Multidimensional frequency models can be used for modeling number of claims from different branches which are somehow dependent on each other. As in the one-dimensional case Poisson distribution and negative binomial distribution are primarily used for modeling multidimensional claim counts data, only they are extended to higher dimensions. The generalization of multi- dimensional distributions is often done using so-called shock variables, where one random variable is included in all dimensions of a random vector which models claim counts. The more comprehensive approach to modeling dependence uses copulas. Comparison of these models is done on a simulated data of number of claims from two different car insurance guarantees.

<p>Analyzing the ever-increasing data of unprecedented scale, dimensionality, diversity, and complexity poses considerable challenges to conventional approaches of statistical modeling. Bayesian nonparametrics constitute a promising research direction, in that such techniques can fit the data with a model that can grow with complexity to match the data. In this dissertation we consider nonparametric Bayesian modeling with completely random measures, a family of pure-jump stochastic processes with nonnegative increments. In particular, we study dictionary learning for sparse image representation using the beta process and the dependent hierarchical beta process, and we present the negative binomial process, a novel nonparametric Bayesian prior that unites the seemingly disjoint problems of count and mixture modeling. We show a wide variety of successful applications of our nonparametric Bayesian latent variable models to real problems in science and engineering, including count modeling, text analysis, image processing, compressive sensing, and computer vision.</p> / Dissertation

We study the properties of thinning and Markov chain thinning of renewal processes. Among others, we investigate whether some special renewal processes can be obtained through Markov chain thinning.

The main role of roadway lighting is to produce quick, accurate and comfortable visibility during nighttime conditions. It is commonly known that good lighting levels enable motorists, pedestrians and bicyclists to obtain necessary visual information in an effective and efficient manner. Many previous studies also proved that roadway lighting minimizes the likelihood of crashes by providing better visibility for roadway users. Appropriate and adequate roadway lighting illuminance levels for each roadway classification and pedestrian areas are essential to provide safe and comfortable usage. These levels are usually provided by national, or local standards and guidelines. The Florida Department of Transportation (FDOT) Plan Preparation Manual recommends a roadway lighting illuminance level average standard of 1.0 horizontal foot candle (fc) for all the roadway segments used in this research. The FDOT Plan Preparation Manual also states that this value should be considered standard, but should be increased if necessary to maintain an acceptable uniformity illuminance ratio. This study aimed to find the relationship between nighttime crash injury severity and roadway lighting illuminance. To accomplish this, the research team analyzed crash data and roadway lighting illuminance measured in roadway segments within the West Central Florida Region. An Ordered Probit Model was developed to understand the relationship between roadway lighting illuminance levels and crash injury severity. Additionally, a Negative Binomial Model was used to determine which roadway lighting illuminance levels can be more beneficial in reducing the counts of crashes resulting in injuries. A comprehensive literature review was conducted using longitudinal studies with and without roadway lighting. Results showed that on the same roadways there was a significant decrease in the number of nighttime crashes with the presence of roadway lighting. In this research, roadway lighting illuminance was measured every 40 feet using an Advanced Lighting Measurement System (ALMS) on a total of 245 centerline miles of roadway segments within the West Central Florida Region. The data were mapped and then analyzed using the existing mile post. During the process of crash data analysis, it was observed that rear-end collisions were the most common first harmful event observed in all crashes, regardless of the lighting conditions. Meanwhile, the average injury severity for all crashes, was found to be possible injury regardless of the lighting conditions (day, dark, dusk, and dawn). Finally, this research presented an Ordered Probit Model, developed to understand the existing relationship between roadway lighting illuminance levels and injury severity within the West Central Florida Region. It was observed that having a roadway lighting average moving illuminance range between 0.4 to 0.6 foot candles (fc) was more likely to have a positive effect in reducing the probability of injury severity during a nighttime crash. A Negative Binomial Model was conducted to determine if the roadway lighting average moving illuminance level, found on the Ordered Probit Model was beneficial in reducing crash injury severity during nighttime, would also be beneficial in reducing the counts of crashes resulting in injuries. It was observed that a roadway lighting average moving illuminance, range between 0.4 to 0.6 fc, was more likely to reduce the count of crashes resulting in injuries during nighttime conditions, thus increasing roadway safety. It was also observed that other factors such as pavement condition, site location (intersection or no intersection), number of lanes, and traffic volume can affect the severity and counts of nighttime crashes. The results of this study suggest that simply adding more roadway lighting does not make the roadway safer. The fact is that a reduction in the amount of roadway lighting illuminance can produce savings in energy consumption and help the environment by reducing light pollution. Moreover, these results show that designing roadway lighting systems go beyond the initial design process, it also requires continuous maintenance. Furthermore, regulations for new developments and the introduction of additional lighting sources near roadway facilities (that are not created with the intent of being used for roadway users) need to be created.

We assessed how estimates of need-expected inpatient hospital use differ depending on whether need-expected use was estimated for a population of all Canadians, Canadian health regions, or a subpopulation of higher income Canadians, who likely had minimal healthcare access problems. Data came from the 2009/2010 Canadian Community Health Survey, a national cross-sectional survey. Using zero-inflated negative binomial regression, we modeled inpatient hospital use separately based on the three aforementioned choices of population. We adjusted for demographic, health behaviour, health status, socioeconomic, and health care supply factors. We then estimated need-expected inpatient hospital use and compared the estimates across individuals and by income and province. The three choices of population that we used in this study had similar results. Our estimates of the average need-expected use by province or income group were not sensitive to the choice of population used to estimate need-expected use.

In this work were considered the maxima and minima structures. Where number of value is random and is distributed by negative binomial distribution. There were theorems that were improved in this work, that helped to find the limit distribute function of this standard structures. These theorems generalize propositions, when set size is geometric random number. Also, there was the concrete distribution analysis done and such distributions were chosen: exponential, general logistic and uniform.

The purpose of this travel cost study is to determine how rider behaviors and site characteristics influence repeat visits for equestrian trail riding in Kentucky. Primary data was collected via a survey developed and administered to trail riders in person and online. The average surveyed trail rider tends to be female, about 46 years old, with some higher education, and an annual household income of $65,000. She makes 11 trips to a specified site per year, 8 of which are daytrips, usually in the fall, and traveling 132 miles round trip. From other information gathered, an index of trail characteristics was developed to identify positive attributes of trails. To account for overdispersion of the number of visits per year, a negative binomial distribution in the estimation was used. The primary variables significant to explaining repeat visits to a site include distance in miles, the index of characteristics, and gender. Given consumer surplus estimates of $800 per equestrian it is recommended that established trails maximize desired characteristics. For new trail development it is recommended that trail characteristics are maximized and that they are built closer to the urban areas of the state since most riders are coming from these areas.

This thesis develops hierarchical spatial models for the analysis of correlated and overdispersed count data based on the negative binomial distribution. Model development is motivated by a large scale study of fire frequency in British Columbia, conducted by the Pacific Forestry Service. Specific to our analysis, the main focus lies in examining the interaction between wildfire and forest insect outbreaks. In particular, we wish to relate the frequency of wildfire to the severity of mountain pine beetle (MPB) outbreaks in the province. There is a widespread belief that forest insect outbreaks lead to an increased frequency of wildfires; however, empirical evidence to date has been limited and thus a greater understanding of the association is required. This is critically important as British Columbia is currently experiencing a historically unprecedented MPB outbreak. We specify regression models for fire frequency incorporating random effects in a generalized linear mixed modeling framework. Within such a framework, both spatial correlation and extra-Poisson variation can be accommodated through random effects that are incorporated into the linear predictor of a generalized linear model. We consider a range of models, and conduct model selection and inference within the Bayesian framework with implementation based on Markov Chain Monte Carlo.

Η παρούσα μεταπτυχιακή διατριβή εντάσσεται ερευνητικά στην περιοχή της Στατιστικής θεωρίας Αποφάσεων και ειδικότερα στη μελέτη της αρνητικής διωνυμικής κατανομής καθώς επίσης και στην εκτίμηση των παραμέτρων της. Στο Κεφάλαιο 1 παρουσιάζονται κάποιοι χρήσιμοι, για την πορεία της μελέτης μας, ορισμοί και θεωρήματα. Στο Κεφάλαιο 2 μελετάται το μοντέλο της αρνητικής διωνυμικής κατανομής, δίνονται τα χαρακτηριστικά μεγέθη αυτής και παρουσιάζονται οι διαφορετικές παραμετρικοποιήσεις της. Στο Κεφάλαιο 3, εξετάζεται το πρόβλημα εκτίμησης των παραμέτρων της αρνητικής διωνυμικής κατανομής και πιο ειδικά η εκτίμηση για τις διάφορες παραμετρικοποιήσης της. Για περισσότερη ανάλυση χρησιμοποιούνται η εκτίμηση μέγιστης πιθανοφάνειας, η εκτίμηση με τη μέθοδο των ροπών και πιο εξειδικευμένες υπολογιστικές μέθοδοι εκτίμησης. Στο Κεφάλαιο 4, και για το ίδιο πρόβλημα εκτίμησης που πραγματεύεται το προηγούμενο κεφάλαιο, επιλέγεται ο βέλτιστος εκτιμητής των παραμέτρων της αρνητικής διωνυμικής κατανομής και παρουσιάζεται ένα παράδειγμα για την κατανόηση των μεθόδων εκτίμησης. / The master thesis we are going to introduce takes place in the region of Statistical Decision Theory and particularly in studying the Negative Binomial Distribution and the estimation of its parameters. In Chapter 1 some useful definitions and theorems are presented. In Chapter 2 the model of negative binomial distribution is studied and its different parameterizations are discussed. In Chapter 3 we examine the problem of estimating the parameters of our model and for its parameterizations. In particular we give the method of Maximum Likelihood Estimation, the Method of Moments and more specified Estimation Methods. In Chapter 4 and for the same estimation problem, as in previous chapter, it’s been chosen the best estimator of the parameters in our model and it’s been derived an example for the better understanding of the above methods.

Com o crescimento da população em áreas urbanas, o transporte coletivo assume um importante papel nos deslocamentos diários nas cidades brasileiras. No entanto, o grande volume de pessoas no entorno desses sistemas podem resultar em pontos críticos de segurança viária. Recentemente, as pesquisas sobre melhorias na segurança viária estão se direcionando para a investigação da relação entre o ambiente construído - isto é, padrões de uso do solo; desenho urbano; e sistemas de transporte - e acidentalidade. Porém, a relação entre os sistemas de transporte coletivo por ônibus e o ambiente construído ainda é pouco explorada. O ponto mais crítico em segurança identificado na literatura em sistemas de transporte coletivo são as estações de embarque e desembarque. Assim, esta dissertação tem como objetivo verificar a existência de relação entre elementos do ambiente construído com a ocorrência e severidade de acidentes viários no entorno de pontos de parada em sistemas prioritários para ônibus. Para isso, aplicou-se os modelos de regressão Binomial Negativo (NB), na análise de frequência de acidentes, e os modelos Logit Ordenado (OL) e Logit Multinomial (MNL), para a análise de severidade. No modelo de frequência de acidentes totais, identificou-se duas variáveis significativas e, no modelo estimado para atropelamentos, três variáveis. O modelo de severidade resultou em 19 variáveis significativas para acidentes totais e oito variáveis significativas para atropelamentos. As variáveis referentes ao envolvimento de motocicletas, automóveis, presença de interseção e uso diversificado do solo foram as mais recorrentes. O estudo possibilitou também avaliar as diferenças e vantagens entre os modelos ordenados e os não-ordenados. O modelo Logit Multinomial teve ajustes levemente melhores do que o Logit Ordenado. Ressalta-se, entretanto, que o ajuste do modelo não deve ser o único critério a se considerar na escolha de modelos para o estudo de severidade de acidentes. / Public transport has played an important role in daily commutes in Brazilian cities as population grows in urban areas. However, high volumes of pedestrians near those systems may result in critical safety issues. Recent studies on road safety improvements have sought for evidence on the relationship between built environment - land use patterns; urban design; and transportation systems - and traffic safety. Nevertheless, the relationship between traffic safety at bus priority systems and the built environment is still incipient. According to studies, the most critical safety issues related to public transport are the stations for boarding and alighting. Thus, this thesis aims to verify the relationship between built environment and incidence and severity of traffic crashes in the surroundings of bus priority systems. To do so, we applied Negative Binomial regression models (NB) for the frequency analysis, and the Ordered Logit (OL) and Multinomial Logit (MNL) models for severity analysis. NB models resulted in two significant variables for traffic crashes in general and three variables for pedestrian crashes. The severity model resulted in 19 significant variables overall crashes and eight significant variables for pedestrian crashes. Variables related to the involvement of motorcycles, automobiles, intersections and mixed land use were significant in majority of model estimations. The results allowed to evaluate the goodness of fit between ordered and unordered models. MNL had a slightly better adjustments compared to OL. Goodness of fit, however, should not be the only criterion for selecting a model to assess crash severity.