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A surge function is a mathematical function of the form f(x)=axpe-bx. We simplify the surge function by holding p constant at 1 and investigate the simplified form as a potential model to represent the full peak of a stream discharge hydrograph. The previously studied Weibull and gamma distributions are included for comparison. We develop an analysis algorithm which produces the best-fit parameters for every peak for each model function, and we process the data with a MATLAB script that uses spectral analysis to filter year-long, 15-minute, stream-discharge data sets. The filtering is necessary to locate the concave-upward inflection points used to separate the data set into its constituent, individual peaks. The Levenberg-Marquardt algorithm is used to iteratively estimate the unknown parameters for each version of the modeled peak by minimizing the sum of squares of residuals. The results allow goodness-of-fit comparisons between the three model functions, as well as a comparison of peaks at the same gage through the year of record. Application of these methods to five rivers from three distinct hydrologic regions shows that the simple surge function is a special case of the gamma distribution, which is known to be useful as a modeling function for a full-peak hydrograph. The study also confirms that the Weibull distribution produces good fits to 15-minute hydrograph data.

This report presents the descriptive data analysis and failure time modeling that can be used to find out the characteristics and pattern of failure time. Descriptive data analysis includes the mean, median, 1st quartile, 3rd quartile, frequency, standard deviation, skewness, kurtosis, minimum, maximum and range. Models like exponential distribution, gamma distribution, normal distribution, lognormal distribution, Weibull distribution and log-logistic distribution have been studied for failure time data. The data in this report comes from the South Texas Project that was collected during the last 40 years. We generated more than 1000 groups for STP failure time data based on Mfg Part Number. In all, the top twelve groups of failure time data have been selected as the study group. For each group, we were able to perform different models and obtain the parameters. The significant level and p-value were gained by Kolmogorov-Smirnov test, which is a method of goodness of fit test that represents how well the distribution fits the data. The In this report, Weibull distribution has been proved as the most appropriate model for STP dataset. Among twelve groups, eight groups come from Weibull distribution. In general, Weibull distribution is powerful in failure time modeling. / text

The statistical inference about the reliability parameter R involving independent gamma stress and exponential strength is considered. Assuming the shape parameter of gamma is a known arbitrary real number and the scale parameters of gamma and exponential are unknown, the UMVUE and MLE of R are obtained. A pivot is proposed. Some inference about R derived from this pivot is presented. It will be shown that the pivot can be used for testing hypothesis and constructing condence interval. A procedure of constructing the condence interval for R is derived. The performances of the UMVUE and MLE are compared numerically based on extensive Monte Carlo simulation. Simulation studies indicate that the performance of the two estimators is about the same. The MLE is preferred because of the simplicity of its computation.

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Classifying individuals into binary disease categories can be challenging due to complex relationships across different exposures of interest. In this thesis, we investigate three different approaches for disease classification using multiple biomarkers. First, we consider combining information from literature reviews and INTERHEART data set to identify the threshold of ApoB, ApoA1 and the ratio of these two biomarkers to classify individuals at risk of developing myocardial infarction. We develop a Bayesian estimation procedure for this purpose that utilizes the conditional probability distribution of these biomarkers. This method is flexible compared to standard logistic regression approach and allows us to identify a precise threshold of these biomarkers. Second, we consider the problem of disease classification using two dependent biomarkers. An independently identified threshold for this purpose usually leads to a conflicting classification for some individuals. We develop and describe a method of determining the joint threshold of two dependent biomarkers for a disease classification, based on the joint probability distribution function constructed through copulas. This method will allow researchers uniquely classify individuals at risk of developing the disease. Third, we consider the problem of classifying an outcome using a gene and miRNA expression data sets. Linear principal component analysis (PCA) is a widely used approach to reduce the dimension of such data sets and subsequently use it for classification, but many authors suggest using kernel PCA for this purpose. Using real and simulated data sets, we compare these two approaches and assess the performance of components towards genetic data integration for an outcome classification. We conclude that reducing dimensions using linear PCA followed by a logistic regression model for classification seems to be acceptable for this purpose. We also observe that integrating information from multiple data sets using either of these approaches leads to a better performance of an outcome classification. / Thesis / Doctor of Philosophy (PhD)

Lyme disease has been a growing problem in the United States over the last few decades, and is currently the most common vector-borne disease in the country. This research evaluates the land cover within specified counties of northern Virginia to determine if a correlation exists between forest fragmentation, suburbanization, and cases of human Lyme disease as has been demonstrated in other Lyme endemic regions in the United States. Few studies have focused specifically on northern Virginia when considering the impacts of land cover change on Lyme disease. Discovered through the use of geospatial and statistical analysis, the cluster of Lyme disease cases in northern Virginia are associated with forest fragmentation within the study region, which creates an ideal habitat for black-legged ticks and the white-footed mouse, allowing for an increase in Lyme disease transfer from vector to humans. The goal is for the research findings to be applicable to other regions with similar land cover types. Regions with similar characteristics would then be able to recognize the potential risk of human Lyme disease and implement ways to reduce the Lyme disease risk associated with suburban development. The purpose of this study is to answer the following research questions: 1) How has the spatial distribution of Lyme disease in Northern Virginia changed since 2005 with respect to land cover? 2) Which suburban communities are more at risk for Lyme disease when considering their land cover types and the increasing spatial distribution of Lyme disease? / Master of Science / Lyme disease has been a growing problem in the United States over the last few decades, and is currently the most common vector-borne disease in the country. This research evaluates the land cover within specified counties of northern Virginia to determine if a correlation exists between forest fragmentation, suburbanization, and cases of human Lyme disease as has been demonstrated in other Lyme endemic regions in the United States. Few studies have focused specifically on northern Virginia when considering the impacts of land cover change on Lyme disease. Discovered through the use of geospatial and statistical analysis, the cluster of Lyme disease cases in northern Virginia are associated with forest fragmentation within the study region, which creates an ideal habitat for black-legged ticks and the white-footed mouse, allowing for an increase in Lyme disease transfer from vector to humans. The goal is for the research findings to be applicable to other regions with similar land cover types. Regions with similar characteristics would then be able to recognize the potential risk of human Lyme disease and implement ways to reduce the Lyme disease risk associated with suburban development. The purpose of this study is to answer the following research questions: 1) How has the spatial distribution of Lyme disease in Northern Virginia changed since 2005 with respect to land cover? 2) Which suburban communities are more at risk for Lyme disease when considering their land cover types and the increasing spatial distribution of Lyme disease?

Yes / Amid much recent interest we discuss a Variance Gamma model for Rugby Union matches (applications to other sports are possible). Our model emerges as a special case of the recently introduced Gamma Difference distribution though there is a rich history of applied work using the Variance Gamma distribution – particularly in finance. Restricting to this special case adds analytical tractability and computational ease. Our three-dimensional model extends classical two-dimensional Poisson models for soccer. Analytical results are obtained for match outcomes, total score and the awarding of bonus points. Model calibration is demonstrated using historical results, bookmakers’ data and tournament simulations.

Characterization of distributions has been an important topic in statistical theory for decades. Although there have been many well known results already developed, it is still of great interest to find new characterizations of commonly used distributions in application, such as normal or gamma distribution. In practice, sometimes we make guesses on the distribution to be fitted to the data observed, sometimes we use the characteristic properties of those distributions to do so. In this paper we will restrict our attention to the characterizations of gamma distribution as well as some related studies on the corresponding parameter estimation based on the characterization properties. Some simulation studies are also given.

The thesis deals with high-order stochastic dominance of random variables and portfolios. The summary of findings about high-order stochastic dominance and portfolio efficiency is presented. As a main part of the thesis it is proven that under assumption of both normal and gamma distribution the infinite-order stochastic dominance is equivalent to the second-order stochastic dominance. The necessary and sufficient condition for the infinite-order stochastic dominance portfolio efficiency is derived under the assumption of normality. The condition is used in the empirical part of the thesis where parametrical approach to the portfolio efficiency is compared to the nonparametric scenario approach. The derived necessary and sufficient condition is based on the assumption of normality; therefore we use two sets of data, one with fulfilled assumption of normality and the other for which the assumption of normality was unambigously rejected. Consequently, the influence of fulfillment of the normality assumption on the results of the necessary and sufficient condition for portfolio efficiency is estimated.