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O teorema central do limite: um estudo ecológico do saber e do didáticoRodrigues, Chang Kuo 02 December 2009 (has links)
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Previous issue date: 2009-12-02 / This paper refers to the building of mathematical and/or statistical ideas and concepts
around Central Limit Theorem for Mathematics graduates.The investigation focuses
the importance of the theorem in Statistics Inference and its comprehension by the
professionals to be, who will act in Basic Education. Therefore, we chose to research
some books related to the teaching and learning process of the theorem and
emphasised its importance on the Mathematics teacher daily practice. The theoretical
approach is about Mathematics Teaching theories, particularly the Theory of Didactic
Transposition ( CHEVALLARD, 1985), with an echological approach under the
knowlwdge and teaching point of view ( ARTAUD, 1998). We chose methodological
procedures directed to the didactic design (ARTIGUE, 2009), with qualitative nature,
and whose assumptions are linked to Teaching Engineering (ARTIGUE, 1988). The
subjects of this investigation are the graduates who had some knowledge about
Basic Statistics and, from a previous analysis about the kind of knowledge they had
about the theme, we presented some activities in a problem-situation context
connected to the Mathematics teachers daily practice. The analysis of these results
allowed us to relate the existing problems between the subject and the students from
Basic Education, which involved statistics literacy. After these activities, there was a
dialogue, with discussions about the theme, allowing us to analyse how the ideas
and concepts around the Central Limit Theorem were built, being its comprehension
the main aim for the graduates. Besides that, we analysed some textbooks for higher
education, based on the Anthropological Theory of Didactic (CHEVALLARD, 1996,
1999), which also showed us the essential knowledge for the theorem to live ,
because the approach is under the knowledge and teaching echological point of
view. On the other hand, we detected what kind of limitations, or restrictions, exist in
the books analysed, interfering in the elaboration of the activities by the teacher.
Thus, our investigation reaffirms the importance of teaching and learning Statistics in
the various applications for the Mathematics teachers to be formation in a world
controlled by the technological advances, which interfere directly on the
understanding of the information we receive every moment / O presente trabalho refere-se à construção das ideias e dos conceitos matemáticos
e/ou estatísticos em torno do Teorema Central do Limite para os Licenciandos de
Matemática. O cerne da investigação limita-se à importância do teorema na
Inferência Estatística e à sua compreensão pelos futuros profissionais que atuarão
na Educação Básica. Nesse sentido, optamos por revisar algumas bibliografias que
têm relação com o processo de ensino e de aprendizagem do teorema e
enfatizamos sua importância na pratica do dia a dia do professor de Matemática. O
quadro teórico incide sobre as teorias da Didática da Matemática, particularmente, a
Teoria da Transposição Didática (CHEVALLARD, 1985), munido de uma abordagem
ecológica sob o ponto de vista do saber e do didático (ARTAUD, 1998). Optamos por
procedimentos metodológicos voltados para o design didático (ARTIGUE, 2009), de
cunho qualitativo e, cujos pressupostos estão aliados à Engenharia Didática
(ARTIGUE, 1988). Os sujeitos dessa investigação são os licenciandos que já
predispunham de conhecimentos sobre a Estatística Básica e, a partir de uma
análise prévia sobre que tipos de conhecimento eles já detinham sobre o tema,
apresentamos algumas atividades no contexto de uma situação-problema pertinente
ao cotidiano dos professores de Matemática. A análise desses resultados nos
propiciou interrelacionar as problemáticas existentes na disciplina de Matemática
com alunos da Educação Básica, envolvendo assim, a literacia estatística. Após a
realização dessas atividades, ocorreu também um diálogo, com discussões acerca
do tema, o que nos permitiu analisar como foram construídos as ideias e os
conceitos no entorno do Teorema Central do Limite, de modo que sua compreensão
fosse o principal alvo para os licenciandos. Além disso, analisamos alguns livrostexto
do ensino superior, à luz da Teoria Antropológica do Didático (CHEVALLARD,
1996, 1999), o que também nos indicou que saberes são indispensáveis de modo
que o teorema viva , já que a abordagem é sob o ponto de vista ecológico do saber
e do didático. Por outro lado, detectamos que tipos de limitações, ou restrições,
existem nas obras consultadas, interferindo assim, a elaboração das atividades por
parte do professor. Portanto, a nossa investigação reitera a importância do ensino e
da aprendizagem da Estatística nas diversas aplicações na formação dos futuros
professores de Matemática num mundo ditado pelos avanços tecnológicos, que
interferem diretamente na leitura de informações que recebemos a todo instante
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Chaotické náhodné veličiny v aplikované pravděpodobnosti / Chaotic random variables in applied probabilityVečeřa, Jakub January 2019 (has links)
This thesis deals with modeling of particle processes. In the first part we ex- amine Gibbs facet process on a bounded window with discrete orientation distri- bution and we derive central limit theorem (CLT) for U-statistics of facet process with increasing intensity. We calculate all asymptotic joint moments for interac- tion U-statistics and use the method of moments for deriving the CLT. Moreover we present an alternative proof which makes use of the CLT for U-statistics of a Poisson facet process. In the second part we model planar segment processes given by a density with respect to the Poisson process. Parametric models involve reference distributions of directions and/or lengths of segments. Statistical methods are presented which first estimate scalar parameters by known approaches and then the reference distribution is estimated non-parametrically. We also introduce the Takacs-Fiksel estimate and demonstrate the use of estimators in a simulation study and also using data from actin fibres from stem cells images. In the third part we study a stationary Gibbs particle process with determin- istically bounded particles on Euclidean space defined in terms of a finite range potential and an activity parameter. For small activity parameters, we prove the CLT for certain statistics of this...
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Gaussian structures and orthogonal polynomialsLarsson-Cohn, Lars January 2002 (has links)
<p>This thesis consists of four papers on the following topics in analysis and probability: analysis on Wiener space, asymptotic properties of orthogonal polynomials, and convergence rates in the central limit theorem. The first paper gives lower bounds on the constants in the Meyer inequality from the Malliavin calculus. It is shown that both constants grow at least like <i>(p-1)</i><sup>-1</sup> or like <i>p</i> when <i>p</i> approaches 1 or ∞ respectively. This agrees with known upper bounds. In the second paper, an extremal problem on Wiener chaos motivates an investigation of the <i>L</i><sup>p</sup>-norms of Hermite polynomials. This is followed up by similar computations for Charlier polynomials in the third paper. In both cases, the <i>L</i><sup>p</sup>-norms present a peculiar behaviour with certain threshold values of p, where the growth rate and the dominating intervals undergo a rapid change. The fourth paper analyzes a connection between probability and numerical analysis. More precisely, known estimates on the convergence rate of finite difference equations are "translated" into results on convergence rates of certain functionals in the central limit theorem. These are also extended, using interpolation of Banach spaces as a main tool. Besov spaces play a central role in the emerging results.</p>
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Gaussian structures and orthogonal polynomialsLarsson-Cohn, Lars January 2002 (has links)
This thesis consists of four papers on the following topics in analysis and probability: analysis on Wiener space, asymptotic properties of orthogonal polynomials, and convergence rates in the central limit theorem. The first paper gives lower bounds on the constants in the Meyer inequality from the Malliavin calculus. It is shown that both constants grow at least like (p-1)-1 or like p when p approaches 1 or ∞ respectively. This agrees with known upper bounds. In the second paper, an extremal problem on Wiener chaos motivates an investigation of the Lp-norms of Hermite polynomials. This is followed up by similar computations for Charlier polynomials in the third paper. In both cases, the Lp-norms present a peculiar behaviour with certain threshold values of p, where the growth rate and the dominating intervals undergo a rapid change. The fourth paper analyzes a connection between probability and numerical analysis. More precisely, known estimates on the convergence rate of finite difference equations are "translated" into results on convergence rates of certain functionals in the central limit theorem. These are also extended, using interpolation of Banach spaces as a main tool. Besov spaces play a central role in the emerging results.
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On the error-bound in the nonuniform version of Esseen's inequality in the Lp-metricPaditz, Ludwig 25 June 2013 (has links) (PDF)
The aim of this paper is to investigate the known nonuniform version of Esseen's inequality in the Lp-metric, to get a numerical bound for the appearing constant L.
For a long time the results given by several authors constate the impossibility of a nonuniform estimation in the most interesting case δ=1, because the effect L=L(δ)=O(1/(1-δ)), δ->1-0, was observed, where 2+δ, 0<δ<1, is the order of the assumed moments of the considered independent random variables X_k, k=1,2,...,n. Again making use of the method of conjugated distributions, we improve the well-known technique to show in the most interesting case δ=1 the finiteness of the absolute constant L and to prove L=L(1)=<127,74*7,31^(1/p), p>1.
In the case 0<δ<1 we only give the analytical structure of L but omit numerical calculations. Finally an example on normal approximation of sums of l_2-valued random elements demonstrates the application of the nonuniform mean central limit bounds obtained here. / Das Anliegen dieses Artikels besteht in der Untersuchung einer bekannten Variante der Esseen'schen Ungleichung in Form einer ungleichmäßigen Fehlerabschätzung in der Lp-Metrik mit dem Ziel, eine numerische Abschätzung für die auftretende absolute Konstante L zu erhalten.
Längere Zeit erweckten die Ergebnisse, die von verschiedenen Autoren angegeben wurden, den Eindruck, dass die ungleichmäßige Fehlerabschätzung im interessantesten Fall δ=1 nicht möglich wäre, weil auf Grund der geführten Beweisschritte der Einfluss von δ auf L in der Form L=L(δ)=O(1/(1-δ)), δ->1-0, beobachtet wurde, wobei 2+δ, 0<δ<1, die Ordnung der vorausgesetzten Momente der betrachteten unabhängigen Zufallsgrößen X_k, k=1,2,...,n, angibt.
Erneut wird die Methode der konjugierten Verteilungen angewendet und die gut bekannte Beweistechnik verbessert, um im interessantesten Fall δ=1 die Endlichkeit der absoluten Konstanten L nachzuweisen und um zu zeigen, dass L=L(1)=<127,74*7,31^(1/p), p>1, gilt.
Im Fall 0<δ<1 wird nur die analytische Struktur von L herausgearbeitet, jedoch ohne numerische Berechnungen. Schließlich wird mit einem Beispiel zur Normalapproximation von Summen l_2-wertigen Zufallselementen die Anwendung der gewichteten Fehlerabschätzung im globalen zentralen Grenzwertsatz demonstriert.
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On Parametric and Nonparametric Methods for Dependent DataBandyopadhyay, Soutir 2010 August 1900 (has links)
In recent years, there has been a surge of research interest in the analysis of time series
and spatial data. While on one hand more and more sophisticated models are being
developed, on the other hand the resulting theory and estimation process has become
more and more involved. This dissertation addresses the development of statistical
inference procedures for data exhibiting dependencies of varied form and structure.
In the first work, we consider estimation of the mean squared prediction error
(MSPE) of the best linear predictor of (possibly) nonlinear functions of finitely many
future observations in a stationary time series. We develop a resampling methodology
for estimating the MSPE when the unknown parameters in the best linear predictor
are estimated. Further, we propose a bias corrected MSPE estimator based on the
bootstrap and establish its second order accuracy. Finite sample properties of the
method are investigated through a simulation study.
The next work considers nonparametric inference on spatial data. In this work
the asymptotic distribution of the Discrete Fourier Transformation (DFT) of spatial
data under pure and mixed increasing domain spatial asymptotic structures are
studied under both deterministic and stochastic spatial sampling designs. The deterministic
design is specified by a scaled version of the integer lattice in IRd while
the data-sites under the stochastic spatial design are generated by a sequence of independent
random vectors, with a possibly nonuniform density. A detailed account
of the asymptotic joint distribution of the DFTs of the spatial data is given which, among other things, highlights the effects of the geometry of the sampling region and
the spatial sampling density on the limit distribution. Further, it is shown that in
both deterministic and stochastic design cases, for "asymptotically distant" frequencies,
the DFTs are asymptotically independent, but this property may be destroyed if
the frequencies are "asymptotically close". Some important implications of the main
results are also given.
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Some remarks on the central limit theorem for stationary Markov processes / Einige Bermerkungen zum zentralen Grenzwertsatz für stationäre Markoffsche ProzesseHolzmann, Hajo 21 April 2004 (has links)
No description available.
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Estudo de séries de tempo financeiras sob a perspectiva do teorema das seções de Lévy / Finalcial time series analysis based on Lévy's section theorem perspectiveRanciaro Neto, Adhemar 25 June 2013 (has links)
This study aimed to analyze financial time series grounded on a perspective of time measure
changing, based on accumulation of volatility of returns relative to the prices observed. Such
a scale was used for two reasons: the first one is related to Ludwig Von Mises’ proposition of
time concept in an economic system and the second one is related to the acceleration of
convergence in Gaussian distribution of a sequence of random variables, according to Lévy
sections theorem. By means of implementation of this new timeline, we designed a type of
trading asset strategy which its resulting average returns and risk were compared to a strategy
using daily time unit. Results suggested reflection about statistical and measurement
procedures applied to the data. / O objetivo deste trabalho foi o de estudar séries temporais financeiras fundamentadas em uma
perspectiva de alteração de medida de tempo, baseada no acúmulo de volatilidade dos
retornos relativos aos preços observados. Esta escala foi utilizada por dois motivos: o
primeiro está relacionado à proposta de Ludwig von Mises sobre a ideia de tempo em um
sistema econômico e o segundo está associado à capacidade que tal medida tem de acelerar o
processo de convergência de distribuição de uma sequência de variáveis aleatórias para a
Gaussiana, de acordo com o teorema das seções de Lévy. Com base nesta nova escala
temporal, foi elaborado um tipo de estratégia de negociação de ativos tendo seus retornos
médios e risco sido avaliados em comparação com uma estratégia utilizando o tempo em
unidades diárias. Os resultados obtidos motivaram a reflexão sobre as estatísticas utilizadas e
os procedimentos para a mensuração de desempenho de cada estratégia.
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Théorèmes limites pour des fonctionnelles de clusters d'extrêmes et applications / Limit theorems for functionals of clusters of extremes and applicationsGomez Garcia, José Gregorio 13 November 2017 (has links)
Cette thèse traite principalement des théorèmes limites pour les processus empiriques de fonctionnelles de clusters d'extrêmes de séquences et champs aléatoires faiblement dépendants. Des théorèmes limites pour les processus empiriques de fonctionnelles de clusters d'extrême de séries temporelles stationnaires sont donnés par Drees & Rootzén [2010] sous des conditions de régularité absolue (ou "ß-mélange"). Cependant, ces conditions de dépendance de type mélange sont très restrictives : elles sont particulièrement adaptées aux modèles dans la finance et dans l'histoire, et elles sont de plus compliquées à vérifier. Généralement, pour d'autres modèles fréquemment rencontré dans les domaines applicatifs, les conditions de mélange ne sont pas satisfaites. En revanche, les conditions de dépendance faible, selon Doukhan and Louhichi [1999] et Dedecker & Prieur [2004a], sont des conditions qui généralisent les notions de mélange et d'association. Elles sont plus simple à vérifier et peuvent être satisfaites pour de nombreux modèles. Plus précisément, sous des conditions faibles, tous les processus causals ou non causals sont faiblement dépendants: les processus Gaussien, associés, linéaires, ARCH(∞), bilinéaires et notamment Volterra entrent dans cette liste. À partir de ces conditions favorables, nous étendons certains des théorèmes limites de Drees & Rootzén [2010] à processus faiblement dépendants. En outre, comme application des théorèmes précédents, nous montrons la convergence en loi de l'estimateur de l'extremogramme de Davis & Mikosch [2009] et l'estimateur fonctionnel de l'indice extrémal de Drees [2011] sous dépendance faible. Nous démontrons un théorème de la valeur extrême pour les champs aléatoires stationnaires faiblement dépendants et nous proposons, sous les mêmes conditions, un critère du domaine d'attraction d'une loi d'extrêmes. Le document se conclue sur des théorèmes limites pour les processus empiriques de fonctionnelles de clusters d’extrêmes de champs aléatoires stationnaires faiblement dépendants, et met en évidence la convergence en loi de l'estimateur d'un extremogramme de processus spatio-temporels stationnaires faiblement dépendants en tant qu'application. / This thesis deals mainly with limit theorems for empirical processes of extreme cluster functionals of weakly dependent random fields and sequences. Limit theorems for empirical processes of extreme cluster functionals of stationnary time series are given by Drees & Rootzén [2010] under absolute regularity (or "ß-mixing") conditions. However, these dependence conditions of mixing type are very restrictive: on the one hand, they are best suited for models in finance and history, and on the other hand, they are difficult to verify. Generally, for other models common in applications, the mixing conditions are not satisfied. In contrast, weak dependence conditions, as defined by Doukhan & Louhichi [1999] and Dedecker & Prieur [2004a], are dependence conditions which generalises the notions of mixing and association. These are easier to verify and applicable to a wide list of models. More precisely, under weak conditions, all the causal or non-causal processes are weakly dependent: Gaussian, associated, linear, ARCH(∞), bilinear and Volterra processes are some included in this list. Under these conveniences, we expand some of the limit theorems of Drees & Rootzén [2010] to weakly dependent processes. These latter results are used in order to show the convergence in distribution of the extremogram estimator of Davis & Mikosch [2009] and the functional estimator of the extremal index introduced by Drees [2011] under weak dependence. We prove an extreme value theorem for weakly dependent stationary random fields and we propose, under the same conditions, a domain of attraction criteria of a law of extremes. The document ends with limit theorems for the empirical process of extreme cluster functionals of stationary weakly dependent random fields, deriving also the convergence in distribution of the estimator of an extremogram for stationary weakly dependent space-time processes.
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Understanding the Functional Central Limit Theorems with Some Applications to Unit Root Testing with Structural Change / El Teorema del Límite Central Funcional con algunas aplicaciones a raíces unitarias con cambios estructuralesAquino, Juan Carlos, Rodríguez, Gabriel 10 April 2018 (has links)
The application of different unit root statistics is by now a standard practice in empirical work. Even when it is a practical issue, these statistics have complex nonstandard distributions depending on functionals of certain stochastic processes, and their derivations represent a barrier even for many theoretical econometricians. These derivations are based on rigorous and fundamental statistical tools which are not (very) well known by standard econometricians. This paper aims to fill this gap by explaining in a simple way one of these fundamental tools: namely, the Functional Central Limit Theorem. To this end, this paper analyzes the foundations and applicability of two versions of the Functional Central Limit Theorem within the framework of a unit root with a structural break. Initial attention is focused on the probabilistic structure of the time series to be considered. Thereafter, attention is focused on the asymptotic theory for nonstationary time series proposed by Phillips (1987a), which is applied by Perron (1989) to study the effects of an (assumed) exogenous structural break on the power of the augmented Dickey-Fuller test and by Zivot and Andrews (1992) to criticize the exogeneity assumption and propose a method for estimating an endogenous breakpoint. A systematic method for dealing with efficiency issues is introduced by Perron and Rodriguez (2003), which extends the Generalized Least Squares detrending approach due to Elliot et al. (1996). An empirical application is provided. / Hoy en día es una práctica estándar de trabajo empírico la aplicación de diferentes estadísticos de contraste de raíz unitaria. A pesar de ser un aspecto práctico, estos estadísticos poseen distribuciones complejas y no estándar que dependen de funcionales de ciertos procesos estocásticos y sus derivaciones representan una barrera incluso para varios econometristas teóricos. Estas derivaciones están basadas en herramientas estadísticas fundamentales y rigurosas que no son (muy) bien conocidas por econometristas estándar. El presente artículo completa esta brecha al explicar en una forma simple una de estas herramientas fundamentales la cual es el Teorema del Límite Central Funcional. Por lo tanto, este documento analiza los fundamentos y la aplicabilidad de dos versiones del Teorema del Límite Central Funcional dentro del marco de una raíz unitaria con un quiebre estructural. La atención inicial se centra en la estructura probabilística de las series de tiempo propuesta por Phillips (1987a), la cual es aplicada por Perron (1989) para estudiar los efectos de un quiebre estructural (asumido) exógeno sobre la potencia de las pruebas Dickey-Fuller aumentadas y por Zivot y Andrews (1992) para criticar el supuesto de exogeneidad y proponer un método para estimar un punto de quiebre endógeno. Un método sistemático para tratar con aspectos de eficiencia es introducido por Perron y Rodríguez (2003), el cual extiende el enfoque de Mínimos Cuadrados Generalizados para eliminar los componentes determinísticos de Elliot et al. (1996). Se presenta además una aplicación empírica.
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