In the present study the cross sectional areas of individual muscle fibers were investigated with focus on statistical methodology. This thesis includes data from two studies; Resistance Study and Method Study. The Resistance Study analyzes the effect of exercise by comparing muscle fiber area before and after eight weeks of resistance training. Muscle biopsies from the vastus lateralis muscle were obtained from young male participants. The purpose of the Method Study was to examine the variation between right and left leg. Contrary to previous studies, this thesis focuses on individual data rather than on group-based data, and therefore takes a different approach than the previously published articles. This is proven to be successful since information is lost when analyzing group-wise, as the increase in small muscle fibers did not show when analyzing as a group. The results of the Resistance Study is similar to the results of the Method Study. Means and standard deviations have a wide spread both between subjects and between biopsies taken from the same subject. Inference on the 10th and 90th percentiles shows a positive pattern in the Resistance Study, in the sense that both the smallest and the largest muscle fibers have grown as a result of the resistance training. If muscle fiber area is used as a proxy for training effect, the conclusion is that many people seem to have responded well to the training.

Title: Stochastical inference in the model of extreme events Author: Jan Dienstbier Department/Institute: Department of probability and mathematical statistics Supervisor of the doctoral thesis: Doc. RNDr. Jan Picek, CSc. Abstract: The thesis deals with extremal aspects of linear models. We provide a brief explanation of extreme value theory. The attention is then turned to linear models Yn×1 = Xn×pβp×1 + En×1 with the errors Ei ∼ F, i = 1, . . . , n fulfilling the do- main of attraction condition. We examine the properties of the regression quantiles of Koenker and Basset (1978) under this setting we develop theory dealing with extremal characteristics of linear models. Our methods are based on an approximation of the regression quantile process for α ∈ [0, 1] expanding older results of Gutenbrunner et al. (1993). Our result holds in [α∗ n, 1 − α∗ n] with a better rate of α∗ n → 0 than the other approximations described previously in the literature. Consecutively we provide an ap- proximation of the tails of regression quantile. The approximations of the tails enable to develop theory of the smooth functionals, which are used to establish a new class of estimates of extreme value index. We prove T(F−1 n (1 − knt/n)) is consistent and asymp- totically normal estimate of extreme for any T member of the class....

Context: Today it is easy to do banking transaction digitally, both on a computer or by using a mobile phone. As the banking-services increases and gets implemented to multi-platforms it makes it easier for a fraudster to commit financial fraud. This thesis will focus on investigating log-files from a Mobile Money system that makes it possible to do banking transactions with a mobile phone.  Objectives: The objectives in this thesis is to evaluate if it is possible to combine two statistical methods, Benford's law together with statistical quantiles, to find a statistical way to find fraudsters within a Mobile Money system. Methods: Rules was extracted from a case study with focus on a Mobile Money system and limits was calculated by using quantiles. A fraud detector was implemented that use these rules together with limits and Benford's law in order to detect fraud.The fraud detector used the methods both independently and combined.The performance was then evaluated. Results: The results show that it is possible to use the Benford's law and statistical quantiles within the studied Mobile Money system. It is also shown that there is only a very small difference when the two methods are combined or not both in detection rate and accuracy precision. Conclusions: We conclude that by combining the chosen methods it is possible to get a medium-high true positive rates and very low false positive rates. The most effective method to find fraudsters is by only using quantiles. However, combining Benford's law with quantiles gives the second best result.

La description et l’estimation des modèles aussi bien univariés que multivariés impliquantdes distributions à queue lourde est un enjeu applicatif majeur. Les L-moments sontdevenus des outils classiques alternatifs aux moments centraux pour décrire les comportementsen dispersion, asymétrie, kurtosis d’une distribution univariée à queue lourde. Eneffet, contrairement aux moments centraux correspondants, ils sont bien définis dès quel’espérance de la distribution d’intérêt est finie. Les L-moments peuvent être vus comme laprojection de la fonction quantile sur une famille orthogonale de polynômes, récupérant lalinéarité inhérente aux quantiles. Nous estimerons dans un premier temps les paramètresde modèles semi paramétriques définis par des contraintes sur ces L-moments par des méthodesde minimisation de divergences.Nous proposons dans un second temps une généralisation des L-moments aux distributionsmultivariées qui passe par la définition d’un quantile multivarié défini comme untransport entre la distribution uniforme sur [0; 1]d et la distribution d’intérêt. Cela nouspermet de proposer des descripteurs pour des distributions multivariées adaptés à l’étudedes queues lourdes. Nous détaillons leurs expressions dans le cadre de modèles possédantdes paramètres de rotation.Enfin, nous proposons des M-estimateurs de la matrice de dispersion des distributions complexeselliptiques. Ces dernières forment un modèle multivarié semi-paramétrique contenantnotamment des distributions à queue lourde. Des M-estimateurs spécifiques adaptésaux distributions elliptiques avec une hypothèse supplémentaire de stationnarité sont égalementproposés. Les performances et la robustesse des estimateurs sont étudiées.Les signaux radar provenant de fouillis tels les fouillis de mer ou les fouillis de sol sontsouvent modélisés par des distributions elliptiques. Nous illustrerons les performances dedétecteurs construits à partir de l’estimation de la matrice de dispersion par les méthodesproposées pour différents scénarios radar pour lesquels la robustesse de la procédure d’estimationest cruciale. / The description and the estimation of univariate and multivariate models whose underlyingdistribution is heavy-tailed is a strategic challenge. L-moments have becomeclassical tools alternative to central moments for the description of dispersion, skewnessand kurtosis of a univariate heavy-tailed distribution. Indeed, contrary to correspondingcentral moments, they are well defined since the expectation of the distribution of interestis finite. L-moments can be seen as projections of the quantile function on a family oforthogonal polynomials. First, we will estimate parameters of semi-parametric modelsdefined by constraints on L-moments through divergence methods.We will then propose a generalization of L-moments for multivariate distributions using amultivariate quantile function defined as a transport of the uniform distribution on [0; 1]dand the distribution of interest. As their univariate versions, these multivariate L-momentsare adapted for the study of heavy-tailed distributions. We explicitly give their formulationsfor models with rotational parameters.Finally, we propose M-estimators of the scatter matrix of complex elliptical distributions.The family of these distributions form a multivariate semi-parametric model especiallycontaining heavy-tailed distributions. Specific M-estimators adapted to complex ellipticaldistribution with an additional assumption of stationarity are proposed. Performancesand robustness of introduced estimators are studied.Ground and sea clutters are often modelized by complex elliptical distributions in the fieldof radar processing. We illustrate performances of detectors built from estimators of thescatter matrix through proposed methods for different radar scenarios.

The g - and - h distributional family is generated from a relatively simple transformation of the standard normal. By changing the skewness and elongation parameters g and h, this distributional family can approximate a broad spectrum of commonly used distributional shapes, such as normal, lognormal, Weibull and exponential. Consequently, it is easy to use in simulation studies and has been applied in multiple areas, including risk management, stock return analysis and missing data imputation studies. The current available methods to estimate the g - and - h distributional family include: letter value based method (LV), numerical maximal likelihood method (NMLE), and moment methods. Although these methods work well when no outliers or contaminations exist, they are not resistant to a moderate amount of contaminated observations or outliers. Meanwhile, NMLE is a computational time consuming method when data sample size is large. In this dissertation a quantile based least squares (QLS) estimation method is proposed to fit the g - and - h distributional family parameters and then derive its basic properties. Then QLS method is extended to a robust version (rQLS). Simulation studies are performed to compare the performance of QLS and rQLS methods with LV and NMLE methods to estimate the g - and - h parameters from random samples with or without outliers. In random samples without outliers, QLS and rQLS estimates are comparable to LV and NMLE in terms of bias and standard error. On the other hand, rQLS performs better than other non-robust method to estimate the g - and - h parameters when moderate amount of contaminated observations or outliers exist. The flexibility of the g - and - h distribution and the robustness of rQLS method make it a useful tool in various fields. The boxplot (BP) method had been used in multiple outlier detections by controlling the some-outside rate, which is the probability of one or more observations, in an outlier-free sample, falling into the outlier region. The BP method is distribution dependent. Usually the random sample is assumed normally distributed; however, this assumption may not be valid in many applications. The robustly estimated g - and - h distribution provides an alternative approach without distributional assumptions. Simulation studies indicate that the BP method based on robustly estimated g - and - h distribution identified reasonable number of true outliers while controlling number of false outliers and some-outside rate compared to normal distributional assumption when it is not valid. Another application of the robust g - and - h distribution is as an empirical null distribution in false discovery rate method (denoted as BH method thereafter). The performance of BH method depends on the accuracy of the null distribution. It has been found that theoretical null distributions were often not valid when simultaneously performing many thousands, even millions, of hypothesis tests. Therefore, an empirical null distribution approach is introduced that uses estimated distribution from the data. This is recommended as a substitute to the currently used empirical null methods of fitting a normal distribution or another member of the exponential family. Similar to BP outlier detection method, the robustly estimated g - and - h distribution can be used as empirical null distribution without any distributional assumptions. Several real data examples of microarray are used as illustrations. The QLS and rQLS methods are useful tools to estimate g - and - h parameters, especially rQLS because it noticeably reduces the effect of outliers on the estimates. The robustly estimated g - and - h distributions have multiple applications where distributional assumptions are required, such as boxplot outlier detection or BH methods. / Statistics

Cette thèse s'inscrit dans le contexte de la statistique des valeurs extrêmes. Elle y apporte deux contributions principales. Dans la littérature récente en statistique des valeurs extrêmes, un modèle de queues de distributions a été introduit afin d'englober aussi bien les lois de type Pareto que les lois à queue de type Weibull. Les deux principaux types de décroissance de la fonction de survie sont ainsi modélisés. Un estimateur des quantiles extrêmes a été déduit de ce modèle mais il dépend de deux paramètres inconnus, le rendant inutile dans des situations pratiques. La première contribution de cette thèse est de proposer des estimateurs de ces paramètres. Insérer nos estimateurs dans l'estimateur des quantiles extrêmes précédent permet alors d'estimer des quantiles extrêmes pour des lois de type Pareto aussi bien que pour des lois à queue de type Weibull d'une façon unifiée. Les lois asymptotiques de nos trois nouveaux estimateurs sont établies et leur efficacité est illustrée sur des données simulées et sur un jeu de données réelles de débits de la rivière Nidd se situant dans le Yorkshire en Angleterre. La seconde contribution de cette thèse consiste à introduire et estimer une nouvelle mesure de risque appelé Conditional Tail Moment. Elle est définie comme le moment d'ordre a>0 de la loi des pertes au-delà du quantile d'ordre p appartenant à ]0,1[ de la fonction de survie. Estimer le Conditional Tail Moment permet d'estimer toutes les mesures de risque basées sur les moments conditionnels telles que la Value-at-Risk, la Conditional Tail Expectation, la Conditional Value-at-Risk, la Conditional Tail Variance ou la Conditional Tail Skewness. Ici, on s'intéresse à l'estimation de ces mesures de risque dans le cas de pertes extrêmes c'est-à-dire lorsque p tend vers 0 lorsque la taille de l'échantillon augmente. On suppose également que la loi des pertes est à queue lourde et qu'elle dépend d'une covariable. Les estimateurs proposés combinent des méthodes d'estimation non-paramétrique à noyau avec des méthodes issues de la statistique des valeurs extrêmes. Le comportement asymptotique de nos estimateurs est établi et illustré aussi bien sur des données simulées que sur des données réelles de pluviométrie provenant de la région Cévennes-Vivarais. / This thesis can be viewed within the context of extreme value statistics. It provides two main contributions to this subject area. In the recent literature on extreme value statistics, a model on tail distributions which encompasses Pareto-type distributions as well as Weibull tail-distributions has been introduced. The two main types of decreasing of the survival function are thus modeled. An estimator of extreme quantiles has been deduced from this model, but it depends on two unknown parameters, making it useless in practical situations. The first contribution of this thesis is to propose estimators of these parameters. Plugging our estimators in the previous extreme quantiles estimator allows us to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unified way. The asymptotic distributions of our three new estimators are established and their efficiency is illustrated on a simulation study and on a real data set of exceedances of the Nidd river in the Yorkshire (England). The second contribution of this thesis is the introduction and the estimation of a new risk measure, the so-called Conditional Tail Moment. It is defined as the moment of order a>0 of the loss distribution above the quantile of order p in (0,1) of the survival function. Estimating the Conditional Tail Moment permits to estimate all risk measures based on conditional moments such as the Value-at-Risk, the Conditional Tail Expectation, the Conditional Value-at-Risk, the Conditional Tail Variance or the Conditional Tail Skewness. Here, we focus on the estimation of these risk measures in case of extreme losses i.e. when p converges to 0 when the size of the sample increases. It is moreover assumed that the loss distribution is heavy-tailed and depends on a covariate. The estimation method thus combines nonparametric kernel methods with extreme-value statistics. The asymptotic distribution of the estimators is established and their finite sample behavior is illustrated both on simulated data and on a real data set of daily rainfalls in the Cévennes-Vivarais region (France).

Correctly specified models to forecast returns of indices are important for in- vestors to minimize risk on financial markets. This thesis focuses on conditional Value at Risk modeling, employing flexible quantile regression framework and hence avoiding the assumption on the return distribution. We apply semi- parametric linear quantile regression (LQR) models with realized variance and also models with positive and negative semivariance which allows for direct modelling of the quantiles. Four European stock price indices are taken into account: Czech PX, Hungarian BUX, German DAX and London FTSE 100. The objective is to investigate how the use of realized variance influence the VaR accuracy and the correlation between the Central & Eastern and Western European indices. The main contribution is application of the LQR models for modelling of conditional quantiles and comparison of the correlation between European indices with use of the realized measures. Our results show that linear quantile regression models on one-step-ahead forecast provide better fit and more accurate modelling than classical VaR model with assumption of nor- mally distributed returns. Therefore LQR models with realized variance can be used as accurate tool for investors. Moreover we show that diversification benefits are...

Este trabalho tem como objetivo calcular intervalos de confiança para altos quantis oriundos de distribuições de caudas pesadas. Para isso, utilizamos os métodos da aproximação pela distribuição normal, razão de verossimilhanças, {\\it data tilting} e gama generalizada. Obtivemos, através de simulações, que os intervalos calculados a partir do método da gama generalizada apresentam probabilidades de cobertura bem próximas do nível de confiança, com amplitudes médias menores do que os outros três métodos, para dados gerados da distribuição Weibull. Todavia, para dados gerados da distribuição Fréchet, o método da razão de verossimilhanças fornece os melhores intervalos. Aplicamos os métodos utilizados neste trabalho a um conjunto de dados reais, referentes aos pagamentos de indenizações, em reais, de seguros de incêndio, de um determinado grupo de seguradoras no Brasil, no ano de 2003 / In this work, confidence intervals for high quantiles from heavy-tailed distributions were computed. More specifically, four methods, namely, normal approximation method, likelihood ratio method, data tilting method and generalised gamma method are used. A simulation study with data generated from Weibull distribution has shown that the generalised gamma method has better coverage probabilities with the smallest average length intervals. However, from data generated from Fréchet distribution, the likelihood ratio method gives the better intervals. Moreover, the methods used in this work are applied on a real data set from 1758 Brazilian fire claims

A importância do Estado de São Paulo (ESP) é notável no desenvolvimento do Brasil, seja no setor econômico ou energético, o que justifica o estudo do comportamento do clima nessa região. O conhecimento da variabilidade da precipitação é imprescindível na gestão de recursos hídricos e possui grande impacto na agricultura e geração de energia por meio de fontes hidrelétricas. Estudos anteriores apontaram efeitos não-lineares do El Niño-Oscilação Sul (ENOS) sobre a precipitação no ESP; entretanto, nenhum estudo específico acerca da influência da Oscilação interdecadal do Pacífico (ODP) nesta área foi ainda realizado, embora haja alguns impactos conhecidos na América do Sul. Deste modo, este trabalho estudou a relação entre anomalias de precipitação no ESP e a ODP, no período de 1901 a 2007, de forma a auxiliar as pesquisas na linha da previsão climática nessa região do Brasil. Na primeira etapa, foram descritos os regimes de precipitação tanto para a América do Sul como localmente, para o ESP, onde se destacaram fatores como a topografia e a influência do Oceano Atlântico. Posteriormente, foram calculados quantis anuais e mensais que permitiram classificar cada evento quanto ao total de precipitação. Regiões pluviometricamente homogêneas foram determinadas no ESP com base na climatologia e nos quantis de precipitação. Notou-se a relação construtiva entre eventos ENOS e as fases da ODP, com máximo durante o verão austral. Os sinais da ODP são percebidos em todo o ESP principalmente na primavera e no verão austrais. Uma análise complementar mostrou que as fases da Oscilação Multidecadal do Atlântico (AMO) também contribuem para a precipitação no ESP durante o verão e a primavera austrais no litoral, durante o verão no interior, e ao longo da primavera na região da Serra da Mantiqueira. Aparentemente, não há relação entre os eventos ENOS e a AMO. / São Paulo State (SPS) is remarkably important to the development of Brazil, economically or energetically, and this justifies climate studies on that region. Knowing rainfall variability is essential to water resources management and it has a great impact on agriculture an power production by hydroelectric power plants. Previous studies have detected non-linear effects of El Niño-Southern Oscillation (ENSO) on SPS rainfall however no specific work deals with PDO influence in this area besides some impacts on South America are known. Therefore this work has studied the relationship between rainfall anomalies in SPS and PDO from 1901 to 2007 to contribute to the climate forecasting improvement. First it was described the rainfall regime in South America, and locally in SPS where topography and the Atlantic Ocean influences were of special importance. Then annual and monthly quantiles were calculated to allow the classification of events according to rainfall totals. Rainfall homogeneous regions were established in SPS using climatology and quantiles. It was observed the constructive relationship between ENSO events and PDO phases, mainly on austral summer. PDO signals were noticed all over the SPS mostly on austral spring and summer. An additional analysis showed that Atlantic Multidecadal Oscillation (AMO) phases also contribute to SPS rainfall during austral summer and spring at the coast, only on summer at the country and during spring at the Mantiqueira Slopes. Apparently, there is no relation between ENSO events and AMO phases.

In this dissertation, we investigate jackknife empirical likelihood methods motivated by recent statistics research and other related fields. Computational intensity of empirical likelihood can be significantly reduced by using jackknife empirical likelihood methods without losing computational accuracy and stability. We demonstrate that proposed jackknife empirical likelihood methods are able to handle several challenging and open problems in terms of elegant asymptotic properties and accurate simulation result in finite samples. These interesting problems include ROC curves with missing data, the difference of two ROC curves in two dimensional correlated data, a novel inference for the partial AUC and the difference of two quantiles with one or two samples. In addition, empirical likelihood methodology can be successfully applied to the linear transformation model using adjusted estimation equations. The comprehensive simulation studies on coverage probabilities and average lengths for those topics demonstrate the proposed jackknife empirical likelihood methods have a good performance in finite samples under various settings. Moreover, some related and attractive real problems are studied to support our conclusions. In the end, we provide an extensive discussion about some interesting and feasible ideas based on our jackknife EL procedures for future studies.