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Thesis (M.S.)--Brigham Young University. Dept. of Statistics, 2007. / Includes bibliographical references (p. 57-60).

A través de una actividad colaborativa, se trabajan diferentes conceptos del tema de fluidos en hidrodinámica e hidrostática. Se presenta el diseño de la actividad, así como el material para los alumnos.

In this thesis I describe the essential steps of developing a credit rating system. This comprises the credit scoring process that assigns a credit score to each credit, the forming of rating classes by the k-means algorithm and the assignment of a probability of default (PD) for the rating classes. The main focus is on the PD estimation for which two approaches are presented. The first and simple approach in form of a calibration curve assumes independence of the defaults of different corporate credits. The second approach with mixture models is more realistic as it takes default dependence into account. With these models we can use an estimate of a country’s GDP to calculate an estimate for the Value-at-Risk of some credit portfolio.

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<p> We study two objects: an ideal of a Hecke algebra, and a pairing in Galois cohomology.</p> <p> Let h be the Hecke algebra of cusp forms of weight 2, level n, and a fixed Dirichlet character modulo n generated by all Hecke operators, where n is an odd prime p or a product of two distinct odd primes N and p. We study the Eisenstein I ideal of h. We wrote a computer program to test whether Up - 1 generates this ideal, where Up is the pth Hecke operator in h. We found many cases of n and the character so that Up - 1 alone generates I. On the other hand, we found one example with N = 3 and p = 331 where Up - 1 does not generate I.</p> <p> Let K = Q(μn) be the nth cyclotomic field. Let S be the set of primes above p in K, and let G_K,S be the Galois group of the maximal extension of K unramified outside S. We study a pairing on cyclotomic p-units that arises from the cup product on H1(G_K,S, μp). This pairing takes values in a Gal(K/Q)-eigenspace of the p-part of the class group of K. Sharifi has conjectured that this pairing is surjective. We studied this pairing in detail by imposing linear relations on the possible pairing values. We discovered many values of n and the character such that these relations single out a unique nontrivial possibility for the pairing, up to a possibly zero scalar.</p> <p> Sharifi showed in [S2] that, under an assumption on Bernoulli numbers, the element Up - 1 generates the Eisenstein ideal I if and only if pairing with the single element p is surjective. In particular, in the instances for which we found a unique nontrivial possibility for the pairing, then if Up - 1 generates I, we know that the scalar up to which it is determined cannot be zero.</p> / Thesis / Master of Science (MSc)

Dans la première partie de cette thèse, on s'intéresse à la convergence de certaines séries aléatoires. On détermine des conditions suffisantes sur la suite $(a_k)_{k\geq 1}$ et sur les variables aléatoires indépendantes $(X_k)_{k\geq 1}$, afin que pour presque tout $\omega\in\Omega$, on ait la convergence uniforme ou pour presque tout $x$ de la série $\sum_{k\geq 1}a_k f(x\cdot(X_1+\cdots+X_k)(\omega))$ pour une certaine classe de fonctions $f$. On trouve, aussi, des conditions suffisantes sur la suite $(a_k)_{k\geq 1}$ et sur les variables aléatoires indépendantes $(X_k)_{k\geq 1}$, afin que pour presque tout $\omega\in\Omega$, on ait la convergence dans $L^2(\mu)$ ou $\mu$-presque partout de la série $\sum_{k\geq 1}a_k T^{(X_1+\cdots+X_k)(\omega)}(g)(x)$ pour certaines classes de fonctions $g\in L^2(\mu)$ et de flot $\{T^{t},t\in G\}$ d'opérateurs de $L^2(\mu)$ dans $L^2(\mu)$ (où $G$ est un semi-groupe de $\mathbb{R}$). La deuxième partie porte sur des propriétés de mesures : quasi-Bernoulli et quasi-Bernoulli faible. On trouve notamment des conditions nécessaires et suffisantes pour qu'une mesure de Markov inhomogène soit quasi-Bernoulli ou quasi-Bernoulli faible. On caractèrise à l'aide de ces conditions les mesures de Bernoulli qui sont quasi-Bernoulli ou quasi-Bernoulli faible. On prouve que si une mesure de Bernoulli n'ayant que des probabilités non nulles est quasi-Bernoulli faible alors elle est quasi-Bernoulli. La dernière partie est consacrée à l'étude du problème du temps de retour uniforme dans les systèmes exponentiellement mélangeant. Il s'agit d'avoir un recouvrement aléatoire de l'espace engendré par un processus exponentiellement mélangeant. Etant donné un recouvrement aléatoire, on obtient une estimation du nombre de recouvrements en fonction de la dimension maximale locale de la mesure

In many applications, the success probability 𝜋 of a Bernoulli distributed variable 𝑌 is influenced by another variable 𝑋. For example for loans granted, it is necessary to rate debtors in different rating classes, where the probability of default (PD) 𝜋 of 𝑌 is assumed to be homogeneous within and heterogeneous between the rating classes. The PD of a debtor is largely influenced by macroeconomic and individual variables (𝑋). In this work, we study a Bernoulli mixture model for 𝑌, where the success probability of 𝑌 changes systematically at the breakpoints. We focus on cross-sectional data and our main objective is to estimate all 𝑘 breakpoints with 𝑘 either known or unknown and their corresponding success probabilities between each pair of neighbouring breakpoints. To the best of our knowledge, an estimator for estimating multiple breakpoints has not yet been developed in this context. Thus, we develop an approach with a view to closing this research gap. We show that our estimator works for independent and identically distributed (i.i.d.) 𝑋 as well as for a linear one-factor model for 𝑋. A theoretical foundation for this estimator is also presented. In practice, the number of breakpoints 𝑘 is often unknown a priori. As the multiple estimator is based on an iterative procedure, we propose stopping criteria for estimating 𝑘 correctly. We conduct a simulation study in the context of credit rating to demonstrate the performance of the developed estimator. Furthermore, we apply the new estimator on credit risk data from the Sächsische Aufbaubank, the Development Bank of Saxony. To simplify the use of the new estimator, we also develop an R package called MultipleBreakpoints.

Random sampling has become a crucial component of modern data management systems. Although the literature on database sampling is large, there has been relatively little work on the problem of maintaining a sample in the presence of arbitrary insertions and deletions to the underlying dataset. Most existing maintenance techniques apply either to the insert-only case or to datasets that do not contain duplicates. In this paper, we provide a scheme that maintains a Bernoulli sample of an underlying multiset in the presence of an arbitrary stream of updates, deletions, and insertions. Importantly, the scheme never needs to access the underlying multiset. Such Bernoulli samples are easy to manipulate, and are well suited to parallel processing environments. Our method can be viewed as an enhancement of the 'counting sample' scheme developed by Gibbons and Matias for estimating the frequency of highly frequent items. We show how the 'tracking counters' used by our maintenance scheme can be exploited to estimate population frequencies, sums, and averages in an unbiased manner, with lower variance than the usual estimators based on a Bernoulli sample. The number of distinct items in the multiset can also be estimated without bias. Finally, we discuss certain problems of subsampling and merging that a rise in systems with limited memory resources or distributed processing, respectively.

Made available in DSpace on 2016-04-27T16:58:14Z (GMT). No. of bitstreams: 1 Marcelo Rivelino Rodrigues.pdf: 515906 bytes, checksum: 0b54cbbd6b99bd2a08b96bb98277ace9 (MD5) Previous issue date: 2007-05-31 / Secretaria da Educação do Estado de São Paulo / In this work I am considering the use of the Urn of Bernoulli, as a basic model in the education of Probability. For the concrete representation of the urn I will use the so called activity The Bottle of Brousseau . The theoretical basis present in this research is composed by two main theories: The Conceptual Fields Theory, which belongs to Gerard Vergnaud, and the Theory of the Situations, developed by Guy Brousseau. The present research was accomplished with students of the last series of the room cycle of Basic Education in order to verify if they, through an education sequence, can be able to construct the probabilistic concepts of base, when these concepts are boarded for two approaches: the Laplaciano approach and the frequency approach / Neste trabalho propomos a utilização da Urna de Bernoulli como modelo fundamental no ensino de Probabilidade. Para a representação concreta da urna, usarei a atividade denominada Garrafa de Brousseau . A base que permea este trabalho foi composta por duas teorias: a de Campos Conceituais , de Gerard Vergnaud, e a Teoria das Situações , de Guy Brousseau. Realizamos esta pesquisa com alunos da última série do quarto ciclo do Ensino Fundamental, com o intuito de verificar se esses alunos, por meio de uma seqüência de ensino, puderam construir os conceitos probabilísticos de base quando estes são abordados por dois enfoques: o laplaciano e o freqüentista