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Master of Science / Department of Statistics / Suzanne Dubnicka / To help with identifying distributions to effectively model univariate continuous data, the R function diagnostic is proposed. The function will aid in determining reasonable candidate distributions that the data may have come from. It uses a combination of the Pearson goodness of fit statistic, Anderson-Darling statistic, Lin’s concordance correlation between the theoretical quantiles and observed quantiles, and the maximum difference between the theoretical quantiles and the observed quantiles. The function generates reasonable candidate distributions, QQ plots, and histograms with superimposed density curves. When a simulation study was done, the function worked adequately; however, it was also found that many of the distributions look very similar if the parameters are chosen carefully. The function was then used to attempt to decipher which distribution could be used to model weekly grocery expenditures of a family household.

Statistically more test samples obtained from a single chip would give a better picture of the various noise processes present. Increasing the number of samples while testing one chip would however lead to an increase in the testing time, decreasing the overall throughput. The aim of this report is to investigate the detection of non-Gaussian noise (burst noise) in a random set of data with a small number of samples. In order to determine whether a given set of noise samples has non-Gaussian noise processes present, a Chi-Squared ‘Goodness of Fit’ test on a modeled set of random data is presented. A discussion of test methodologies using a single test measurement pass as well as two passes is presented from the obtained simulation results. / text

Discrete-time survival analysis as an educational research technique has focused on analysing and interpretating parameter estimates. The purpose of this study was to examine the effects of certain data characteristics on the hazard estimates and goodness of fit statistics. Fifty-four simulated data sets were crossed with four conditions in a 2 (time period) by 3 (distribution of Y = 1) by 3 (distribution of Y = 0) by 3 (sample size) design.

One of the most useful and best known goodness of fit test is the Kolmogorov one-sample test. The assumptions for the Kolmogorov (one-sample test) test are: 1. A random sample; 2. A continuous random variable; 3. F(x) is a completely specified hypothesized cumulative distribution function. The Kolmogorov one-sample test has a wide range of applications. Knowing the effect fromusing the test when an assumption is not met is of practical importance. The purpose of this research is to analyze the robustness of the Kolmogorov one-sample test to sampling from a finite discrete distribution. The standard tables for the Kolmogorov test are derived based on sampling from a theoretical continuous distribution. As such, the theoretical distribution is infinite. The standard tables do not include a method or adjustment factor to estimate the effect on table values for statistical experiments where the sample stems from a finite discrete distribution without replacement. This research provides an extension of the Kolmogorov test when the hypothesized distribution function is finite and discrete, and the sampling distribution is based on sampling without replacement. An investigative study has been conducted to explore possible tendencies and relationships in the distribution of Dn when sampling with and without replacement for various parameter settings. In all, 96 sampling distributions were derived. Results show the standard Kolmogorov table values are conservative, particularly when the sample sizes are small or the sample represents 10% or more of the population.

This thesis deals with the goodness of fit tests in nonparametric model in the presence of unknown parameters of the probability distribution. The first part is devoted to understanding of the theoretical basis. We compare two methodologies for the construction of test statistics with application of empirical characteristic and empirical distribution functions. We use kernel estimates of regression functions and parametric bootstrap method to approximate the critical values of the tests. In the second part of the thesis, the work is complemented with the simulation study for different choices of weighting functions and parameters. Finally we illustrate the use and the comparison of goodness of fit tests on the example with the real data set. Powered by TCPDF (www.tcpdf.org)

By testing a fiber beard, certain cotton fiber length parameters can be obtained rapidly. This is the method used by the High Volume Instrument (HVI). This study is aimed to explore the approaches and obtain the inference of length distributions of HVI beard sam- ples in order to develop new methods that can help us find the distribution of original fiber lengths and further improve HVI length measurements. At first, the mathematical functions were searched for describing three different types of length distributions related to the beard method as used in HVI: cotton fiber lengths of the original fiber population before picked by the HVI Fibrosampler, fiber lengths picked by HVI Fibrosampler, and fiber beard's pro-jecting portion that is actually scanned by HVI. Eight sets of cotton samples with a wide range of fiber lengths are selected and tested on the Advanced Fiber Information System (AFIS). The measured single fiber length data is used for finding the underlying theoreti-cal length distributions, and thus can be considered as the population distributions of the cotton samples. In addition, fiber length distributions by number and by weight are dis- cussed separately. In both cases a mixture of two Weibull distributions shows a good fit to their fiber length data. To confirm the findings, Kolmogorov-Smirnov goodness-of-fit tests were conducted. Furthermore, various length parameters such as Mean Length (ML) and Upper Half Mean Length (UHML) are compared between the original distribution from the experimental data and the fitted distributions. The results of these obtained fiber length distributions are discussed by using Partial Least Squares (PLS) regression, where the dis-tribution of the original fiber length from the distribution of the projected one is estimated.

<div>Networks and point processes provide flexible tools for representing and modeling complex dependencies in data arising from various social and physical domains. Graphs, or networks, encode relational dependencies between entities, while point processes characterize temporal or spatial interactions among events.</div><div><br></div><div>In the first part of this dissertation, we consider dynamic network data (such as communication networks) in which links connecting pairs of nodes appear continuously over time. We propose latent space point process models to capture two different aspects of the data: (i) communication occurs at a higher rate between individuals with similar latent attributes (i.e., homophily); and (ii) individuals tend to reciprocate communications from others, but in a varied manner. Our framework marries ideas from point process models, including Poisson and Hawkes processes, with ideas from latent space models of static networks. We evaluate our models on several real-world datasets and show that a dual latent space model, which accounts for heterogeneity in both homophily and reciprocity, significantly improves performance in various link prediction and network embedding tasks.</div><div><br></div><div>In the second part of this dissertation, we develop nonparametric goodness-of-fit tests for discrete distributions and point processes that contain intractable normalization constants, providing the first generally applicable and computationally feasible approaches under those circumstances. Specifically, we propose and characterize Stein operators for discrete distributions, and construct a general Stein operator for point processes using the Papangelou conditional intensity function. Based on the proposed Stein operators, we establish kernelized Stein discrepancy measures for discrete distributions and point processes, which enable us to develop nonparametric goodness-of-fit tests for un-normalized density/intensity functions. We apply the kernelized Stein discrepancy tests to discrete distributions (including network models) as well as temporal and spatial point processes. Our experiments demonstrate that the proposed tests typically outperform two-sample tests based on the maximum mean discrepancy, which, unlike our goodness-of-fit tests, assume the availability of exact samples from the null model.</div><div><br></div>

Este trabalho trata de modelos de regressão quantílica. Foi feita uma introdução a essa classe de modelos para motivar a discussão. Em seguida, conceitos inferenciais, como estimação, intervalos de confiança, testes de hipóteses para os parâmetros são discutidos, acompanhados de alguns estudos de simulação. Para analisar a qualidade do ajuste, são apresentados o coeficiente de determinação e um teste de falta de ajuste para modelos de regressão quantílica. Também é proposta a utilização de gráficos para análise da qualidade do ajuste considerando a distribuição Laplace Assimétrica. Uma aplicação utilizando um banco de dados com informação sobre renda no Brasil foi utilizado para exemplificar os tópicos discutidos durante o texto. / This work is about quantile regression models. An introduction was made to this class of models to motivate the discussion. Then, inferential concepts, like estimation, confidence intervals, tests of hypothesis for the parameters are discussed, followed by some simulation studies. To analyse goodness of fit, a coefficient of determination and a lack-of-fit test for quantile regression models are presented. Its also proposed the use of graphs for the goodness of fit analysis considering the Asymmetric Laplace Distribution. An application using a data base with information about income in Brazil was used to exemplify the topics discussed during the text.