Browsing Mathematics by Title
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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 (Belgrade)[more][less]
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Laković, Bosiljka (Titograd , 1979)[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 ) -
Nešić, Slobodan (Belgrade , 1980)[more][less]
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Baralić, Đorđe (Beograd , 2013)[more][less]
Abstract: The main objects studied in this doctoral thesis are quasitoric manifolds and spaces arising as the images of polyhedral product functors. Quasitoric manifolds are particularly interesting as topological generalization of non-singular toric varieties. They are a research topic of many mathematical disciplines including toric geometry, symplectic geometry, toric topology, algebraic geometry, algebraic topology, theory of convex polytopes, and topological combinatorics. These objects have already found numerous applications in mathematics and sciences and they continue to be intensively studied. In this thesis we put some emphasis on combinatorial methods, focusing on the interaction of the geometry of toric actions and combinatorics of simple polytopes. This connection of geometry and combinatorics is based on the fundamental observation that convex polytopes naturally arise as orbit spaces of toric actions on quasitoric manifolds. Our main original contributions in this thesis are related to classical topological questions about degrees of maps between manifolds as well as their embeddings and immersions into Euclidean spaces. We follow the general scheme characteristic for Algebraic Topology where a topological problem is reduced, often by non-trivial reductions, to a question of arithmetical, algebraic, or combinatorial nature. We believe that the novel applications of this scheme developed in the thesis, especially the new techniques and calculations, have a potential to be applied on other problems about quasitoric manifods. Here is a summary of the content of the thesis. For the reader’s convenience and for completeness, in the first three chapters we give an elementary exposition of the basic theory of simplicial complexes, convex polytopes, toric varieties and quasitoric manifolds. The emphasis is on the fundamental constructions and central results, however the combinatorial approach, utilized in the thesis, allows us present the theory in a direct and concrete way, with a minimum of topological prerequisites. The mapping degrees of maps between quasitoric manifolds are studied in Chapter 4 with a particular emphasis on quasitoric 4-manifolds. Utilizing the technique pioneered by Haibao Duan and Shicheng Wang, which is based on the intersection form and the cohomology ring calculations, we demonstrate that a complete information about mapping degrees can be obtained in many concrete situations. The theorems and the corresponding criteria for the existence of mapping degrees are formulated in the language of elementary number theory. It is amusing that the question whether a number appears as a mapping degree between concrete 4-manifolds is directly linked with classical results from number theory such as whether a number can be expressed as a sum of two or three squares, etc. This approach allows us to analyze many concrete 4-manifolds, including CP2, CP2♯CP2, S2×S2, etc. In Chapter 5 we calculate the Stiefel-Whitney classes of some concrete quasitoric manifolds and their duals. This information is used to determine cohomological obstructions to embeddings and immersions of these manifolds in Euclidean spaces. As an initial observation we showed that the calculations are highly dependent on the action of torus. Indeed, there are examples of quasitoric manifolds over the same polytope which exhibit a very different behavior and different complexity of the associated characteristic classes. Focusing on the quasitoric manifolds over the n-dimensional cube, we are able to produce quasitoric manifolds which are very complex in the sense that they almost attain the theoretical minimum dimension for their embedding or (totally skew) immersion in Euclidean spaces. The thesis ends with an appendix with an outline of the theory of group actions and equivariant topology. URI: http://hdl.handle.net/123456789/4232 Files in this item: 1
phdDjordjeBaralic.pdf ( 8.102Mb )