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Tópicos em penalidades exatas diferenciáveis; Topics in differentiable exact penalties

Fukuda, Ellen Hidemi
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Tese de Doutorado Formato: application/pdf
Publicado em 11/03/2011 Português
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Durante as décadas de 70 e 80, desenvolveram-se métodos baseados em penalidades exatas diferenciáveis para resolver problemas de otimização não linear com restrições. Uma desvantagem dessas penalidades é que seus gradientes contêm termos de segunda ordem em suas fórmulas, o que impede a utilização de métodos do tipo Newton para resolver o problema. Para contornar essa dificuldade, utilizamos uma ideia de construção de penalidade exata para desigualdades variacionais, introduzida recentemente por André e Silva. Essa construção consiste em incorporar um estimador de multiplicadores, proposto por Glad e Polak, no lagrangiano aumentado para desigualdades variacionais. Nesse trabalho, estendemos o estimador de multiplicadores para restrições gerais de igualdade e desigualdade, e enfraquecemos a hipótese de regularidade. Como resultado, obtemos uma função penalidade exata continuamente diferenciável e uma nova reformulação do sistema KKT associado a problemas não lineares. A estrutura dessa reformulação permite a utilização do método de Newton semi-suave, e a taxa de convergência local superlinear pode ser provada. Além disso, verificamos que a penalidade exata construída pode ser usada para globalizar o método...

Time in the stair-climbing test as a predictor of thoracotomy postoperative complications

Ambrozin, Alexandre Ricardo Pepe; Cataneo, Daniele Cristina; Arruda, Karine Aparecida; Cataneo, Antônio José Maria
Fonte: Universidade Estadual Paulista Publicador: Universidade Estadual Paulista
Tipo: Artigo de Revista Científica Formato: 1093-1097
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Objectives: The stair-climbing test as measured in meters or number of steps has been proposed to predict the risk of postoperative complications. The study objective was to determine whether the stair-climbing time can predict the risk of postoperative complications. Methods: Patients aged more than 18 years with a recommendation of thoracotomy for lung resection were included in the study. Spirometry was performed according to the criteria by the American Thoracic Society. The stair-climbing test was performed on shaded stairs with a total of 12.16 m in height, and the stair-climbing time in seconds elapsed during the climb of the total height was measured. The accuracy test was applied to obtain stair-climbing time predictive values, and the receiver operating characteristic curve was calculated. Variables were tested for association with postoperative cardiopulmonary complications using the Student t test for independent populations, the Mann-Whitney test, and the chi-square or Fisher exact test. Logistic regression analysis was performed. Results: Ninety-eight patients were evaluated. Of these, 27 showed postoperative complications. Differences were found between the groups for age and attributes obtained from the stair-climbing test. The cutoff point for stair-climbing time obtained from the receiver operating characteristic curve was 37.5 seconds. No differences were found between the groups for forced expiratory volume in 1 second. In the logistic regression...

The weak patch test for nonhomogeneous materials modeled with graded finite elements

Paulino,Glaucio H.; Kim,Jeong-Ho
Fonte: Associação Brasileira de Engenharia e Ciências Mecânicas - ABCM Publicador: Associação Brasileira de Engenharia e Ciências Mecânicas - ABCM
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/03/2007 Português
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Functionally graded materials have an additional length scale associated to the spatial variation of the material property field which competes with the usual geometrical length scale of the boundary value problem. By considering the length scale of nonhomogeneity, this paper presents the weak patch test (rather than the standard one) of the graded element for nonhomogeneous materials to assess convergence of the finite element method (FEM). Both consistency (as the size of elements approach zero, the FEM approximation represents the exact solution) and stability (spurious mechanisms are avoided) conditions are addressed. The specific graded elements considered here are isoparametric quadrilaterals (e.g. 4, 8 and 9-node) considering two dimensional plane and axisymmetric problems. The finite element approximate solutions are compared with exact solutions for nonhomogeneous materials.

Exact Nonparametric Two-Sample Homogeneity Tests for Possibly Discrete Distributions.

DUFOUR, Jean-Marie; FARHAT, Abdeljelil
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 236458 bytes; application/pdf
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In this paper, we study several tests for the equality of two unknown distributions. Two are based on empirical distribution functions, three others on nonparametric probability density estimates, and the last ones on differences between sample moments. We suggest controlling the size of such tests (under nonparametric assumptions) by using permutational versions of the tests jointly with the method of Monte Carlo tests properly adjusted to deal with discrete distributions. We also propose a combined test procedure, whose level is again perfectly controlled through the Monte Carlo test technique and has better power properties than the individual tests that are combined. Finally, in a simulation experiment, we show that the technique suggested provides perfect control of test size and that the new tests proposed can yield sizeable power improvements.; Dans ce texte, nous étudions plusieurs tests pour l’égalité de deux distributions inconnues. Deux de ces tests sont basés sur des fonctions de distribution empiriques, trois autres sur des estimateurs non paramétriques de fonctions de densité et les trois derniers sur des moments empiriques. Nous proposons de contrôler la taille des tests (sous des hypothèses non paramétriques) en employant des versions permutationnelles de ces tests conjointement avec la méthode des tests de Monte Carlo ajustée pour tenir compte de la possibilité de distributions discontinues. Nous proposons aussi une méthode pour combiner plusieurs de ces tests...

Méthodes d’inférence exactes pour un modèle de régression avec erreurs AR(2) gaussiennes

DUFOUR, Jean-Marie; NEIFAR, Malika
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 4601120 bytes; application/pdf
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Ce texte propose des méthodes d’inférence exactes (tests et régions de confiance) sur des modèles de régression linéaires avec erreurs autocorrélées suivant un processus autorégressif d’ordre deux [AR(2)], qui peut être non stationnaire. L’approche proposée est une généralisation de celle décrite dans Dufour (1990) pour un modèle de régression avec erreurs AR(1) et comporte trois étapes. Premièrement, on construit une région de confiance exacte pour le vecteur des coefficients du processus autorégressif (φ). Cette région est obtenue par inversion de tests d’indépendance des erreurs sur une forme transformée du modèle contre des alternatives de dépendance aux délais un et deux. Deuxièmement, en exploitant la dualité entre tests et régions de confiance (inversion de tests), on détermine une région de confiance conjointe pour le vecteur φ et un vecteur d’intérêt M de combinaisons linéaires des coefficients de régression du modèle. Troisièmement, par une méthode de projection, on obtient des intervalles de confiance «marginaux» ainsi que des tests à bornes exacts pour les composantes de M. Ces méthodes sont appliquées à des modèles du stock de monnaie (M2) et du niveau des prix (indice implicite du PNB) américains; In this paper...

Exact side effects for interprocedural dependence analysis

Tang, Peiyi
Fonte: Universidade Nacional da Austrália Publicador: Universidade Nacional da Austrália
Tipo: Working/Technical Paper Formato: 284346 bytes; 356 bytes; application/pdf; application/octet-stream
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Exact side effects of array references in subroutines are essential for exact interprocedural dependence analysis. To summarize the side effects of multiple array references, a collective representation of all the array elements accessed is needed. So far all existing forms of collective summary of side effects of multiple array references are approximate. In this paper, we present a method to represent the exact side effects of multiple array references in the form of the projection of a single integer programming problem. Since the representation is collective, it dramatically reduces the number of pairs of dependences checking compared with other methods of exact interprocedural analysis. The representation of the exact side effects proposed in this paper can be used by the Omega test to support the exact interprocedural dependence analysis in parallelizing compilers.; no

Rank test of location optimal for hyperbolic secant distribution

Kravchuk, O.
Fonte: Marcel Dekker Inc Publicador: Marcel Dekker Inc
Tipo: Artigo de Revista Científica
Publicado em //2005 Português
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There are at least two reasons for a symmetric, unimodal, diffuse tailed hyperbolic secant distribution to be interesting in real-life applications. It displays one of the common types of non normality in natural data and is closely related to the logistic and Cauchy distributions that often arise in practice. To test the difference in location between two hyperbolic secant distributions, we develop a simple linear rank test with trigonometric scores. We investigate the small-sample and asymptotic properties of the test statistic and provide tables of the exact null distribution for small sample sizes. We compare the test to the Wilcoxon two-sample test and show that, although the asymptotic powers of the tests are comparable, the present test has certain practical advantages over the Wilcoxon test.; O. Y. Kravchuk

Exact optimal and adaptive inference in regression models under heteroskedasticity and non-normality of unknown forms

Dufour, Jean-Marie; Taamouti, Abderrahim
Fonte: Universidade Carlos III de Madrid Publicador: Universidade Carlos III de Madrid
Tipo: Trabalho em Andamento Formato: application/pdf
Publicado em /11/2008 Português
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In this paper, we derive simple point-optimal sign-based tests in the context of linear and nonlinear regression models with fixed regressors. These tests are exact, distribution-free, robust against heteroskedasticity of unknown form, and they may be inverted to obtain confidence regions for the vector of unknown parameters. Since the point-optimal sign tests depend on the alternative hypothesis, we propose an adaptive approach based on split-sample techniques in order to choose an alternative such that the power of point-optimal sign tests is close to the power envelope. The simulation results show that when using approximately 10% of sample to estimate the alternative and the rest to calculate the test statistic, the power of point-optimal sign test is typically close to the power envelope. We present a Monte Carlo study to assess the performance of the proposed “quasi”-point-optimal sign test by comparing its size and power to those of some common tests which are supposed to be robust against heteroskedasticity. The results show that our procedures are superior.

Exact goodness-of-fit tests for censored dats

Grané, Aurea
Fonte: Universidade Carlos III de Madrid Publicador: Universidade Carlos III de Madrid
Tipo: Trabalho em Andamento Formato: text/plain; application/octet-stream; application/octet-stream; application/octet-stream; application/pdf
Publicado em /05/2009 Português
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The statistic introduced in Fortiana and Grané (2003) is modified so that it can be used to test the goodness-of-fit of a censored sample, when the distribution function is fully specified. Exact and asymptotic distributions of three modified versions of this statistic are obtained and exact critical values are given for different sample sizes. Empirical power studies show the good performance of these statistics in detecting symmetrical alternatives.

Finite-Sample Inference Methods for Simultaneous Equations and Models with Unobserved and Generated Regressors

DUFOUR, Jean-Marie; JASIAK, Joanna
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 242575 bytes; application/pdf
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We propose finite sample tests and confidence sets for models with unobserved and generated regressors as well as various models estimated by instrumental variables methods. The validity of the procedures is unaffected by the presence of identification problems or \"weak instruments\", so no detection of such problems is required. We study two distinct approaches for various models considered by Pagan (1984). The first one is an instrument substitution method which generalizes an approach proposed by Anderson and Rubin (1949) and Fuller (1987) for different (although related) problems, while the second one is based on splitting the sample. The instrument substitution method uses the instruments directly, instead of generated regressors, in order to test hypotheses about the \"structural parameters\" of interest and build confidence sets. The second approach relies on \"generated regressors\", which allows a gain in degrees of freedom, and a sample split technique. For inference about general possibly nonlinear transformations of model parameters, projection techniques are proposed. A distributional theory is obtained under the assumptions of Gaussian errors and strictly exogenous regressors. We show that the various tests and confidence sets proposed are (locally) \"asymptotically valid\" under much weaker assumptions. The properties of the tests proposed are examined in simulation experiments. In general...

Testing Mean-Variance Efficiency in CAPM with Possibly Non-Gaussian Errors : An Exact Simulation-Based Approach

BEAULIEU, Marie-Claude; DUFOUR, Jean-Marie; KHALAF, Lynda
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 400691 bytes; application/pdf
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In this paper we propose exact likelihood-based mean-variance efficiency tests of the market portfolio in the context of Capital Asset Pricing Model (CAPM), allowing for a wide class of error distributions which include normality as a special case. These tests are developed in the frame-work of multivariate linear regressions (MLR). It is well known however that despite their simple statistical structure, standard asymptotically justified MLR-based tests are unreliable. In financial econometrics, exact tests have been proposed for a few specific hypotheses [Jobson and Korkie (Journal of Financial Economics, 1982), MacKinlay (Journal of Financial Economics, 1987), Gib-bons, Ross and Shanken (Econometrica, 1989), Zhou (Journal of Finance 1993)], most of which depend on normality. For the gaussian model, our tests correspond to Gibbons, Ross and Shanken’s mean-variance efficiency tests. In non-gaussian contexts, we reconsider mean-variance efficiency tests allowing for multivariate Student-t and gaussian mixture errors. Our framework allows to cast more evidence on whether the normality assumption is too restrictive when testing the CAPM. We also propose exact multivariate diagnostic checks (including tests for multivariate GARCH and mul-tivariate generalization of the well known variance ratio tests) and goodness of fit tests as well as a set estimate for the intervening nuisance parameters. Our results [over five-year subperiods] show the following: (i) multivariate normality is rejected in most subperiods...

Exact Skewness-Kurtosis Tests for Multivariate Normality and Goodness-of-fit in Multivariate Regressions with Application to Asset Pricing Models

DUFOUR, Jean-Marie; KHALAF, Lynda; BEAULIEU, Marie-Claude
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 225374 bytes; application/pdf
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We study the problem of testing the error distribution in a multivariate linear regression (MLR) model. The tests are functions of appropriately standardized multivariate least squares residuals whose distribution is invariant to the unknown cross-equation error covariance matrix. Empirical multivariate skewness and kurtosis criteria are then compared to simulation-based estimate of their expected value under the hypothesized distribution. Special cases considered include testing multivariate normal, Student t; normal mixtures and stable error models. In the Gaussian case, finite-sample versions of the standard multivariate skewness and kurtosis tests are derived. To do this, we exploit simple, double and multi-stage Monte Carlo test methods. For non-Gaussian distribution families involving nuisance parameters, confidence sets are derived for the the nuisance parameters and the error distribution. The procedures considered are evaluated in a small simulation experi-ment. Finally, the tests are applied to an asset pricing model with observable risk-free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over five-year subperiods from 1926-1995.; Dans cet article, nous proposons des tests sur la forme de la distribution des erreurs dans un modèle de régression linéaire multivarié (RLM). Les tests que nous développons sont fonction des résidus obtenus par moindres carrés multivariés...

Simulation-Based Finite and Large Sample Tests in Multivariate Regressions.

DUFOUR, Jean-Marie; KHALAF, Lynda
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 762828 bytes; application/pdf
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In the context of multivariate linear regression (MLR) models, it is well known that commonly employed asymptotic test criteria are seriously biased towards overrejection. In this paper, we propose a general method for constructing exact tests of possibly nonlinear hypotheses on the coefficients of MLR systems. For the case of uniform linear hypotheses, we present exact distributional invariance results concerning several standard test criteria. These include Wilks' likelihood ratio (LR) criterion as well as trace and maximum root criteria. The normality assumption is not necessary for most of the results to hold. Implications for inference are two-fold. First, invariance to nuisance parameters entails that the technique of Monte Carlo tests can be applied on all these statistics to obtain exact tests of uniform linear hypotheses. Second, the invariance property of the latter statistic is exploited to derive general nuisance-parameter-free bounds on the distribution of the LR statistic for arbitrary hypotheses. Even though it may be difficult to compute these bounds analytically, they can easily be simulated, hence yielding exact bounds Monte Carlo tests. Illustrative simulation experiments show that the bounds are sufficiently tight to provide conclusive results with a high probability. Our findings illustrate the value of the bounds as a tool to be used in conjunction with more traditional simulation-based test methods (e.g....

Exact Tests for Contemporaneous Correlation of Disturbances in Seemingly Unrelated Regressions.

DUFOUR, Jean-Marie; KHALAF, Lynda
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 425709 bytes; application/pdf
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This paper proposes finite-sample procedures for testing the SURE specification in multi-equation regression models, i.e. whether the disturbances in different equations are contemporaneously uncorrelated or not. We apply the technique of Monte Carlo (MC) tests [Dwass (1957), Barnard (1963)] to obtain exact tests based on standard LR and LM zero correlation tests. We also suggest a MC quasi-LR (QLR) test based on feasible generalized least squares (FGLS). We show that the latter statistics are pivotal under the null, which provides the justification for applying MC tests. Furthermore, we extend the exact independence test proposed by Harvey and Phillips (1982) to the multi-equation framework. Specifically, we introduce several induced tests based on a set of simultaneous Harvey/Phillips-type tests and suggest a simulation-based solution to the associated combination problem. The properties of the proposed tests are studied in a Monte Carlo experiment which shows that standard asymptotic tests exhibit important size distortions, while MC tests achieve complete size control and display good power. Moreover, MC-QLR tests performed best in terms of power, a result of interest from the point of view of simulation-based tests. The power of the MC induced tests improves appreciably in comparison to standard Bonferroni tests and...

Markovian Processes, Two-Sided Autoregressions and Finite-Sample Inference for Stationary and Nonstationary Autoregressive Processes.

DUFOUR, Jean-Marie; TORRES, Olivier
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 665597 bytes; application/pdf
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In this paper, we develop finite-sample inference procedures for stationary and nonstationary autoregressive (AR) models. The method is based on special properties of Markov processes and a split-sample technique. The results on Markovian processes (intercalary independence and truncation) only require the existence of conditional densities. They are proved for possibly nonstationary and/or non-Gaussian multivariate Markov processes. In the context of a linear regression model with AR(1) errors, we show how these results can be used to simplify the distributional properties of the model by conditioning a subset of the data on the remaining observations. This transformation leads to a new model which has the form of a two-sided autoregression to which standard classical linear regression inference techniques can be applied. We show how to derive tests and confidence sets for the mean and/or autoregressive parameters of the model. We also develop a test on the order of an autoregression. We show that a combination of subsample-based inferences can improve the performance of the procedure. An application to U.S. domestic investment data illustrates the method.; Dans cet article, nous proposons des procédures d’inférence valides à distance finie pour des modèles autorégressifs (AR) stationnaires et non stationnaires. La méthode suggérée est fondée sur des propriétés particulières des processus markoviens combinées à une technique de subdivision d’échantillon. Les résultats sur les processus de Markov (indépendance intercalaire...

Simulation-Based Finite-Sample Tests for Heteroskedasticity and ARCH Effects.

DUFOUR, Jean-Marie; KHALAF, Lynda; BERNARD, Jean-Thomas; GENEST, Ian
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 342447 bytes; application/pdf
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A wide range of tests for heteroskedasticity have been proposed in the econometric and statistics literature. Although a few exact homoskedasticity tests are available, the commonly employed procedures are quite generally based on asymptotic approximations which may not provide good size control in finite samples. There has been a number of recent studies that seek to improve the reliability of common heteroskedasticity tests using Edgeworth, Bartlett, jackknife and bootstrap methods. Yet the latter remain approximate. In this paper, we describe a solution to the problem of controlling the size of homoskedasticity tests in linear regression contexts. We study procedures based on the standard test statistics [e.g., the Goldfeld-Quandt, Glejser, Bartlett, Cochran, Hartley, Breusch-Pagan-Godfrey, White and Szroeter criteria] as well as tests for autoregressive conditional heteroskedasticity (ARCH-type models). We also suggest several extensions of the existing procedures (sup-type of combined test statistics) to allow for unknown breakpoints in the error variance. We exploit the technique of Monte Carlo tests to obtain provably exact p-values, for both the standard and the new tests suggested. We show that the MC test procedure conveniently solves the intractable null distribution problem...

Monte Carlo Tests with Nuisance Parameters: A General Approach to Finite-Sample Inference and Nonstandard Asymptotics

DUFOUR, Jean-Marie
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 322116 bytes; application/pdf
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The technique of Monte Carlo (MC) tests [Dwass (1957), Barnard (1963)] provides an attractive method of building exact tests from statistics whose finite sample distribution is intractable but can be simulated (provided it does not involve nuisance parameters). We extend this method in two ways: first, by allowing for MC tests based on exchangeable possibly discrete test statistics; second, by generalizing the method to statistics whose null distributions involve nuisance parameters (maximized MC tests, MMC). Simplified asymptotically justified versions of the MMC method are also proposed and it is shown that they provide a simple way of improving standard asymptotics and dealing with nonstandard asymptotics (e.g., unit root asymptotics). Parametric bootstrap tests may be interpreted as a simplified version of the MMC method (without the general validity properties of the latter).

Exact Multivariate Tests of Asset Pricing Models with Stable Asymmetric Distributions

BEAULIEU, Marie-Claude; DUFOUR, Jean-Marie; KHALAF, Lynda
Fonte: Université de Montréal Publicador: Université de Montréal
Tipo: Artigo de Revista Científica Formato: 204421 bytes; application/pdf
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In this paper, we propose exact inference procedures for asset pricing models that can be formulated in the framework of a multivariate linear regression (CAPM), allowing for stable error distributions. The normality assumption on the distribution of stock returns is usually rejected in empirical studies, due to excess kurtosis and asymmetry. To model such data, we propose a comprehensive statistical approach which allows for alternative - possibly asymmetric - heavy tailed distributions without the use of large-sample approximations. The methods suggested are based on Monte Carlo test techniques. Goodness-of-fit tests are formally incorporated to ensure that the error distributions considered are empirically sustainable, from which exact confidence sets for the unknown tail area and asymmetry parameters of the stable error distribution are derived. Tests for the efficiency of the market portfolio (zero intercepts) which explicitly allow for the presence of (unknown) nuisance parameter in the stable error distribution are derived. The methods proposed are applied to monthly returns on 12 portfolios of the New York Stock Exchange over the period 1926-1995 (5 year subperiods). We find that stable possibly skewed distributions provide statistically significant improvement in goodness-of-fit and lead to fewer rejections of the efficiency hypothesis.

Spinning massive test particles in cosmological and general static spherically symmetric spacetimes

Asenjo, Felipe A.; Hojman Guiñerman, Sergio Andrés; Zalaquett, Nicolás
Fonte: IOP Publishing Publicador: IOP Publishing
Tipo: Artículo de revista
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Artículo de publicación ISI; A Lagrangian formalism is used to study the motion of a spinning massive particle in Friedmann–Robertson–Walker and G¨odel spacetimes, as well as in a general Schwarzschild-like spacetime and in static spherically symmetric conformally flat spacetimes. Exact solutions for the motion of the particle and general exact expressions for the momenta and velocities are displayed for different cases. In particular, the solution for the motion in spherically symmetric metrics is presented in the equatorial plane. The exact solutions are found using constants of motion of the particle, namely its mass, its spin, its angular momentum, and a fourth constant, which is its energy when the metric is time-independent, and a different constant otherwise. These constants are associated to Killing vectors. In the case of the motion on the Friedmann– Robertson–Walker metric, a new constant of motion is found. This is the fourth constant which generalizes previously known results obtained for spinless particles. In the case of general Schwarzschild-like spacetimes, our results allow for the exploration of the case of the Reissner–Nordstrom–(Anti)de Sitter metric. Finally, for the case of the conformally flat spacetimes...

Simulation Study on Exchangeability and Significant Test on Survey Data

Cao, Yong
Fonte: Universidade Duke Publicador: Universidade Duke
Tipo: Tese de Doutorado
Publicado em //2015 Português
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The two years of Master of Science in Statistical and Economic Modeling program is the most rewarding time ever in my life. This thesis acts as a portfolio of project and applied experience while I am enrolled in the Master of Science in Statistical and Economic Modeling program. This thesis will summarize my graduate study in two parts: Simulation Study of Exchangeability for Binary Data, and Summary of Summer Internship at Center for Responsible Lending. The project of Simulation Study of Exchangeability for Binary Data contains materials from a team project, which jointly performed by Sheng Jiang, Xuan Sun and me. Abstracts for both projects are below in order.

(1) Simulation Study of Exchangeability for Binary Data

To investigate tractable Bayesian tests on exchangeability, this project considers special cases of nonexchangeable random sequences: Markov chains. Asymptotic results of Bayes factor (BF) are derived. When null hypothesis is true, Bayes Factor in favor of the null goes to infinity at geometric rate (true odds is not one half). When null hypothesis is not true, Bayes Factor in favor of the null goes to 0 faster than geometric rate. The results are robust under misspecifications. Simulation studies are employed to see the performance of the test when the sample size is small...