This PhD thesis consists of a summary and five papers which all deal with asymptotic problems on certain homogeneous spaces. In Paper I we prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. All our results are given with precise estimates on the rates of convergence to equidistribution. Papers II and III are concerned with statistical problems on the space of n-dimensional lattices of covolume one. In Paper II we study the distribution of lengths of non-zero lattice vectors in a random lattice of large dimension. We prove that these lengths, when properly normalized, determine a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. In Paper III we complement this result by proving that the asymptotic distribution of the angles between the shortest non-zero vectors in a random lattice is that of a family of independent Gaussians. In Papers IV and V we investigate the value distribution of the Epstein zeta function along the real axis. In Paper IV we determine the asymptotic value distribution and moments of the Epstein zeta function to the right of the critical strip as the dimension of the underlying space of lattices tends to infinity. In Paper V we determine the asymptotic value distribution of the Epstein zeta function also in the critical strip. As a special case we deduce a result on the asymptotic value distribution of the height function for flat tori. Furthermore, applying our results we discuss a question posed by Sarnak and Strömbergsson as to whether there in large dimensions exist lattices for which the Epstein zeta function has no zeros on the positive real line.

Progressive Type-II censoring was introduced by Cohen (1963) and has since been the topic of much research. The question stands whether it is sensible to use this sampling plan by design, instead of regular Type-II right censoring. We introduce an asymptotic progressive censoring model, and find optimal censoring schemes for location-scale families. Our optimality criterion is the determinant of the 2x2 covariance matrix of the asymptotic best linear unbiased estimators. We present an explicit expression for this criterion, and conditions for its boundedness. By means of numerical optimization, we determine optimal censoring schemes for the extreme value, the Weibull and the normal distributions. In many situations, it is shown that these progressive schemes significantly improve upon regular Type-II right censoring.

This thesis presents a class of graphical models for directly representing the joint cumulative distribution function (CDF) of many random variables, called cumulative distribution networks (CDNs). Unlike graphical models for probability density and mass functions, in a CDN, the marginal probabilities for any subset of variables are obtained by computing limits of functions in the model. We will show that the conditional independence properties in a CDN are distinct from the conditional independence properties of directed, undirected and factor graph models, but include the conditional independence properties of bidirected graphical models. As a result, CDNs are a parameterization for bidirected models that allows us to represent complex statistical dependence relationships between observable variables. We will provide a method for constructing a factor graph model with additional latent variables for which graph separation of variables in the corresponding CDN implies conditional independence of the separated variables in both the CDN and in the factor graph with the latent variables marginalized out. This will then allow us to construct multivariate extreme value distributions for which both a CDN and a corresponding factor graph representation exist. In order to perform inference in such graphs, we describe the `derivative-sum-product' (DSP) message-passing algorithm where messages correspond to derivatives of the joint cumulative distribution function. We will then apply CDNs to the problem of learning to rank, or estimating parametric models for ranking, where CDNs provide a natural means with which to model multivariate probabilities over ordinal variables such as pairwise preferences. We will show that many previous probability models for rank data, such as the Bradley-Terry and Plackett-Luce models, can be viewed as particular types of CDN. Applications of CDNs will be described for the problems of ranking players in multiplayer team-based games, document retrieval and discovering regulatory sequences in computational biology using the above methods for inference and estimation of CDNs.

Currently, financial institutions are supposed to analyze and quantify a new type of banking risk, known as operational risk. Financial institutions are exposed to this risk in their everyday activities. The main objective of this work is to construct an acceptable statistical model of capital requirement computation. Such a model must respect specificity of losses arising from operational risk events. The fundamental task is represented by searching for a suitable distribution, which describes the probabilistic behavior of losses arising from this type of risk. There is a strong utilization of the Pickands-Balkema-de Haan theorem used in extreme value theory. Roughly speaking, distribution of a random variable exceeding a given high threshold, converges in distribution to generalized Pareto distribution. The theorem is subsequently used in estimating the high percentile from a simulated distribution. The simulated distribution is considered to be a compound model for the aggregate loss random variable. It is constructed as a combination of frequency distribution for the number of losses random variable and the so-called severity distribution for individual loss random variable. The proposed model is then used to estimate a fi -nal quantile, which represents a searched amount of capital requirement. This capital requirement is constituted as the amount of funds the bank is supposed to retain, in order to make up for the projected lack of funds. There is a given probability the capital charge will be exceeded, which is commonly quite small. Although a combination of some frequency distribution and some severity distribution is the common way to deal with the described problem, the final application is often considered to be problematic. Generally, there are some combinations for severity distribution of two or three, for instance, lognormal distributions with different location and scale parameters. Models like these usually do not have any theoretical background and in particular, the connecting of distribution functions has not been conducted in the proper way. In this work, we will deal with both problems. In addition, there is a derivation of maximum likelihood estimates of lognormal distribution for which hold F_LN(u) = p, where u and p is given. The results achieved can be used in the everyday practices of financial institutions for operational risks quantification. In addition, they can be used for the analysis of a variety of sample data with so-called heavy tails, where standard distributions do not offer any help. As an integral part of this work, a CD with source code of each function used in the model is included. All of these functions were created in statistical programming language, in S-PLUS software. In the fourth annex, there is the complete description of each function and its purpose and general syntax for a possible usage in solving different kinds of problems.

The concern of this diploma thesis is extreme value distributions. The first part formulates and proves the limit theorem for distribution of maximum. Further there are described basic properties of class of extreme value distributions. The key role of this thesis is on non-parametric estimations of extreme value index. Primarily, Hill and moment estimator are derived, for which is, based on the results of mathematical analysis, suggested an alternative choice of optimal sample fraction using a bootstrap based method. The estimators of extreme value index are compared based on simulations from proper chosen distributions, being close to distribution of given rain-fall data series. This time series is recommended a suitable estimator and suggested choice of optimal sample fraction, which belongs to the most difficult task in the area of extreme value theory.

The thesis is focused on extreme value theory and its applications. Initially, extreme value distribution is introduced and its properties are discussed. At this basis are described two models mostly used for an extreme value analysis, i.e. the block maxima model and the Pareto-distribution threshold model. The first one takes advantage in its robustness, however recently the threshold model is mostly preferred. Although the threshold choice strongly affects estimation quality of the model, an optimal threshold selection still belongs to unsolved issues of this approach. Therefore, the thesis is focused on techniques for proper threshold identification, mainly on adaptive methods suitable for the use in practice. For this purpose a simulation study was performed and acquired knowledge was applied for analysis of precipitation records from South-Moravian region. Further on, the thesis also deals with extreme value estimation within a stationary series framework. Usually, an observed time series needs to be separated to obtain approximately independent observations. The use of the advanced theory for stationary series allows to avoid the entire separation procedure. In this context the commonly applied separation techniques turn out to be quite inappropriate in most cases and the estimates based on theory of stationary series are obtained with better precision.

Progressive Type-II censoring was introduced by Cohen (1963) and has since been the topic of much research. The question stands whether it is sensible to use this sampling plan by design, instead of regular Type-II right censoring. We introduce an asymptotic progressive censoring model, and find optimal censoring schemes for location-scale families. Our optimality criterion is the determinant of the 2x2 covariance matrix of the asymptotic best linear unbiased estimators. We present an explicit expression for this criterion, and conditions for its boundedness. By means of numerical optimization, we determine optimal censoring schemes for the extreme value, the Weibull and the normal distributions. In many situations, it is shown that these progressive schemes significantly improve upon regular Type-II right censoring.

>Magister Scientiae - MSc / This study sought to identify and fit the appropriate extreme value distribution to flood data, using the method of maximum likelihood. To examine the uncertainty of the estimated parameters and evaluate the goodness of fit of the model identified. The study revealed that the three parameter Weibull and the generalised extreme value (GEV) distributions fit the data very well. Standard errors for the estimated parameters were calculated from the empirical information matrix. An upper limit to the flood levels followed from the fitted distribution.

Modeling of extreme events is a challenging statistical task. Firstly, there is always a limit number of observations and secondly therefore no experience to back test the result. One way of estimating higher quantiles is to fit one of theoretical distributions to the data and extrapolate to the tail. The shortcoming of this approach is that the estimate of the tail is based on the observations in the center of distribution. Alternative approach to this problem is based on idea to split the data into two sub-populations and model body of the distribution separately from the tail. This methodology is applied to non-life insurance losses, where extremes are particularly important for risk management. Never the less, even this approach is not a conclusive solution of heavy tail modeling. In either case, estimated 99.5% percentiles have such high standard errors, that the their reliability is very low. On the other hand this approach is theoretically valid and deserves to be considered as one of the possible methods of extreme value analysis.

The thesis is focused on extreme value distributions and their applications. Firstly, basics of the extreme value theory for one-dimensional observations are summarized. Using the limit theorem for distribution of maximum, three extreme value distributions (Gumbel, Fréchet, Weibull) are introduced and their domains of attraction are described. Two models for parametric functions estimation based on the generalized extreme value distribution (block maxima model) and the generalized Pareto distribution (threshold model) are introduced. Parameters estimates of these distributions are derived using the method of maximum likelihood and the probability weighted moment method. Described methods are used for analysis of the rainfall data in the Brno Region. Further attention is paid to Gumbel class of distributions, which is frequently used in practice. Methods for statistical inference of multiply left-censored samples from exponential and Weibull distribution considering the type I censoring are developed and subsequently used in the analysis of synthetic musk compounds concentrations. The last part of the thesis deals with the extreme value theory for two-dimensional observations. Demonstrational software for the extreme value distributions was developed as a part of this thesis.