Browsing Mathematics by Title
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Aranđelović, Ivan (Beograd , 1999)[more][less]
URI: http://hdl.handle.net/123456789/4137 Files in this item: 1
Stavovi_o_presecanju.PDF ( 2.592Mb ) -
Merkle, Ana (Beograd , 2023)[more][less]
Abstract: Many new developments in the filed of probability and statistics focus on finding causal connections between observed processes. This leads to considering dependence relations and investigating how the past influence the present and the future. The well known concept of Granger (1969) causality is closely related to the idea of local dependence introduced by Schweder (1970). Granger studied time series, while Schweder considered Markov chains. The concept was later extended to more general stochastic processes by Mykland (1986). All this concepts incorporate the time into consideration dependence. The dissertation consist of four chapters. New results are presented in the fourth chap- ter. The main aim of this doctoral dissertation is to determine di↵erent concepts of stochastic predictability using the well known tool of conditional independence. Follow Granger’s idea, relationships between family of sigma - algebras (filtrations) and between processes in continuous ti- me were considered since continuous time models dependence represent the first step in various applications, such as in finance, econometric practice, neuroscience, epidemiology, climatology, demographic, etc. In this dissertation the concept of dependence between stochastic processes and filtration is introduced. This concept is named causal predictability since it is focused on prediction. Some major characteristics of the given concept are shown and connections with known concept of dependence are explained. Finally, the concept of causal predictability is applied to the processes of di↵usion type, more precisely, to the uniqueness of weak solutions of Ito stochastic di↵erential equations and stochastic di↵erential equations with driving semi- martingales. Also, the representation theorem in terms of causal predictability is established and numerous examples of applications of the given concept are presented such as application in financial mathematics in the view of modeling default risk, in Bayesian statistics. The idea for the future might be to deal with the case of progressive stochastic predictability, i.e. the generalization of stochastic predictability from fixed time to stopping time. URI: http://hdl.handle.net/123456789/5572 Files in this item: 1
DOKTORAT_finalnaVerzija.pdf ( 1.785Mb ) -
Jocković, M. Jelena (Belgrade , 2012)[more][less]
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Jocković, Jelena (Beograd , 2012)[more][less]
Abstract: Statistical methodology for dealing with extremes depend on how extreme values are defined. One way to extract extremes from a given sample x1, x2, ..., xN is to consider maxima (minima). The other way is to consider values y1 = x1 − u, y2 = x2 − u, . . . , yn = xn − u, where y1, y2, . . . , yn are sample members above (below) a given predetermined threshold u. These two methods lead to two different approaches in extreme value theory. This doctoral dissertation has two main goals. One of them is to apply the techniques from extreme value framework to certain type of combinatorial problems. The other goal is to contribute to the field of statistical modeling of extremes. The dissertation consists of three chapters. In the first chapter, we introduce generalized extreme value distributions and generalized Pareto distributions (GPD). These two families play key roles in the two approaches to modeling extremes. We set out the theoretical background for both approaches. In the second chapter, we apply the extremal techniques to combinatorial waiting time problems. Precisely, we consider Coupon collector’s problem, defined as follows: elements are sampled with replacement from the set Nn = {1, 2, . . . , n} under assumption that each element has probability 1/n of being drawn. The subject of interest is the waiting time Mn until all elements of Nn or some other pattern are sampled. We focus our attention to the following two cases: 1. Mn is the waiting time until all elements of Nn are sampled at least r times, where r is a positive integer; 2. Mn is the waiting time until all pairs of elements jj, j ∈ Nn are sampled. We present new results related to the asymptotic behavior of the waiting time Mn, if it is known that a large number of trials was performed and the experiment is not over. For both cases, we determine the limiting distribution of exceedances of Mn over high thresholds, and answer some related questions: how to choose a suitable high threshold (depending on n) in order to obtain a limiting distribution; under what conditions the limit does not depend on the threshold; are the generalized Pareto distributions the only possible limits. We also estimate the speed of convergence in both cases. The third chapter of the dissertation is devoted to estimation of parameters and quantiles of the generalized Pareto distributions. We restrict the attention to the two-parameter version of GPD, defined as: Wγ,σ(x) = 1 − e−x , x ≥ 0, γ = 0 1 − 1 + γ σx −1 , x ≥ 0, γ > 0 1 − 1 + γ σx −1 , x ∈ h 0,−σ γ i , γ < 0. Well known problem with this model is inconsistency with the sample data, which is that one or more sample observations exceed the estimated upper bound in case when γ < 0. We propose a new, general technique to overcome the inconsistency problem and improve performance of the existing GPD estimation methods. We apply the proposed technique to methodof- moments and method-of-probability-weighted-moments estimates, investigate its performance through computer simulation and provide some real data examples. Finally, we address the problem of estimating high GPD quantiles. We evaluate the robustness of some estimation methods through simulation study and present a case study from finance (value-at-risk estimation), with special emphasis to certain difficulties related to this field of application. URI: http://hdl.handle.net/123456789/4271 Files in this item: 1
phdJockovic_Jelena.pdf ( 1.687Mb ) -
Petruševski, Ljiljana (Belgrade , 1986)[more][less]
URI: http://hdl.handle.net/123456789/51 Files in this item: 1
phdLjiljanaPetrusevski.pdf ( 1.651Mb ) -
Blagojević, Dragan (Belgrade)[more][less]
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Mihnjević, Danilo (Belgrade)[more][less]
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Stamenković, Blagoje (Belgrade)[more][less]
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Gjergji, Rexhep (Priština)[more][less]
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Petrović, Mihailo (Paris)[more][less]
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Adamović, Dušan (Belgrade , 1965)[more][less]
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Gajić, Ljiljana (Novi Sad , 1982)[more][less]
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Laković, Bosiljka (Titograd , 1979)[more][less]
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Laković, Bosiljka (Belgrade)[more][less]
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Jablan, Slavik (Belgrade , 1984)[more][less]
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Spasić, Slađana (Belgrade)[more][less]
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Cvetković, Ljiljana (Novi Sad)[more][less]
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Teorija onfinitezimalnih transformacija i njihova primena na integraljenje diferencijalnih jednačinaOkiljević, Blažo (Belgrade , 1986)[more][less]
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Đaja, Časlav (Belgrade , 1967)[more][less]
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Cuparić, Marija (Beograd , 2021)[more][less]
Abstract: The goal of this dissertation is the construction of new goodness-of-fit tests,analysis of their properties, as well as to obtain new theoretical findings regarding the limitingdistributions of weakly degeneratedV−statistics with estimated parameters. New goodness-of-fit tests are based on equidistributional type characterizations of two sample functions.Test statistics are formed asL2distances betweenV−empirical distribution functions ofstatistics from characterization, and also asL2andL∞distances betweenV−empiricalLaplace transformations of those statistics. In the latter case, resulting test statistics can beobserved asV−statistics with an estimated parameter or as functions of those statistics.Until now, limiting results were known for non-degenerateV−statistics with estimatedparameters, as well as for weakly degenerateV−statistics of degree two with estimatedparameters. Limiting results for the appropriate class of weakly degenerateV−statistics withan estimated parameter of degreem, wheremis even number, are derived in this dissertation.Owing to these results, asymptotic properties for presented tests are determined. To assessthe quality of these tests, empirical powers were determined using Monte Carlo simulations, aswell as approximate Bahadur efficiency. New results are presented regarding the approximateBahadur efficiency in case of close alternatives, which is applicable also when the limitingdistribution of statistics under the null hypothesis is not normal. In this sense, the comparisonbetween many tests is performed, both classical tests and recently developed tests.All previously mentioned results were obtained for complete samples. Additional, modifi-cation of previously introduced tests for randomly censored data was also proposed. In sucha case, the new theoretically justified bootstrap method is proposed for the approximation ofp−value. URI: http://hdl.handle.net/123456789/5212 Files in this item: 1
marijacuparicdr.pdf ( 1.771Mb )