<p>A stochastic differential equationdriven by a Brownian motion where the dispersion is determined by a parameter is considered. The parameter undergoes a change at a certain time point. Estimates of the time change point and the parameter, before and after that time, is considered.The estimates were presented in Lacus 2008. Two cases are considered: (1) the drift is known, (2) the drift is unknown and the dispersion space-independent. Applications to Dow-Jones index 1971-1974  and Goldmann-Sachs closings 2005-- May 2009 are given.</p>

<p>We apply some recent numerical solutions to diffusion bridges written in Iacus (2008). One is an approximate scheme from Bladt and S{\o}rensen (2007), another one, from  Beskos et al (2006), is an algorithm which is exact: no numerical error at given grid points!</p>

In this thesis we investigate a model of nonlinearly perturbed continuous-time renewal equation. Some characteristics of the renewal equation are assumed to have non-polynomial perturbations, more specifically they can be expanded with respect to a non-polynomial asymptotic scale. The main result of the present study is exponential asymptotic expansions for the solution of the perturbed renewal equation. These asymptotic results are also applied to various applied probability models like perturbed risk processes, perturbed M/G/1 queues and perturbed dam/storage processes. The thesis is based on five papers where the model described above is successively studied.

According to the Joint United Nations Programme against HIV/AIDS 2009 AIDS epidemic update, there were a total of 33.3 million (31.4 million–35.3 million) people living with HIV worldwide in 2009. The majority of the epidemic occurs in Sub-Saharan Africa. Of the 33.3 million people living with HIV worldwide in 2009, a vast majority of 22.5 million (20.9 million-24.2 million) were from Sub-Saharan Africa. There were 1.8 million (1.6 million-2.0 million) new infections and 1.3 million (1.1 million-1.5 million) AIDS-related deaths in Sub-Saharan Africa in 2009 (UNAIDS, 2009). Statistical models and analysis are required in order to further understand the dynamics of HIV/AIDS and in the design of intervention and control strategies. Despite the prevalence of this disease, its pathogenesis is still poorly understood. A thorough understanding of HIV and factors that influence progression of the disease is required in order to prevent the further spread of the virus. Modelling provides us with a means to understand and predict the progression of the disease better. Certain genetic factors play a key role in the way the disease progresses in a human body. For example HLA-B types and IL-10 genotypes are some of the genetic factors that have been independently associated with the control of HIV infection. Both HLA-B and IL-10 may influence the quality and magnitude of immune responses and IL-10 has also been shown to down regulate the expression of certain HLA molecules. Studies are therefore required to investigate how HLA-B types and IL-10 genotypes may interact to affect HIV infection outcomes. This dissertation uses the Sinikithemba study data from the HIV Pathogenesis Programme (HPP) at the Medical School, University of KwaZulu-Natal involving 450 HIV positive and treatment naive individuals to model how certain outcome biomarkers (CD4+ counts and viral loads) are associated with immuno genetic parameters (HLA-B types and IL-10 genotypes). The work also seeks to exploit novel longitudinal data methods in Statistics in order to efficiently model longitudinally measured HIV outcome data. Statistical techniques such as linear mixed models and generalized estimating equations were used to model this data. The findings from the current work agree quite closely with what is expected from the biological understanding of the disease. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

With the increasing amount of collected data, large-scale high-dimensional data analysis is becoming essential in many areas. These data can be analyzed either by using fully computational methods or by leveraging human capabilities via interactive visualization. However, each method has its drawbacks. While a fully computational method can deal with large amounts of data, it lacks depth in its understanding of the data, which is critical to the analysis. With the interactive visualization method, the user can give a deeper insight on the data but suffers when large amounts of data need to be analyzed. Even with an apparent need for these two approaches to be integrated, little progress has been made. As ways to tackle this problem, computational methods have to be re-designed both theoretically and algorithmically, and the visual analytics system has to expose these computational methods to users so that they can choose the proper algorithms and settings. To achieve an appropriate integration between computational methods and visual analytics, the thesis focuses on essential computational methods for visualization, such as dimension reduction and clustering, and it presents fundamental development of computational methods as well as visual analytic systems involving newly developed methods. The contributions of the thesis include (1) the two-stage dimension reduction framework that better handles significant information loss in visualization of high-dimensional data, (2) efficient parametric updating of computational methods for fast and smooth user interactions, and (3) an iteration-wise integration framework of computational methods in real-time visual analytics. The latter parts of the thesis focus on the development of visual analytics systems involving the presented computational methods, such as (1) Testbed: an interactive visual testbed system for various dimension reduction and clustering methods, (2) iVisClassifier: an interactive visual classification system using supervised dimension reduction, and (3) VisIRR: an interactive visual information retrieval and recommender system for large-scale document data.

Solar radiation is a primary driving force behind a number of solar energy applications such as photovoltaic systems for electricity generation amongst others. Hence, the accurate modelling and prediction of the solar flux incident at a particular location, is essential for the design and performance prediction of solar energy conversion systems. In this regard, literature shows that time series models such as the Box-Jenkins Seasonal/Non-seasonal Autoregressive Integrated Moving Average (S/ARIMA) stochastic models have considerable efficacy to describe, monitor and forecast solar radiation data series at various sites on the earths surface (see e.g. Reikard, 2009). This success is attributable to their ability to capture the stochastic component of the irradiance series due to the effects of the ever-changing atmospheric conditions. On the other hand at the top of the atmosphere, there are no such conditions and deterministic models which have been used successfully to model extra-terrestrial solar radiation. One such modelling procedure is the use of a sinusoidal predictor at determined harmonic (Fourier) frequencies to capture the inherent periodicities (seasonalities) due to the diurnal cycle. We combine this deterministic model component and SARIMA models to construct harmonically coupled SARIMA (HCSARIMA) models to model the resulting mixture of stochastic and deterministic components of solar radiation recorded at the earths surface. A comparative study of these two classes of models is undertaken for the horizontal global solar irradiance incident on the solar panels at UKZN Howard College (UKZN HC), located at 29.9º South, 30.98º East with elevation, 151.3m. The results indicated that both SARIMA and HCSARIMA models are good in describing the underlying data generating processes for all data series with respect to different diagnostics. In terms of the predictive ability, the HCSARIMA models generally had a competitive edge over the SARIMA models in most cases. Also, a tentative study of long range dependence (long memory) shows this phenomenon to be inherent in high frequency data series. Therefore autoregressive fractionally integrated moving average (ARFIMA) models are recommended for further studies on high frequency irradiance. / M.Sc. University of KwaZulu-Natal, Durban 2014.

Evidence for a human contribution to climatic changes during the past century is accumulating rapidly. Given the strength of the evidence, it seems natural to ask whether forcing projections can be used to forecast climate change. A Bayesian method for post-processing forced climate model simulations that produces probabilistic hindcasts of inter-decadal temperature changes on large spatial scales is proposed. Hindcasts produced for the last two decades of the 20th century are shown to be skillful. The suggestion that skillful decadal forecasts can be produced on large regional scales by exploiting the response to anthropogenic forcing provides additional evidence that anthropogenic change in the composition of the atmosphere has influenced our climate. In the absence of large negative volcanic forcing on the climate system (which cannot presently be forecast), the global mean temperature for the decade 2000-2009 is predicted to lie above the 1970-1999 normal with probability 0.94. The global mean temperature anomaly for this decade relative to 1970-1999 is predicted to be 0.35C (5-95% confidence range: 0.21C-0.48C). Reconstruction of temperature variability of the past centuries using climate proxy data can also provide important information on the role of anthropogenic forcing in the observed 20th century warming. A state-space model approach that allows incorporation of additional non-temperature information, such as the estimated response to external forcing, to reconstruct historical temperature is proposed. An advantage of this approach is that it permits simultaneous reconstruction and detection analysis as well as future projection. A difficulty in using this approach is that estimation of several unknown state-space model parameters is required. To take advantage of the data structure in the reconstruction problem, the existing parameter estimation approach is modified, resulting in two new estimation approaches. The competing estimation approaches are compared based on theoretical grounds and through simulation studies. The two new estimation approaches generally perform better than the existing approach. A number of studies have attempted to reconstruct hemispheric mean temperature for the past millennium from proxy climate indicators. Different statistical methods are used in these studies and it therefore seems natural to ask which method is more reliable. An empirical comparison between the different reconstruction methods is considered using both climate model data and real-world paleoclimate proxy data. The proposed state-space model approach and the RegEM method generally perform better than their competitors when reconstructing interannual variations in Northern Hemispheric mean surface air temperature. On the other hand, a variety of methods are seen to perform well when reconstructing decadal temperature variability. The similarity in performance provides evidence that the difference between many real-world reconstructions is more likely to be due to the choice of the proxy series, or the use of difference target seasons or latitudes, than to the choice of statistical method.

Benford’s Law states that the first significant digit for most data is not uniformly distributed. Instead, it follows the distribution: P(d = d1) = log10(1 + 1/d1) for d1 ϵ {1, 2, …, 9}. In 2006, my supervisor, Dr. Mary Lesperance et. al tested the goodness-of-fit of data to Benford’s Law using the first significant digit. Here we extended the research to the first two significant digits by performing several statistical tests – LR-multinomial, LR-decreasing, LR-generalized Benford, LR-Rodriguez, Cramѐr-von Mises Wd2, Ud2, and Ad2 and Pearson’s χ2; and six simultaneous confidence intervals – Quesenberry, Goodman, Bailey Angular, Bailey Square, Fitzpatrick and Sison. When testing compliance with Benford’s Law, we found that the test statistics LR-generalized Benford, Wd2 and Ad2 performed well for Generalized Benford distribution, Uniform/Benford mixture distribution and Hill/Benford mixture distribution while Pearson’s χ2 and LR-multinomial statistics are more appropriate for the contaminated additive/multiplicative distribution. With respect to simultaneous confidence intervals, we recommend Goodman and Sison to detect deviation from Benford’s Law.

This dissertation called The Statistical Investigation of Time and Economical Development of the Extraordinary Incident Kind Selected deals with traffic accidents on roads within the Czech Republic. The goal of this work is to investigate a time and economical connection of traffic accidents. The theoretical part specifies basic terms which are related to road transport in the Czech Republic and which are mainly based on valid laws in the field of road transportation. There is also deeply analysed the term "traffic accident" which has to gain specific features in order to be classified as a traffic accident. Subsequently, there are also presented and discussed the most common causes of accidents. Impacts of economic damage to the state budget were described by a certified methodology for calculating losses from a traffic accident rate on roads. Within possibilities of a bigger kind of an extraordinary incident during which emergency could be announced, there is described a system of crisis management in transport and an operation of Crisis Management Department of the Ministry of Transport. In the last section of the theoretical part there are summarized the basic methods of descriptive and mathematical statistics which were used for processing of the practical part of the dissertation. In the practical part three hypotheses were being tested based on the collected data, which was drawn from statistical yearbooks of a traffic accident rate published on the website of the Police of the Czech Republic. The methodology provides basic methods of descriptive and mathematical statistics to whom the gained data has been subjected. To verify the hypothesis H2 the amounts of damage per a time unit have a distribution close to the normal distribution. It was used within the mathematical statistics of nonparametric testing of hypothesis. Through the non-parametric testing was allowed to prove normality of damage amount in traffic accidents in the period 2009 - 2013. For verification of the hypothesis H1 number of traffic accidents and the amount of damage are positively correlated and hypotheses H3 time development of traffic accidents per one time unit can be expressed by linear regression with negative correlation. Simple linear and correlation dependence was used in the context of mathematical statistics. The aim of the hypotheses H1 and H3 mentioned above was to find out a kind of regression, depending on given statistical features, to find suitable regression function and to determine tightness of correlation using an appropriate coefficient. A negative correlation coefficient was calculated for the hypothesis H1 so it was rejected. This result is possible to comment on the fact that, although there is less material damage, the number of incidents in particular time units does not reflect this fact. For the hypotheses H3 we can conclude that traffic accidents have been declining in a longer time interval and this leads to fulfilment of the goal of the national road safety strategy 2011 - 2020. For the short time period 2009 - 2013 it is not possible to come to the same conclusion and the hypotheses H3 has to be rejected. In the context of a deeper research of the impact of economic damage to our state, it would be good to work e.g. with the amount of money which insurance companies have to pay to persons involved in an accident. We could also work with the amount of costs which the state has to spend on activities of organs of emergency service for dealing with all consequences of any emergencies associated with an accident. Determination of the total amount of economic damage from a traffic accident rate and from a number of traffic accidents helps us to realize the seriousness of this issue. Just as other indicators of traffic safety, economic damage and a number of traffic accidents are important indicators of prevention and effectiveness of traffic-safety measures.

Submitted by Luciano Monteiro de Castro (lucianogmcastro@gmail.com) on 2017-06-26T23:35:31Z No. of bitstreams: 1 TeseTRIv6.pdf: 2619923 bytes, checksum: 7852109eaf466514e20db551b7aa312b (MD5) / Approved for entry into archive by Janete de Oliveira Feitosa (janete.feitosa@fgv.br) on 2017-08-23T17:10:47Z (GMT) No. of bitstreams: 1 TeseTRIv6.pdf: 2619923 bytes, checksum: 7852109eaf466514e20db551b7aa312b (MD5) / Made available in DSpace on 2017-08-25T12:27:14Z (GMT). No. of bitstreams: 1 TeseTRIv6.pdf: 2619923 bytes, checksum: 7852109eaf466514e20db551b7aa312b (MD5) Previous issue date: 2017-05-09 / Since 2009 the Brazilian National High School Examination (Exame Nacional do Ensino Médio - ENEM) started utilising the Item-Response Theory (IRT) to determine students scores. Since 2009 the Brazilian National High School Examination (Exame Nacional do Ensino Médio - ENEM) started utilising the Item-Response Theory (IRT) to determine students scores. The ENEM micro data published by INEP (Instituto Nacional de Estudos e Pesquisas Educacionais Anísio Teixeira) contains detailed information about all instalments of the examination. But there is no data directly related to the actual IRT implementation besides the scores themselves. In this work we analyse the ENEM micro data under the the light of the three parameter logistic model of IRT, with two main goals: 1) To generate more information on the implementation of IRT used in ENEM, particularly estimating the parameters used for each question of the examination; 2) Treat the information gathered and suggest different ways of presenting it in order to improve the understanding of the effect of utilising IRT over the examination scores. The item estimation methods may be used, as an example, to create simulated examinations and calculate the scores on the same scale used by ENEM, with the same methodology, allowing for a better assessment of students preparing for the examination. / A partir de 2009 o Exame Nacional do Ensino Médio (ENEM) passou a utilizar a Teoria da Resposta ao Item (TRI) para o cálculo das notas dos candidatos. Os microdados do ENEM divulgados pelo INEP (Instituto Nacional de Estudos e Pesquisas Educacionais Anísio Teixeira) contêm informação detalhada sobre todas as aplicações do exame. No entanto, além das próprias notas, não há dados diretamente ligados à implementação da TRI. No presente trabalho analisamos os microdados do ENEM sob o ponto de vista do modelo logístico de três parâmetros da TRI, com dois principais objetivos: 1) Gerar mais informações sobre a implementação da TRI no ENEM, em particular estimando os parâmetros utilizados em cada questão da prova; 2) Tratar a informação obtida e propor diferentes formas de apresentá-la de maneira a melhorar a compreensão sobre os efeitos da utilização da TRI sobre as notas do exame. Os métodos de estimação dos parâmetros das questões podem ser utilizados, por exemplo, para criar exames simulados cujas notas sejam calculadas na mesma escala utilizada pelo ENEM, e com a mesma metodologia, permitindo uma melhor avaliação dos candidatos durante sua preparação para o exame.