating bargaining model, there is a unique SPNE, in which the player who offers last is always at an advantageous position. In the two-period contract, the assumption considered attenuates the last mover advantage and leads to the efficiency. We have shown that one possible way to achieve efficiency, without the assumption, is to make the agents uncertain about the period of interaction. This possibility can be included in an infinite horizon contract. Hence, next, we design an infinite horizon contract for a single consumer. It is proved that this contract is economically efficient and provides revenue sufficiency. The sensitivity analysis of the contract shows that the discounting rate measures the aversion to conservation characteristics of the consumer. The analysis of the contract shows that a sufficiently time-patient consumer is not penalized for the deviation, as the consumer himself is aware of conservation requirements. This result is similar to the results for the present-biased preferences in behavioral economics. Lastly, the infinite horizon contract is extended to two consumers case which internalizes the externality a consumer causes to another. In the two consumer case, consumers are strategically noninteracting; and it is shown that the producer acts as a budget balancer. These contracts are also shown to be economically efficient. The demand management contracts achieve both the procedural and end-state fairness. Also, the infinite horizon contracts are homeomorphic to infinite horizon alternating bargaining model. The efficiency of infinite horizon contracts is due to their homeomorphism with the alternating bargaining process as they exhaust all possible mutual gains from exchange. In the two-period model, the bargaining process is constrained and hence all possible mutual gains are not eliminated, leading to the inefficiency. In part II of the thesis, we discuss the notions of exchangeability in the Shapley value. The Shapley value is a probabilistic value for the transferable utility (TU) cooperative games, in which each player subjectively assigns probabilities to the events which define their positions in the game. In this part, the objective have been to explore the aspect of subjective probability which leads to the uniqueness of the Shapley value. This aspect of subjective probability is known as exchangeability. We derive the Shapley value using de Finetti’s theorem. We also show that, in the Shapley value, each player’s prospects of joining a t-player game as the last member of the game is a moment sequence of the uniquely determined uniform distribution. We stress on finite exchangeability; and deduce that, with finite exchangeability, the Shapley value is the only value in which the probability assignment is a unique mixture of independent and identical distributions. It is concluded that, in both the finite and infinite exchangeable cases, the uniqueness of probability assignment in the Shapley value is due to exchangeability and the mixing with the uniform distribution.

Esta tese trata de dois assuntos importantes na teoria de risco: o fenômeno da dependência local e a identificação e mensuração de assimetrias apresentadas pelos dados. A primeira parte trata de dependência local, sendo abordadas algumas medidas já analisadas na literatura. Versões locais dos coeficientes de Kendall e Spearman , baseadas na distribuição condicional dos dados, são propostas. São apresentadas algumas propriedades dessas medidas e a aplicação das mesmas a algumas cópulas. Na segunda parte são apresentados resultados sobre cópulas bivariadas que são as menos associativas e menos bi-simétricas segundo o critério de máxima distância modular. A última parte trata da não-permutabilidade e assimetria radial dos dados. Uma medida de não-permutabilidade baseada nos coeficientes de correlação condicional é proposta e aplicada a algumas distribuições. No final, o conceito de quantil bivariado é aplicado nas definições de medidas para avaliar o grau de permutabilidade e de simetria radial presentes na estrutura de dependência dos dados e de testes de hipóteses para verificar se a cópula subjacente aos dados é permutável ou radialmente simétrica. / In this thesis two important fields in risk theory are studied: the local dependence phenomenon and the identification and measuring of asymmetries contained in data. The first part deals with local dependence: some measures already studied in the literature are presented and discussed, and local versions of the coefficients Kendall and Spearman , based on the conditional distribution of data, are proposed. Properties of these measures and some examples concerning its application are treated. In the second part are presented some results about bivariate copulas which are the least associative and the least bi-symmetric according to the maximum modular distance. The last part analyses the nonexchangeability and the radial asymmetry of data. A measure of nonexchangeability based on the conditional correlation coefficient is proposed and applied to some distribution functions. At the end, the concept of bivariate quantile is applied in the definitions of measures for evaluating the degree of exchangeability and radial symmetry present in data and of hypothesis tests proposed for verifying whether the underlying copula is exchangeable or radially symmetric.

This thesis is divided into three main parts. In the first, we discuss that, although permutation tests can provide exact control of false positives under the reasonable assumption of exchangeability, there are common examples in which global exchangeability does not hold, such as in experiments with repeated measurements or tests in which subjects are related to each other. To allow permutation inference in such cases, we propose an extension of the well known concept of exchangeability blocks, allowing these to be nested in a hierarchical, multi-level definition. This definition allows permutations that retain the original joint distribution unaltered, thus preserving exchangeability. The null hypothesis is tested using only a subset of all otherwise possible permutations. We do not need to explicitly model the degree of dependence between observations; rather the use of such permutation scheme leaves any dependence intact. The strategy is compatible with heteroscedasticity and can be used with permutations, sign flippings, or both combined. In the second part, we exploit properties of test statistics to obtain accelerations irrespective of generic software or hardware improvements. We compare six different approaches using synthetic and real data, assessing the methods in terms of their error rates, power, agreement with a reference result, and the risk of taking a different decision regarding the rejection of the null hypotheses (known as the resampling risk). In the third part, we investigate and compare the different methods for assessment of cortical volume and area from magnetic resonance images using surface-based methods. Using data from young adults born with very low birth weight and coetaneous controls, we show that instead of volume, the permutation-based non-parametric combination (NPC) of thickness and area is a more sensitive option for studying joint effects on these two quantities, giving equal weight to variation in both, and allowing a better characterisation of biological processes that can affect brain morphology.