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.

This dissertation proposes the development of a new quantile-based generalized logistic distribution GLDQB, by using the quantile function of the generalized logistic distribution (GLO) as the basic building block. This four-parameter distribution is highly flexible with respect to distributional shape in that it explains extensive levels of skewness and kurtosis through the inclusion of two shape parameters. The parameter space as well as the distributional shape properties are discussed at length. The distribution is characterized through its -moments and an estimation algorithm is presented for estimating the distribution’s parameters with method of -moments estimation. This new distribution is then used to fit and approximate the probability of a data set. / Dissertation (MSc)--University of Pretoria, 2014. / Statistics / MSc / Unrestricted

L-moments are defined as linear combinations of expected values of order statistics of a variable.(Hosking 1990) L-moments are estimated from samples using functions of weighted means of order statistics. The advantages of L-moments over classical moments are: able to characterize a wider range of distributions; L-moments are more robust to the presence of outliers in the data when estimated from a sample; and L-moments are less subject to bias in estimation and approximate their asymptotic normal distribution more closely. Hosking (1990) obtained an asymptotic result specifying the sample L-moments have a multivariate normal distribution as n approaches infinity. The standard deviations of the estimators depend however on the distribution of the variable. So in order to be able to build confidence intervals we would need to know the distribution of the variable. Bootstrapping is a resampling method that takes samples of size n with replacement from a sample of size n. The idea is to use the empirical distribution obtained with the subsamples as a substitute of the true distribution of the statistic, which we ignore. The most common application of bootstrapping is building confidence intervals without knowing the distribution of the statistic. The research question dealt with in this work was: How well do bootstrapping confidence intervals behave in terms of coverage and average width for estimating L-moments and ratios of L-moments? Since Hosking's results about the normality of the estimators of L-moments are asymptotic, we are particularly interested in knowing how well bootstrap confidence intervals behave for small samples. There are several ways of building confidence intervals using bootstrapping. The most simple are the standard and percentile confidence intervals. The standard confidence interval assumes normality for the statistic and only uses bootstrapping to estimate the standard error of the statistic. The percentile methods work with the (α/2)th and (1-α/2)th percentiles of the empirical sampling distribution. Comparing the performance of the three methods was of interest in this work. The research question was answered by doing simulations in Gauss. The true coverage of the nominal 95% confidence interval for the L-moments and ratios of L-moments were found by simulations.

Frequency analysis is a vital tool when nding a well-suited probability distributionin order to predict extreme rainfall. The regional frequency approach have beenused for determination of homogeneous regions, using 11 sites in Skane, Sweden. Todescribe maximum annual daily rainfall, the Generalized Logistic (GLO), GeneralizedExtreme Value (GEV), Generalized Normal (GNO), Pearson Type III (PE3),and Generalized Pareto (GPA) distributions have been considered. The method ofL-moments have been used in order to nd parameter estimates for the candidatedistributions. Heterogeneity measures, goodness-of-t tests, and accuracy measureshave been executed in order to accurately estimate quantiles for 1-, 5-, 10-, 50- and100-year return periods. It was found that the whole province of Skane could beconsidered as homogeneous. The GEV distribution was the most consistent withthe data followed by the GNO distribution and they were both used in order toestimate quantiles for the return periods. The GEV distribution generated the mostprecise estimates with the lowest relative RMSE, hence, it was concluded to be thebest-t distribution for maximum annual daily rainfall in the province.

The Burr families (Type III and Type XII) of distributions are traditionally used in the context of statistical modeling and for simulating non-normal distributions with moment-based parameters (e.g., Skew and Kurtosis). In educational and psychological studies, the Burr families of distributions can be used to simulate extremely asymmetrical and heavy-tailed non-normal distributions. Conventional moment-based estimators (i.e., the mean, variance, skew, and kurtosis) are traditionally used to characterize the distribution of a random variable or in the context of fitting data. However, conventional moment-based estimators can (a) be substantially biased, (b) have high variance, or (c) be influenced by outliers. In view of these concerns, a characterization of the Burr Type III and Type XII distributions through the method of L-moments is introduced. Specifically, systems of equations are derived for determining the shape parameters associated with user specified L-moment ratios (e.g., L-Skew and L-Kurtosis). A procedure is also developed for the purpose of generating non-normal Burr Type III and Type XII distributions with arbitrary L-correlation matrices. Numerical examples are provided to demonstrate that L-moment based Burr distributions are superior to their conventional moment based counterparts in the context of estimation, distribution fitting, and robustness to outliers. Monte Carlo simulation results are provided to demonstrate that L-moment-based estimators are nearly unbiased, have relatively small variance, and are robust in the presence of outliers for any sample size. Simulation results are also provided to show that the methodology used for generating correlated non-normal Burr Type III and Type XII distributions is valid and efficient. Specifically, Monte Carlo simulation results are provided to show that the empirical values of L-correlations among simulated Burr Type III (and Type XII) distributions are in close agreement with the specified L-correlation matrices.

京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第18562号 / 工博第3923号 / 新制||工||1603(附属図書館) / 31462 / 京都大学大学院工学研究科社会基盤工学専攻 / (主査)教授 寶 馨, 教授 中北 英一, 教授 田中 茂信 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DFAM

Made available in DSpace on 2017-07-10T18:56:25Z (GMT). No. of bitstreams: 1 Wagner Alessandro Pansera.pdf: 570958 bytes, checksum: 6984d964a2c6069ce6ee03e50131b7a1 (MD5) Previous issue date: 2010-02-01 / The rainfall is the only way of entry of water in river basins, therefore the knowledge of their spatial and temporal behavior as well as their frequency is of vital importance for the planning, operation and design of hydraulic works. To this end, probabilistic models are built based on observations made in the place of interest in order to obtain future risks. It often happens that a network of hydrological monitoring data does not show the place of interest is necessary to use procedures that allow the transport of this information. Hydrological regionalization is a statistical procedure that allows, within a homogeneous region, estimate probabilities of occurrence, by data from neighboring stations, in site of interest. The problem is how to define homogeneous region. The solution proposed in this paper was the use of multivariate clustering methods, k-means and hierarquical cluster, validated by quality index. The clusters were then subjected to measures of discordancy and heterogeneity to evaluate homogeneity of groups. The clustering methodology that obtained the best performance was a hybrid approach between k-means and ward, getting errors of at most 10% in the estimation of quantiles regional dimensionless. / O conhecimento do comportamento espacial e temporal da precipitação pluvial e sua frequência são vitais para o planejamento, operação e dimensionamento de obras hidráulicas. Para este fim, são construídos modelos probabilísticos baseados nas observações efetuadas nos locais de interesse com a finalidade de obter os riscos de ocorrência futura. No entanto, uma rede de monitoramento hidrológico não apresenta dados de todos os locais de interesse, sendo necessário recorrer a procedimentos que possibilitem o transporte desta informação. A regionalização hidrológica é um procedimento estatístico que permite, dentro de uma região homogênea, estimar probabilidades de ocorrência, pelos dados das estações vizinhas, no local de interesse. O problema está em como determinar uma região homogênea. A solução proposta neste trabalho foi a utilização de metodologias de agrupamento multivariados, k-médias e hierárquicos, validados por índices de qualidade de agrupamento. Foram usadas 227 estações localizadas no estado do Paraná com dados mensais referentes ao período de 1976-2006. Os agrupamentos obtidos foram submetidos às medidas de discordância e heterogeneidade para avaliar a homogeneidade dos grupos. A metodologia de agrupamento que obteve o melhor desempenho foi a metodologia híbrida entre k-médias e ward, obtendo erros de, no máximo, 10% na estimativa de quantis regionais adimensionais.

L’étude des modèles de mélanges est un champ très vaste en statistique. Nous présentons dans la première partie de la thèse les phi-divergences et les méthodes existantes qui construisent des estimateurs robustes basés sur des phi-divergences. Nous nous intéressons en particulier à la forme duale des phi-divergences et nous construisons un nouvel estimateur robuste basant sur cette formule. Nous étudions les propriétés asymptotiques de cet estimateur et faisons une comparaison numérique avec les méthodes existantes. Dans un seconde temps, nous introduisons un algorithme proximal dont l’objectif est de calculer itérativement des estimateurs basés sur des critères de divergences statistiques. La convergence de l’algorithme est étudiée et illustrée par différents exemples théoriques et sur des données simulées. Dans la deuxième partie de la thèse, nous construisons une nouvelle structure pour les modèles de mélanges à deux composantes dont l’une est inconnue. La nouvelle approche permet d’incorporer une information a priori linéaire de type moments ou L-moments. Nous étudions les propriétés asymptotiques des estimateurs proposés. Des simulations numériques sont présentées afin de montrer l’avantage de la nouvelle approche en comparaison avec les méthodes existantes qui ne considèrent pas d’information a priori à part une hypothèse de symétrie sur la composante inconnue. / The study of mixture models constitutes a large domain of research in statistics. In the first part of this work, we present phi-divergences and the existing methods which produce robust estimators. We are more particularly interested in the so-called dual formula of phi-divergences. We build a new robust estimator based on this formula. We study its asymptotic properties and give a numerical comparison with existing methods on simulated data. We also introduce a proximal-point algorithm whose aim is to calculate divergence-based estimators. We give some of the convergence properties of this algorithm and illustrate them on theoretical and simulated examples. In the second part of this thesis, we build a new structure for two-component mixture models where one component is unknown. The new approach permits to incorporate a prior linear information about the unknown component such as moment-type and L-moments constraints. We study the asymptotic properties of the proposed estimators. Several experimental results on simulated data are illustrated showing the advantage of the novel approach and the gain from using the prior information in comparison to existing methods which do not incorporate any prior information except for a symmetry assumption over the unknown component.

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In the Canadian Prairies recurring droughts are one of the realities which can have significant economical, environmental, and social impacts. For example, droughts in 1997 and 2001 cost over $100 million on different sectors. Drought frequency analysis is a technique for analyzing how frequently a drought event of a given magnitude may be expected to occur. In this study the state of the science related to frequency analysis of droughts is reviewed and studied. The main contributions of this thesis include development of a model in Matlab which uses the qualities of Fuzzy C-Means (FCMs) clustering and corrects the formed regions to meet the criteria of effective hydrological regions. In FCM each site has a degree of membership in each of the clusters. The algorithm developed is flexible to get number of regions and return period as inputs and show the final corrected clusters as output for most case scenarios. While drought is considered a bivariate phenomena with two statistical variables of duration and severity to be analyzed simultaneously, an important step in this study is increasing the complexity of the initial model in Matlab to correct regions based on L-comoments statistics (as apposed to L-moments). Implementing a reasonably straightforward approach for bivariate drought frequency analysis using bivariate L-comoments and copula is another contribution of this study. Quantile estimation at ungauged sites for return periods of interest is studied by introducing two new classes of neural network and machine learning: Radial Basis Function (RBF) and Support Vector Machine Regression (SVM-R). These two techniques are selected based on their good reviews in literature in function estimation and nonparametric regression. The functionalities of RBF and SVM-R are compared with traditional nonlinear regression (NLR) method. As well, a nonlinear regression with regionalization method in which catchments are first regionalized using FCMs is applied and its results are compared with the other three models. Drought data from 36 natural catchments in the Canadian Prairies are used in this study. This study provides a methodology for bivariate drought frequency analysis that can be practiced in any part of the world.