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The publisher of the Degrees of Reading Power test of reading comprehension (DRP) calibrate their test using an item response model called the Rasch or one-parameter logistic model. The relationship between the use of the Rasch model in calibration of the DRP and the use of the DRP as a component of the Virginia Literacy Passport Testing Program (LPT) is addressed. Analyses concentrate on sixth grade students who were administered the DRP in 1991. The question that arises is whether the Rasch model is the appropriate model to use to calibrate the DRP in this high-stakes setting. The majority of research that has been reported by the publisher of the DRP to assess the adequacy of the Rasch model have not included direct checks on model assumptions, model features or model predictions. Instead, they have relied almost exclusively on statistical tests in assessment of model fit. This study will assess the adequacy of fitting DRP test data to the Rasch model through direct examination of the assumptions, features and predictions of the IRT model. This is accomplished by comparing the Rasch model to the less restrictive two- and three-parameter logistic models. Robust IRT-based goodness-of-fit techniques are conducted. When the DRP is used in a high stakes setting, guessing is likely for those in jeopardy of failing. Under these circumstances, we must attend to the possibility that guessing may be a factor and thereby calibrate the DRP with the three-parameter model, as this model takes guessing into account. / Ph. D.

This thesis aims to analyze some of the available methods that aid in risk estimation based on econometric models, as well as to propose some new ones. Some of the questions that are expected to be answered include which distribution to choose to obtain better risk estimates for series with abnormal behaviours, how to determine whether the distribution in parametric conditional models is a Student’s t, and how to assess whether an asset’s risk helps predict the risk of another asset. In Chapter 1, we estimate several cryptocurrencies’ Expected Shortfall using different error distributions and GARCH-type models for conditional variance. ur goal is to examine which distributions perform better and to check which component of the specification plays a more crucial role in estimating Expected Shortfall. The performance of the estimations is conducted using a backtesting technique with a rolling-window approach. Results show that, in the case of Bitcoin, it is important to use a distribution with at least two parameters that control its shape and an extension of the GARCH model, whether it be the NGARCH or the CGARCH model. On the other hand, other smaller cryptocurrencies yield good enough risk predictions with the Student’s t distribution and a GARCH model. The fact that the main measures of financial risk are focused on the tail of the distribution of returns highlights the importance of the choice of an appropriate distribution model. Chapter 2 develops a procedure for consistently testing the specification of a Student’s t distribution for the innovations of a dynamic model. This contributes to the existing literature by providing a test for Student’s t distributions in conditional mean and variance models with a parameter-free test statistic and, thus, a known asymptotic distribution, avoiding the use of more computationally costly resampling techniques such as bootstrapping. The specific expressions needed for the computation of the test statistic are obtained by adapting the generic test of Bai (2003), which is based on the Khmaladze (1988) transformation of the model residuals. Finally, in Chapter 3, the concept of Granger causality in Expected Shortfall (ES) is introduced, along with a testing procedure to detect this type of predictive relationship between return series. Granger causality in Expected Shortfall is here defined as the predictive ability of tail values of a series over future tail values of another series on average. This definition may help in analyzing whether past values of an asset in extreme risk affect future extreme risk values of another asset. The main contribution of this chapter is a test for detecting this type of causality, based on the test for Granger causality in VaR by Hong et al. (2009). An empirical application on financial institutions from different industries (banking, insurance, and diversified financials) is presented to analyze the risk spillovers in the US financial market. The contribution of this thesis to the field of financial econometrics focuses on the market risk of financial assets, both in its modeling through the metric known as Expected Shortfall suggested in the Basel III Accords and in its utility beyond capital requirements. The results highlight the importance of a good specification of the chosen distribution model for risk estimation - especially in high-risk assets such as cryptocurrencies - and a test is proposed to verify if the conditional distribution in parametric models used for risk predictions is or is not a Student’s t distribution. Finally, a Granger causality test in Expected Shortfall is proposed, which allows for studying risk propagation in tails of return distributions. The proposed test can be used to investigate interconnections within and between markets as a complement when evaluating systemic risk. Other potential applications include improving Expected Shortfall forecasts by including causing variables as regressors in estimations, studying the inclusion of certain asset pairs in the same portfolio based on how they interact in the riskiest situations, or constructing networks of extreme risk propagation. / Esta tesis doctoral ha sido financiada mediante una ayuda FPU por el Ministerio de Educación, Cultura y Deporte (FPU17/06227).

The global loss of biodiversity calls for robust large-scale diversity assessment. Biological diversity is a multi-faceted concept; defined as the “variety of life”, answering questions such as “How much is there?” or more precisely “Have we succeeded in reducing the rate of its decline?” is not straightforward. While various aspects of biodiversity give rise to numerous ways of quantification, we focus on temporal (and spatial) trends and their changes in species diversity. Traditional diversity indices summarise information contained in the species abundance distribution, i.e. each species' proportional contribution to total abundance. Estimated from data, these indices can be biased if variation in detection probability is ignored. We discuss differences between diversity indices and demonstrate possible adjustments for detectability. Additionally, most indices focus on the most abundant species in ecological communities. We introduce a new set of diversity measures, based on a family of goodness-of-fit statistics. A function of a free parameter, this family allows us to vary the sensitivity of these measures to dominance and rarity of species. Their performance is studied by assessing temporal trends in diversity for five communities of British breeding birds based on 14 years of survey data, where they are applied alongside the current headline index, a geometric mean of relative abundances. Revealing the contributions of both rare and common species to biodiversity trends, these "goodness-of-fit" measures provide novel insights into how ecological communities change over time. Biodiversity is not only subject to temporal changes, but it also varies across space. We take first steps towards estimating spatial diversity trends. Finally, processes maintaining biodiversity act locally, at specific spatial scales. Contrary to abundance-based summary statistics, spatial characteristics of ecological communities may distinguish these processes. We suggest a generalisation to a spatial summary, the cross-pair overlap distribution, to render it more flexible to spatial scale.

O Modelo de Equações Estruturais (MEE) é habitualmente ajustado para realizar uma análise confirmatória sobre as conjecturas de um pesquisador acerca do relacionamento entre as variáveis observadas e latentes de algum estudo. Na prática, a maneira mais recorrente de avaliar a qualidade das estimativas de um MEE é a partir de medidas que buscam mensurar o quanto a usual matriz de covariâncias clássicas ou ordinárias se distancia da matriz de covariâncias do modelo ajustado, ou a magnitude do afastamento entre as funções de discrepância do modelo hipotético e do modelo saturado. Entretanto, elas podem não captar problemas no ajuste quando há muitos parâmetros a estimar ou bastantes observações. A fim de detectar irregularidades no ajustamento resultantes do impacto provocado pela presença de outliers no conjunto de dados, este trabalho contemplou alguns indicadores conhecidos na literatura, como também considerou alterações no Índice da Qualidade do Ajuste (ou GFI, de Goodness-of-Fit Index) e no Índice Corrigido da Qualidade do Ajuste (ou AGFI, de Ajusted Goodness-of-Fit Index), ambos nas expressões para estimação de parâmetros pelo método de Máxima Verossimilhança, que consistiram em substituir a tradicional matriz de covariâncias pelas matrizes de covariâncias computadas com os seguintes estimadores: Elipsoide de Volume Mínimo, Covariância de Determinante Mínimo, S, MM e Gnanadesikan-Kettenring Ortogonalizado (GKO). Através de estudos de simulação sobre perturbações de desvio de simetria e excesso de curtose, em baixa e alta frações de contaminação, em diferentes tamanhos de amostra e quantidades de variáveis observadas afetadas, foi possível constatar que as propostas de modificação do GFI e do AGFI adaptadas pelo estimador GKO foram as únicas que conseguiram ser informativas em todas essas situações, devendo-se escolher a primeira ou a segunda respectivamente quando a quantidade de parâmetros a serem estimados é baixa ou elevada. / The Structural Equation Model (SEM) is usually set to perform a confirmatory analysis on the assumptions of a researcher about the relationship between the observed variables and the latent variables of such a study. In practice, the most iterant way of evaluating the quality of the estimates of a SEM comes either from procedures of measuring how distant the usual classic or ordinary covariance matrix is from the covariance matrix of the adjusted model, or from the magnitude of the hiatus in discrepancy functions of both the hypothetical model and the saturated model. Nevertheless, they may fail to capture problems in the adjustment in the occurrence of either several parameters to estimate or several observations. This study included indicators known in the literature in order to detect irregularities in the adjustment resulting from the impact caused by the presence of outliers in the data set. This study has also considered changes in both the Goodness-of-Fit Index (GFI) and the Adjusted Goodness-of-Fit Index (AGFI) in the expressions for parameter estimation by Maximum Likelihood method, which consisted in replacing the traditional covariance matrix by the robust covariance matrices computed through the following estimators: Minimum Volume Ellipsoid, Minimum Covariance Determinant, S, MM and Orthogonalized Gnanadesikan-Kettenring (OGK). Through simulation studies on disturbances of both symmetry deviations and excess kurtosis in both low and high fractions of contamination in different sample sizes and quantities of affected observed variables it has become clear that the proposals of modification of both the GFI and the AGFI adapted by the OGK estimator were the only ones able to be informative in all these situations. It must be considered that GFI or AGFI must be used when the number of parameters to be estimated is either low or high, respectively.

O Modelo de Equações Estruturais (MEE) é habitualmente ajustado para realizar uma análise confirmatória sobre as conjecturas de um pesquisador acerca do relacionamento entre as variáveis observadas e latentes de algum estudo. Na prática, a maneira mais recorrente de avaliar a qualidade das estimativas de um MEE é a partir de medidas que buscam mensurar o quanto a usual matriz de covariâncias clássicas ou ordinárias se distancia da matriz de covariâncias do modelo ajustado, ou a magnitude do afastamento entre as funções de discrepância do modelo hipotético e do modelo saturado. Entretanto, elas podem não captar problemas no ajuste quando há muitos parâmetros a estimar ou bastantes observações. A fim de detectar irregularidades no ajustamento resultantes do impacto provocado pela presença de outliers no conjunto de dados, este trabalho contemplou alguns indicadores conhecidos na literatura, como também considerou alterações no Índice da Qualidade do Ajuste (ou GFI, de Goodness-of-Fit Index) e no Índice Corrigido da Qualidade do Ajuste (ou AGFI, de Ajusted Goodness-of-Fit Index), ambos nas expressões para estimação de parâmetros pelo método de Máxima Verossimilhança, que consistiram em substituir a tradicional matriz de covariâncias pelas matrizes de covariâncias computadas com os seguintes estimadores: Elipsoide de Volume Mínimo, Covariância de Determinante Mínimo, S, MM e Gnanadesikan-Kettenring Ortogonalizado (GKO). Através de estudos de simulação sobre perturbações de desvio de simetria e excesso de curtose, em baixa e alta frações de contaminação, em diferentes tamanhos de amostra e quantidades de variáveis observadas afetadas, foi possível constatar que as propostas de modificação do GFI e do AGFI adaptadas pelo estimador GKO foram as únicas que conseguiram ser informativas em todas essas situações, devendo-se escolher a primeira ou a segunda respectivamente quando a quantidade de parâmetros a serem estimados é baixa ou elevada. / The Structural Equation Model (SEM) is usually set to perform a confirmatory analysis on the assumptions of a researcher about the relationship between the observed variables and the latent variables of such a study. In practice, the most iterant way of evaluating the quality of the estimates of a SEM comes either from procedures of measuring how distant the usual classic or ordinary covariance matrix is from the covariance matrix of the adjusted model, or from the magnitude of the hiatus in discrepancy functions of both the hypothetical model and the saturated model. Nevertheless, they may fail to capture problems in the adjustment in the occurrence of either several parameters to estimate or several observations. This study included indicators known in the literature in order to detect irregularities in the adjustment resulting from the impact caused by the presence of outliers in the data set. This study has also considered changes in both the Goodness-of-Fit Index (GFI) and the Adjusted Goodness-of-Fit Index (AGFI) in the expressions for parameter estimation by Maximum Likelihood method, which consisted in replacing the traditional covariance matrix by the robust covariance matrices computed through the following estimators: Minimum Volume Ellipsoid, Minimum Covariance Determinant, S, MM and Orthogonalized Gnanadesikan-Kettenring (OGK). Through simulation studies on disturbances of both symmetry deviations and excess kurtosis in both low and high fractions of contamination in different sample sizes and quantities of affected observed variables it has become clear that the proposals of modification of both the GFI and the AGFI adapted by the OGK estimator were the only ones able to be informative in all these situations. It must be considered that GFI or AGFI must be used when the number of parameters to be estimated is either low or high, respectively.

This project studies the signal detection and identification problem in high-dimensional noisy data and the possibility of using it on microbiome data. An extensive simulation study was performed on generated data using as well as a microbiome dataset collected on patients with Parkinson's disease, using Donoho and Jin's Higher criticism, Jager and Wellner's phi-divergence-based goodness-of-fit-test and Stepanova and Pavlenko's CsCsHM statistic . We present some novel approaches based on established theory that perform better than existing methods and show that it is possible to use the signal identification framework to detect differentially abundant features in microbiome data. Although the novel approaches produce good results, they lack substantial mathematical foundations and should be avoided if theoretical rigour is needed. We also conclude that while we have found that it is possible to use signal identification methods to find abundant features in microbiome data, further refinement is necessary before it can be properly used in research. / Detta projekt studerar signaldetekterings- och identifieringsproblemet i högdimensionell brusig data och möjligheten att använda det på mikrobiomdata från människor. En omfattande simuleringsstudie utfördes på genererad data samt ett mikrobiomdataset som samlats in på patienter med Parkinsons sjukdom, med hjälp av ett antal goodness-of-fit-metoder: Donoho och Jins Higher criticis , Jager och Wellners phi-divergenser och Stepanova och Pavelenkos CsCsHM. Vi presenterar några nya tillvägagångssätt baserade på vedertagen teori som visar sig fungera bättre än befintliga metoder och visar att det är möjligt att använda signalidentifiering för att upptäcka olika funktioner i mikrobiomdata. Även om de nya metoderna ger goda resultat saknar de betydande matematiska grunder och bör undvikas om teoretisk formalism är nödvändigt. Vi drar också slutsatsen att medan vi har funnit att det är möjligt att använda signalidentifieringsmetoder för att hitta information i mikrobiomdata, är ytterligare experiment nödvändiga innan de kan användas på ett korrekt sätt i forskning.

Thesis (MComm (Statistics and Actuarial Science))--Stellenbosch University, 2008. / The goodness-of- t of a statistical model describes how well it ts a set of observations. Measures of goodness-of- t typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, for example to test for normality, to test whether two samples are drawn from identical distributions, or whether outcome frequencies follow a speci ed distribution. Goodness-of- t for copulas is a special case of the more general problem of testing multivariate models, but is complicated due to the di culty of specifying marginal distributions. In this thesis, the goodness-of- t test statistics for general distributions and the tests for copulas are investigated, but prior to that an understanding of copulas and their properties is developed. In fact copulas are useful tools for understanding relationships among multivariate variables, and are important tools for describing the dependence structure between random variables. Several univariate, bivariate and multivariate test statistics are investigated, the emphasis being on tests for normality. Among goodness-of- t tests for copulas, tests based on the probability integral transform, Rosenblatt's transformation, as well as some dimension reduction techniques are considered. Bootstrap procedures are also described. Simulation studies are conducted to rst compare the power of rejection of the null hypothesis of the Clayton copula by four di erent test statistics under the alternative of the Gumbel-Hougaard copula, and also to compare the power of rejection of the null hypothesis of the Gumbel-Hougaard copula under the alternative of the Clayton copula. An application of the described techniques is made to a practical data set.

Research has shown that effective school leadership has a positive influence on school effectiveness and student achievement. Current reform efforts include teachers, both formally and informally, as leaders of schools. However, there are currently no widely-accepted measurements or models to assess both formal and informal teacher leadership in schools. The purpose of this study was to compare model fit for the four-factor model of teacher leadership to model fit for three alternative models. The four-factor model was developed during the second administration of the Teacher Leadership Inventory (TLI), and the three alternative models were developed from the results and recommendations from the confirmatory factor analysis of that administration. Teacher responses to the second administration of the TLI constituted the data set for this study. Participants included 421 teachers from 23 schools in three East Tennessee school districts. Confirmatory factor analyses were conducted for each of the measurement models under investigation, and model fit indices and parameter estimates of all four models were used for comparison. Model fit indices indicate better model fit for the four-factor model over both the two-factor and five-factor models but not over the three-factor model. However, further evaluation of both parameter estimates and prior research provide support for the acceptability of the four-factor model over the three-factor model.

We consider a nonparametric survival model with random censoring. To test whether the hazard rate has a parametric form the unknown hazard rate is estimated by a kernel estimator. Based on a limit theorem stating the asymptotic normality of the quadratic distance of this estimator from the smoothed hypothesis an asymptotic ®-test is proposed. Since the test statistic depends on the maximum likelihood estimator for the unknown parameter in the hypothetical model properties of this parameter estimator are investigated. Power considerations complete the approach.