Neste trabalho, alguns resultados, tais como, função geradora de momentos, relações de recorrência para os momentos e alguns teoremas da classe de distribuições em séries de potencias modificadas (MPSD) proposta por Gupta (1974) e da classe de distribuições em séries de potências modificadas inflacionadas (IMPSD) tanto em um ponto diferente de zero como no ponto zero são apresentados. Uma aplicação do Modelo Poisson padrão, do modelo binomial negativo padrão e dos modelos inflacionados de zeros para dados de contagem, ZIP e ZINB, utilizando-se as técnicas dos MLGs, foi realizada para dois conjuntos de dados reais juntamente com o gráfico normal de probabilidade com envelopes simulados. Também foi proposta a distribuição Weibull binomial negativa (WNB) que é bastante flexível em análise de dados positivos e foram estudadas algumas de suas propriedades matemáticas. Esta é uma importante alternativa para os modelos Weibull e Weibull geométrica, sub-modelos da WNB. A demostração de que a densidade da distribuição Weibull binomial negativa pode ser expressa como uma mistura de densidades Weibull é apresentada. Fornecem-se, também, seus momentos, função geradora de momentos, gráficos da assimetria e curtose, expressoes expl´citas para os desvios médios, curvas de Bonferroni e Lorenz, função quantílica, confiabilidade e entropia, a densidade da estat´stica de ordem e expressões explícita para os momentos da estatística de ordem. O método de máxima verossimilhança é usado para estimar os parametros do modelo. A matriz de informação esperada ´e derivada. A utilidade da distribuição WNB está ilustrada na an´alise de dois conjuntos de dados reais. / In this paper, some result such as moments generating function, recurrence relations for moments and some theorems of the class of modified power series distributions (MPSD) proposed by Gupta (1974) and of the class of inflated modified power series distributions (IMPSD) both at a different point of zero as the zero point are presented. The standard Poisson model, the standard negative binomial model and zero inflated models for count data, ZIP and ZINB, using the techniques of the GLMs, were used to analyse two real data sets together with the normal plot with simulated envelopes. The new distribution Weibull negative binomial (WNB) was proposed. Some mathematical properties of the WNB distribution which is quite flexible in analyzing positive data were studied. It is an important alternative model to the Weibull, and Weibull geometric distributions as they are sub-models of WNB. We demonstrate that the WNB density can be expressed as a mixture of Weibull densities. We provide their moments, moment generating function, plots of the skewness and kurtosis, explicit expressions for the mean deviations, Bonferroni and Lorenz curves, quantile function, reliability and entropy, the density of order statistics and explicit expressions for the moments of order statistics. The method of maximum likelihood is used for estimating the model parameters. The expected information matrix is derived. The usefulness of the new distribution is illustrated in two analysis of real data sets.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-04-23T14:13:46Z No. of bitstreams: 2 Dissertação - Tiago Carvalhais de Oliveira - 2014.pdf: 2283233 bytes, checksum: 6e0744d8d8b25db17ecedc5ed707eee8 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-04-23T14:16:09Z (GMT) No. of bitstreams: 2 Dissertação - Tiago Carvalhais de Oliveira - 2014.pdf: 2283233 bytes, checksum: 6e0744d8d8b25db17ecedc5ed707eee8 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-04-23T14:16:09Z (GMT). No. of bitstreams: 2 Dissertação - Tiago Carvalhais de Oliveira - 2014.pdf: 2283233 bytes, checksum: 6e0744d8d8b25db17ecedc5ed707eee8 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2014-02-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Caterpillars defoliators of soybean occurs in different times and feed on different parts of the plant, mainly in flowers and pods. During the growing seasons of 2011/2012 and 2012/2013, soybean fields were sampled in different crop phenological stages with the objective of knowing the population fluctuation of the main species of caterpillar defoliating soybeans and to determine the within field spatial distribution of the species of the Plusiinae complex, to describe how species colonize the soybean field. Five commercial fields submitted to different management practices were monitored. Sampling meshes were fabricated with equidistant points 50 meters. The areas were sampled weekly from the occurrence of the first defoliating caterpillars, until close to harvest. The sampling method used was ground cloth, with one row-meter. It was quantified all caterpillars. Data were submitted to statistical analysis using the following SAS procedures: Proc Univariate, Proc Means, Proc Freq. To test what statistical distribution fit to caterpillars population the procedure Proc Genmode was used. Maps representing the localization of samples of caterpillars in soybean fields were constructed. The species of Plusiinae predominated in the area, corresponding to 52, 04% in 2011/2012 e 48, 53% in 2012/2013 of the total of caterpillars found in the reproductive phase of soybean. The distribution of the complex of Plusiinae within field is random and do not present border effect. In this study, while the population fits the negative binomial distribution maps clearly indicate that the caterpillars are randomly dispersed in soybean fields. / As lagartas desfolhadoras da soja ocorrem em épocas distintas e atacam várias partes da planta, principalmente as folhas, flores e frutos. Durante os anos agrícolas de 2011/2012 e 2012/2013, campos de soja foram amostrados em diversos estádios fenológicos da planta com o objetivo de conhecer a flutuação populacional das principais espécies de lagartas desfolhadoras da cultura, e determinar a distribuição espacial em lavoura, espécies pertencentes ao complexo de Plusiinae, para compreender a forma que este complexo coloniza os campos de soja. Foram monitoradas cinco lavouras comerciais, submetidas a diferentes condições de manejo. Foram confeccionadas malhas de amostragem, com pontos eqüidistantes de 50 metros. As áreas foram amostradas semanalmente a partir da ocorrência das primeiras lagartas desfolhadoras, até próximo à colheita. O método de amostragem utilizado foi o pano de batida, com o qual um metro linear foi amostrado. Foram quantificadas todas as lagartas presentes nas amostragens dos campos selecionados. Para a realização das inferências, dos dados foram submetidos á análises estatísticas descritivas utilizando os seguintes procedimentos dos Sas, Proc Univariate, Proc Means e Proc Freq. Para verificar qual a distribuição estatística a que a população de lagartas melhor se ajustava foi utilizado o procedimento Proc Genmode. Mapas representativos da localização das lagartas do complexo Plusiinae no campo foram confeccionados. As espécies de Plusiinae foram as lagartas predominantes, compondo 52,04% em 2011/12 e 48,53% em 2012/13, do total de lagartas encontradas. As flutuações populacionais demonstraram que o pico das lagartas desfolhadoras ocorre principalmente na fase reprodutiva da cultura da soja. A distribuição de lagartas do complexo Plusiinae em lavoura é aleatória, não apresentando efeitos de bordas. Neste estudo, enquanto a população se ajusta à distribuição binomial negativa os mapas indicam claramente que as lagartas estão aleatoriamente dispersas nos campos de soja.

The transformed rejection method, a combination of inversion and rejection, which can be applied to various continuous distributions, is well suited to generate binomial random variates as well. The resulting algorithms are simple and fast, and need only a short set-up. Among the many possible variants two algorithms are described and tested: BTRS a short but nevertheless fast rejection algorithm and BTRD which is more complicated as the idea of decomposition is utilized. For BTRD the average number of uniforms required to return one binomial deviate lies between 2.5 and 1.4 which is considerably lower than for any of the known uniformly fast algorithms. Timings for a C-implementation show that for the case that the parameters of the binomial distribution vary from call to call BTRD is faster than the current state of the art algorithms. Depending on the computer, the speed of the uniform generator used and the binomial parameters the savings are between 5 and 40 percent. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

We propose a simple adjustment to a Wald confidence interval to estimate a linear function of binomial proportions. This method is an extension to the adjusted Wald confidence intervals for proportions and their differences that have recently been proposed.

A thesis submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfillment of the requirements for the degree of Doctor of Philosophy. Johannesburg 2016 / Quantifying risk in the absence of hard data presents a significant challenge. Onshore mining of the diamondiferous linear beach deposit along the south western coast of Namibia has been ongoing for more than 80 years. A historical delineated campaign from the 1930s to 1960s used coast perpendicular trenches spaced 500 m apart, comprising a total of 26 000 individual samples, to identify 6 onshore raised beaches. These linear beaches extend offshore and are successfully mined in water depths deeper than 30 m. There is, however, a roughly 4 km wide submerged coast parallel strip adjacent to the mostly mined out onshore beaches for which no real hard data is available at present. The submerged beaches within the 4 km coast parallel strip hold great potential for being highly diamondiferous. To date hard data is not yet available to quantify or validate this potential. The question is how to obtain sufficient hard data within the techno economic constraints to enable a resource with an acceptable level of confidence to be developed. The work presented in this thesis illustrates how virtual orebodies (VOBs) are created based on geological proxies in order to have a basis to assess and rank different sampling and drilling strategies. Overview of 4 papers Paper I demonstrates the challenge of obtaining a realistic variogram that can be used in variogram-based geostatistical simulations. Simulated annealing is used to unfold the coastline and improve the detectable variography for a number of the beaches. Paper II shows how expert opinion interpretation is used to supplement sparse data that is utilised to create an indicator simulation to study the presence and absence of diamondiferous gravel. When only the sparse data is used the resultant simulation is unsuitable as a VOB upon which drilling strategies can be assessed. Paper III outlines how expert opinion hand sketches are used to create a VOB. The composite probability map based on geological proxies is adjusted using a grade profile based on adjacent onshore data before it is seeded with stones and used as a VOB for strategy testing. Paper IV illustrates how the Nachman model based on a Negative Binomial Distribution (NBD) is used to predict a minimum background grade by considering only the zero proportions (Zp) of the grade data. v Conclusions and future work In the realm of creating spatial simulations that can serve as VOBs it is very difficult to attempt to quantify uncertainty when no hard data is available. In the absence of hard data, geological proxies and expert opinion are the only inputs that can be used to create VOBs. Subsequently these VOBs are used as a base to be analysed in order to evaluate and rank different sampling and drilling strategies based on techno economic constraints. VOBs must be updated and reviewed as hard data becomes available after which sampling strategies should be reassessed. During early stage exploration projects the Zp of sample results can be used to predict a minimum background grade and rank different targets for further sampling and valuation. The research highlights the possibility that multi point statistics (MPS) can be used. Higher order MPS should be further investigated as an additional method for creating VOBs upon which sampling strategies can be assessed. / MT2017

The objective of this thesis is to examine one of the most fundamental and yet important methodologies used in statistical practice, interval estimation of the probability of success in a binomial distribution. The textbook confidence interval for this problem is known as the Wald interval as it comes from the Wald large sample test for the binomial case. It is generally acknowledged that the actual coverage probability of the standard interval is poor for values of p near 0 or 1. Moreover, recently it has been documented that the coverage properties of the standard interval can be inconsistent even if p is not near the boundaries. For this reason, one would like to study the variety of methods for construction of confidence intervals for unknown probability p in the binomial case. The present thesis accomplishes the task by presenting several methods for constructing confidence intervals for unknown binomial probability p. It is well known that the hypergeometric distribution is related to the binomial distribution. In particular, if the size of the population, N, is large and the number of items of interest k is such that k/N tends to p as N grows, then the hypergeometric distribution can be approximated by the binomial distribution. Therefore, in this case, one can use the confidence intervals constructed for p in the case of the binomial distribution as a basis for construction of the confidence intervals for the unknown value k = pN. The goal of this thesis is to study this approximation and to point out several confidence intervals which are designed specifically for the hypergeometric distribution. In particular, this thesis considers several confidence intervals which are based on estimation of a binomial proportion as well as Bayesian credible sets based on various priors.

We examine some of the methodologies implemented by the Centers for Disease Control and Prevention's (CDC) BioSense program. The program uses data from hospitals and public health departments to detect outbreaks using the Early Aberration Reporting System (EARS). The EARS method W2 allows one to monitor syndrome counts (W2count) from each source and the proportion of counts of a particular syndrome relative to the total number of visits (W2rate). We investigate the performance of the W2r method designed using an empiric recurrence interval (RI) in this dissertation research. An adaptive threshold monitoring method is introduced based on fitting sample data to the underlying distributions, then converting the current value to a Z-score through a p-value. We compare the upper thresholds on the Z-scores required to obtain given values of the recurrence interval for different sets of parameter values. We then simulate one-week outbreaks in our data and calculate the proportion of times these methods correctly signal an outbreak using Shewhart and exponentially weighted moving average (EWMA) charts. Our results indicate the adaptive threshold method gives more consistent statistical performance across different parameter sets and amounts of baseline historical data used for computing the statistics. For the power analysis, the EWMA chart is superior to its Shewhart counterpart in nearly all cases, and the adaptive threshold method tends to outperform the W2 rate method. Two modified W2r methods proposed in the dissertation also tend to outperform the W2r method in terms of the RI threshold functions and in the power analysis. / Ph. D.

Highway safety has been a priority for many years now. A system-wide crash analysis is a practical solution when only a limited budget is available for improving safety of highways. A systematic approach, in contrast to a hotspot analysis, allows for a widespread installation of lower-cost countermeasures across the highway network. This study focuses on the safety evaluation of a particular facility type, urban freeway off-ramps, in terms of its geometric and traffic characteristics. 144 off-ramp segments in Richmond, VA were evaluated based on the crash data available from 2011 to 2015. A statistical model was developed that relates crashes to the geometric and traffic characteristics of each off-ramp segment. A test for independence was performed to identify if a statistically significant difference existed between type of collision and severity of crashes with respect to ramp geometry and traffic control. Significant geometric and traffic variables were then identified from the model and independence test to assist in the selection of low-cost countermeasures. AADTs of both freeways and off-ramps were found to be the most statistically significant variables. Installation of advance warning signs for better traffic management near the freeway diverge area and clearing roadsides of fixed objects to reduce rear-end collisions are low-cost solutions for crashes on urban off-ramps in the study area. The results of this study demonstrate an approach to safety evaluations that could support transportation planners and agencies in identifying system-wide locations to install or apply appropriate low-cost countermeasures. / Master of Science

Made available in DSpace on 2016-06-02T20:06:02Z (GMT). No. of bitstreams: 1 2203.pdf: 1300161 bytes, checksum: 2c1f11d939eab9ab849bb04bf2363a53 (MD5) Previous issue date: 2009-02-10 / Financiadora de Estudos e Projetos / This study considers two discrete distributions based on Bernoulli trials: the Binomial and the Negative Binomial. We explore credibility and confidence intervals to estimate the probability of success of each distribution. The main goal is to analyze their performance coverage probability and average range across the parametric space. We also consider point analysis of bayesian estimators and maximum likelihood estimators, whose interest is to confirm through simulation their consistency, bias and mean square error. In this paper the Objective Bayesian Inference is applied through the noninformative Bayes-Laplace prior, Haldane prior, reference prior and least favorable prior. By analyzing the prior distributions in the minimax decision theory context we verified that the least favorable prior distribution has every other considered prior distributions as particular cases when a quadratic loss function is applied, and matches the Bayes-Laplace prior in considering the quadratic weighed loss function for the Binomial model (which was never found in literature). We used the noninformative Bayes-Laplace prior and Jeffreys prior for the Negative Binomial model. Our findings show through coverage probability, average range of bayesian intervals and point estimation that the Objective Bayesian Inference has good frequentist properties for the probability of success of Binomial and Negative Binomial models. The last stage of this study discusses the presence of correlated proportions in matched-pairs (2 × 2 table) of Bernoulli with the goal of obtaining more information in relation of the considered measures for testing the occurrence of correlated proportions. In this sense the Trinomial model and the partial likelihood function were used from the frequentist and bayesian point of view. The Full Bayesian Significance Test (FBST) was used for real data sets and was shown sensitive to parameterization, however, this study was not possible for the frequentist method since distinct methods are needed to be applied to Trinomial model and the partial likelihood function. / Neste estudo são abordadas duas distribuições discretas baseadas em ensaios de Bernoulli, a Binomial e a Binomial Negativa. São explorados intervalos de credibilidade e confiança para estimação da probabilidade de sucesso de ambas as distribuições. A principal finalidade é analisar nos contextos clássico e bayesiano o desempenho da probabilidade de cobertura e amplitude média gerada pelos intervalos de confiança e intervalos de credibilidade ao longo do espaço paramétrico. Considerou-se também a análise dos estimadores pontuais bayesianos e o estimador de máxima verossimilhança, cujo interesse é confirmar por meio de simulação a consistência e calcular o viés e o erro quadrático médio dos mesmos. A Inferência Bayesiana Objetiva é empregada neste estudo por meio das distribuições a priori não-informativas de Bayes-Laplace, de Haldane, de Jeffreys e menos favorável. Ao analisar as distribuições a priori no contexto de teoria de decisões minimax, a distribuição a priori menos favorável resgata as demais citadas ao empregar a função de perda quadrática e coincide com a distribuição a priori de Bayes-Laplace ao considerar a função de perda quadrática ponderada para o modelo Binomial, o que não foi encontrado até o momento na literatura. Para o modelo Binomial Negativa são consideradas as distribuições a priori não-informativas de Bayes-Laplace e de Jeffreys. Com os estudos desenvolvidos pôde-se observar que a Inferência Bayesiana Objetiva para a probabilidade de sucesso dos modelos Binomial e Binomial Negativa apresentou boas propriedades freqüentistas, analisadas a partir da probabilidade de cobertura e amplitude média dos intervalos bayesianos e por meio das propriedades dos estimadores pontuais. A última etapa do trabalho consiste na análise da ocorrência de proporções correlacionadas em pares de eventos de Bernoulli (tabela 2×2) com a finalidade de determinar um possível ganho de informação em relação as medidas consideradas para testar a ocorrência de proporções correlacionadas. Para tanto fez-se uso do modelo Trinomial e da função de verossimilhança parcial tanto numa abordagem clássica quanto bayesiana. Nos conjuntos de dados analisados observou-se a medida de evidência bayesiana (FBST) como sensível à parametrização, já para os métodos clássicos essa comparação não foi possível, pois métodos distintos precisam ser aplicados para o modelo Trinomial e para a função de verossimilhança parcial.

Master of Science / Department of Statistics / Leigh Murray / The southern root-knot nematode (SRKN) and the weedy perennials, yellow nutsedge (YNS) and purple nutsedge (PNS) are simultaneously occurring pests in the irrigated agricultural soils of southern New Mexico. Previous research has very well characterized SRKN, YNS and PNS as a mutually-beneficial pest complex and has revealed their enhanced population growth and survival when they occur together. The density of nutsedge in a field could be used as a predictor of SRKN juveniles in the soil. In addition to SRKN, which is the most harmful of the plant parasitic nematodes, in southern New Mexico, other species or categories of nematodes could be identified and counted. Some of them are not as damaging to the plant as SRKN, and some of them may be essential for soil health. The nematode species could be grouped into categories according to trophic level (what nematodes eat) and herbivore feeding behavior (how herbivore nematodes eat). Subsequently, three ratios of counts were calculated for trophic level and for feeding behavior level to investigate the soil nematode community structure. These proportions were modeled as functions of the weed hosts YNS and PNS by generalized linear regression models using the logit link function and three probability distributions: the Binomial, Zero Inflated Binomial (ZIB) and Binomial Hurdle (BH). The latter two were used to account for potential high proportions of zeros in the data. The SAS NLMIXED procedure was used to fit models for each of the six sampling dates (May, July and September) over the two years of the alfalfa study. General results showed that the Binomial pmf generally provided the best fit, indicating lower zero-inflation than expected. Importance of YNS and PNS predictors varied over time and the different ratios. Specific results illustrate the differences in estimated probabilities between Binomial, ZIB and BH distributions as YNS counts increase for two selected ratios.