The shrinkage estimation has proven to be very useful when dealing with a large number of mean parameters. In this dissertation, we consider the problem of simultaneous estimation of multiple variances and construct a shrinkage type, non-parametric estimator. We take the non-parametric empirical Bayes approach by starting with an arbitrary prior on the variances. Under an invariant loss function, the resultant Bayes estimator relies on the marginal cumulative distribution function of the sample variances. Replacing the marginal cdf by the empirical distribution function, we obtain a Non-parametric Empirical Bayes estimator for multiple Variances (NEBV). The proposed estimator converges to the corresponding Bayes version uniformly over a large set. Consequently, the NEBV works well in a post-selection setting. We then apply the NEBV to construct condence intervals for mean parameters in a post-selection setting. It is shown that the intervals based on the NEBV are shortest among all the intervals which guarantee a desired coverage probability. Through real data analysis, we have further shown that the NEBV based intervals lead to the smallest number of discordances, a desirable property when we are faced with the current "replication crisis". / Statistics

The relative power of goodness-of-fit test statistics has long been debated in the literature. Chi-Square type test statistics to determine 'fit' for categorical data are still dominant in the goodness-of-fit arena. Empirical Distribution Function type goodness-of-fit test statistics are known to be relatively more powerful than Chi-Square type test statistics for restricted types of null and alternative distributions. In many practical applications researchers who use a standard Chi-Square type goodness-of-fit test statistic ignore the rank of ordinal classes. This thesis reviews literature in the goodness-of-fit field, with major emphasis on categorical goodness-of-fit tests. The continued use of an asymptotic distribution to approximate the exact distribution of categorical goodness-of-fit test statistics is discouraged. It is unlikely that an asymptotic distribution will produce a more accurate estimation of the exact distribution of a goodness-of-fit test statistic than a Monte Carlo approximation with a large number of simulations. Due to their relatively higher powers for restricted types of null and alternative distributions, several authors recommend the use of Empirical Distribution Function test statistics over nominal goodness-of-fit test statistics such as Pearson's Chi-Square. In-depth power studies confirm the views of other authors that categorical Empirical Distribution Function type test statistics do not have higher power for some common null and alternative distributions. Because of this, it is not sensible to make a conclusive recommendation to always use an Empirical Distribution Function type test statistic instead of a nominal goodness-of-fit test statistic. Traditionally the recommendation to determine 'fit' for multivariate categorical data is to treat categories as nominal, an approach which precludes any gain in power which may accrue from a ranking, should one or more variables be ordinal. The presence of multiple criteria through multivariate data may result in partially ordered categories, some of which have equal ranking. This thesis proposes a modification to the currently available Kolmogorov-Smirnov test statistics for ordinal and nominal categorical data to account for situations of partially ordered categories. The new test statistic, called the Combined Kolmogorov-Smirnov, is relatively more powerful than Pearson's Chi-Square and the nominal Kolmogorov-Smirnov test statistic for some null and alternative distributions. A recommendation is made to use the new test statistic with higher power in situations where some benefit can be achieved by incorporating an Empirical Distribution Function approach, but the data lack a complete natural ordering of categories. The new and established categorical goodness-of-fit test statistics are demonstrated in the analysis of categorical data with brief applications as diverse as familiarity of defence programs, the number of recruits produced by the Merlin bird, a demographic problem, and DNA profiling of genotypes. The results from these applications confirm the recommendations associated with specific goodness-of-fit test statistics throughout this thesis.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-08-02T21:21:48Z No. of bitstreams: 1 ALINE BARBOSA TSUYUGUCHI - DISSERTAÇÃO PPGMAT 2012..pdf: 613833 bytes, checksum: c354cd90842e461c0fb29b0ee5f925d3 (MD5) / Made available in DSpace on 2018-08-02T21:21:48Z (GMT). No. of bitstreams: 1 ALINE BARBOSA TSUYUGUCHI - DISSERTAÇÃO PPGMAT 2012..pdf: 613833 bytes, checksum: c354cd90842e461c0fb29b0ee5f925d3 (MD5) Previous issue date: 2012-02 / CNPq / Neste trabalho estudamos testes de bondade de ajuste para a distribuição Birnbaum-Saunders. Consideramos testes clássicos baseados em função de distribuição empírica (Anderson-Darling, Cramér-von Mises e Kolmogorov-Sminorv) e baseados em função característica empírica. Nos limitamos ao caso onde o vetor de parâmetros é desconhecido e, portanto deverá ser estimado. Apresentamos estudos de simulação para verificar o desempenho das estatísticas de teste em estudo. Além disso, propomos estudos de simulação de Monte Carlo para testes de bondade de ajuste para a distribuição Birnbaum-Saunders com dados com censura tipo II. / In this work we study goodness-of-fit tests for Birnbaum-Saunders distribution. We consider classical tests based on empirical distribution function (Anderson-Darling, Cramér-von Mises e Kolmogorov-Sminorv) and based on empirical characteristic function. We limited this study to the case in which the vector of parameters is unknown and, therefore, must be estimated. We present the simulation studies to verify the performance of the test statistics in study. Also, we propose simulation studies of Monte Carlo for goodness-of-fit test for Birnbaum-Saunders distribution using Type-II censored data.

Au cours des dernières années, des changements importants dans le domaine des assurances et des finances attirent de plus en plus l’attention sur la nécessité d’élaborer un cadre normalisé pour la mesure des risques. Récemment, il y a eu un intérêt croissant de la part des experts en assurance sur l’utilisation de l’espérance conditionnelle des pertes (CTE) parce qu’elle partage des propriétés considérées comme souhaitables et applicables dans diverses situations. En particulier, il répond aux exigences d’une mesure de risque “cohérente”, selon Artzner [2]. Cette thèse représente des contributions à l’inférence statistique en développant des outils, basés sur la convergence des intégrales fonctionnelles, pour l’estimation de la CTE qui présentent un intérêt considérable pour la science actuarielle. Tout d’abord, nous développons un outil permettant l’estimation de la moyenne conditionnelle E[X|X > x], ensuite nous construisons des estimateurs de la CTE, développons la théorie asymptotique nécessaire pour ces estimateurs, puis utilisons la théorie pour construire des intervalles de confiance. Pour la première fois, l’approche de bootstrap non paramétrique est explorée dans cette thèse en développant des nouveaux résultats applicables à la valeur à risque (VaR) et à la CTE. Des études de simulation illustrent la performance de la technique de bootstrap.

Masters of Science / The multivariate two-sample tests provide a means to test the match between two multivariate distributions. Although many tests exist in the literature, relatively little is known about the relative power of these procedures. The studies reported in the thesis contrasts the effectiveness, in terms of power, of seven such tests with a Monte Carlo study. The relative power of the tests was investigated against location, scale, and correlation alternatives. Samples were drawn from bivariate exponential, normal and uniform populations. Results from the power studies show that there is no single test which is the most powerful in all situations. The use of particular test statistics is recommended for specific alternatives. A possible supplementary non-parametric graphical procedure, such as the Depth-Depth plot, can be recommended for diagnosing possible differences between the multivariate samples, if the null hypothesis is rejected. As an example of the utility of the procedures for real data, the multivariate two-sample tests were applied to photometric data of twenty galactic globular clusters. The results from the analyses support the recommendations associated with specific test statistics.

This thesis concentrates on different approaches of solving decision making problems with an aspect of randomness. The basic methodologies of converting stochastic optimization problems to deterministic optimization problems are described. The proximity of solution of a problem and its empirical counterpart is shown. The empirical counterpart is used when we don't know the distribution of the random elements of the former problem. The distribution with heavy tails, stable distribution and their relationship is described. The stochastic dominance and the possibility of defining problems with stochastic dominance is introduced. The proximity of solution of problem with second order stochastic dominance and the solution of its empirical counterpart is proven. A portfolio management problem with second order stochastic dominance is solved by solving the equivalent empirical problem. Powered by TCPDF (www.tcpdf.org)

In this thesis we present results about some eigenvalue statistics of the beta-Hermite ensembles, both in the classical cases corresponding to beta = 1, 2, 4, that is the Gaussian orthogonal ensemble (consisting of real symmetric matrices), the Gaussian unitary ensemble (consisting of complex Hermitian matrices) and the Gaussian symplectic ensembles (consisting of quaternionic self-dual matrices) respectively. We also look at the less explored general beta-Hermite ensembles (consisting of real tridiagonal symmetric matrices). Specifically we look at the empirical distribution function and two different scalings of the largest eigenvalue. The results we present relating to these statistics are the convergence of the empirical distribution function to the semicircle law, the convergence of the scaled largest eigenvalue to the Tracy-Widom distributions, and with a different scaling, the convergence of the largest eigenvalue to 1. We also use simulations to illustrate these results. For the Gaussian unitary ensemble, we present an expression for its level density. To aid in understanding the Gaussian symplectic ensemble we present properties of the eigenvalues of quaternionic matrices. Finally, we prove a theorem about the symmetry of the order statistic of the eigenvalues of the beta-Hermite ensembles. / I denna kandidatuppsats presenterar vi resultat om några olika egenvärdens-statistikor från beta-Hermite ensemblerna, först i de klassiska fallen då beta = 1, 2, 4, det vill säga den gaussiska ortogonala ensemblen (bestående av reella symmetriska matriser), den gaussiska unitära ensemblen (bestående av komplexa hermitiska matriser) och den gaussiska symplektiska ensemblen (bestående av kvaternioniska själv-duala matriser). Vi tittar även på de mindre undersökta generella beta-Hermite ensemblerna (bestående av reella symmetriska tridiagonala matriser). Specifikt tittar vi på den empiriska fördelningsfunktionen och två olika normeringar av det största egenvärdet. De resultat vi presenterar för dessa statistikor är den empiriska fördelningsfunktionens konvergens mot halvcirkel-fördelningen, det normerade största egenvärdets konvergens mot Tracy-Widom fördelningen, och, med en annan normering, största egenvärdets konvergens mot 1. Vi illustrerar även dessa resultat med hjälp av simuleringar. För den gaussiska unitära ensemblen presenterar vi ett uttryck för dess nivåtäthet. För att underlätta förståelsen av den gaussiska symplektiska ensemblen presenterar vi egenskaper hos egenvärdena av kvaternioniska matriser. Slutligen bevisar vi en sats om symmetrin hos ordningsstatistikan av egenvärdena av beta-Hermite ensemblerna.

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2017-02-21T15:12:08Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação_JoséVianneyMendonçaDeAlencastroJr.pdf: 648691 bytes, checksum: f2fbc98e547f0284f5aef34aee9249ca (MD5) / Made available in DSpace on 2017-02-21T15:12:08Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Dissertação_JoséVianneyMendonçaDeAlencastroJr.pdf: 648691 bytes, checksum: f2fbc98e547f0284f5aef34aee9249ca (MD5) Previous issue date: 2016-09-09 / A lei de Newcomb-Benford, também conhecida como a lei do dígito mais significativo, foi descrita pela primeira vez por Simon Newcomb, sendo apenas embasada estatisticamente após 57 anos pelo físico Frank Benford. Essa lei rege grandezas naturalmente aleatórias e tem sido utilizada por várias áreas como forma de selecionar e validar diversos tipos de dados. Em nosso trabalho tivemos como primeiro objetivo propor o uso de um método substituto ao qui-quadrado, sendo este atualmente o método comumente utilizado pela literatura para verificação da conformidade da Lei de Newcomb-Benford. Fizemos isso pois em uma massa de dados com uma grande quantidade de amostras o método qui-quadrado tende a sofrer de um problema estatístico conhecido por excesso de poder, gerando assim resultados do tipo falso negativo na estatística. Dessa forma propomos a substituição do método qui-quadrado pelo método de Kolmogorov-Smirnov baseado na Função de Distribuição Empírica para análise da conformidade global, pois esse método é mais robusto não sofrendo do excesso de poder e também é mais fiel à definição formal da Lei de Benford, já que o mesmo trabalha considerando as mantissas ao invés de apenas considerar dígitos isolados. Também propomos investigar um intervalo de confiança para o Kolmogorov-Smirnov baseando-nos em um qui-quadrado que não sofre de excesso de poder por se utilizar o Bootstraping. Em dois artigos publicados recentemente, dados de exoplanetas foram analisados e algumas grandezas foram declaradas como conformes à Lei de Benford. Com base nisso eles sugerem que o conhecimento dessa conformidade possa ser usado para uma análise na lista de objetos candidatos, o que poderá ajudar no futuro na identificação de novos exoplanetas nesta lista. Sendo assim, um outro objetivo de nosso trabalho foi explorar diversos bancos e catálogos de dados astronômicos em busca de grandezas, cuja a conformidade à lei do dígito significativo ainda não seja conhecida a fim de propor aplicações práticas para a área das ciências astronômicas. / The Newcomb-Benford law, also known as the most significant digit law, was described for the first time by astronomer and mathematician Simon Newcomb. This law was just statistically grounded after 57 years after the Newcomb’s discovery. This law governing naturally random greatness and, has been used by many knowledge areas to validate several kind of data. In this work, the first goal is propose a substitute of qui-square method. The qui-square method is the currently method used in the literature to verify the Newcomb-Benford Law’s conformity. It’s necessary because in a greatness with a big quantity of samples, the qui-square method can has false negatives results. This problem is named Excess of Power. Because that, we proposed to use the Kolmogorov-Smirnov method based in Empirical Distribution Function (EDF) to global conformity analysis. Because this method is more robust and not suffering of the Excess of Power problem. The Kolmogorov-Smirnov method also more faithful to the formal definition of Benford’s Law since the method working considering the mantissas instead of single digits. We also propose to invetigate a confidence interval for the Kolmogorov-Smirnov method based on a qui-square with Bootstrapping strategy which doesn’t suffer of Excess of Power problem. Recently, two papers were published. I this papaers exoplanets data were analysed and some greatness were declared conform to a Newcomb-Benford distribution. Because that, the authors suggest that knowledge of this conformity can be used for help in future to indentify new exoplanets in the candidates list. Therefore, another goal of this work is explorer a several astronomicals catalogs and database looking for greatness which conformity of Benford’s law is not known yet. And after that , the authors suggested practical aplications for astronomical sciences area.