Browsing Mathematics by Title
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Stanojević, Jelena (Beograd , 2015)[more][less]
Abstract: The main goal of the thesis is the development of a new suggested trans- form con¯dence intervals for the ratio of the variances of the two samples. Since now, the methods based on the F statistic have been suggested in the literature. However, the defect of that intervals is the huge sensitivity in re- lation with assumption of parameters distribution. Suggested statistic could be modi¯ed. Edgeworth expansion of the t-statistic has found the place in the thesis and based on that intervals have been compared. Also, on the base of the simulation it was point out that Johnsons transformation give better result in the sense of probability covering in regard to F interval and interval based on Halls transformation. Moreover, the con¯dence intervals for the mean and variances for the one and two sample problems have been considered in the dissertation. Especially, the problem of the di®erence of the proportions for the two samples, with the numerical results and data from the insurance. In addition, the existing methods for the estimation of the extreme value index and the high quantiles have been reviewed. Particularly, the direct simulation estimation of the quantile and probability covering of its deviation from the rights value, for Pareto and Gamma distributions, and also for general Pareto distribution have been discussed. The results were obtained by large deviation theory and their generalization on the topological spaces is stated. In this research, beside the probability theory and elemen- tary principles of the classical analysis, methods of the statistical theory and statistical conclusions have been applied. URI: http://hdl.handle.net/123456789/4261 Files in this item: 1
phdJelenaStanojevic.pdf ( 1.063Mb ) -
Vrećica, Ilija (Beograd , 2022)[more][less]
Abstract: First part of dissertation examines sumsets hA = {a1 + · · · + ah ∈ Z d : a1, . . . , ah ∈ A}, where A is a finite set in Z d . It is known that there exists a constant h0 ∈ N and a polynomial pA(X) such that pA(h) = |hA| for h ⩾ h0. However, little is known of polynomial pA and constant h0. Cone CA over the set A contains information about hA, for all h ∈ N. When A has d + 2 elements, polynomial pA and constant h0 can be explicitly described. When A has d + 3 elements, an upper bound is found for the number of elements of hA. Second part of dissertation examines Selmer groups of elliptic curves in the con gruent number family. A squarefree natural number is congruent if and only if there exists a right triangle with area n whose sides all have integer lengths. It is known that n is a congruent number if and only if elliptic curve En : y 2 = x 3 − n 2x has nonzero rank as an algebraic group. Selmer groups of isogenies on En are interesting, because their rank is not smaller than the rank of En, so when the Selmer groups have rank zero, then the elliptic curve En also has rank zero. Elements of these Selmer groups can be represented as partitions of a particular graph, from which one may find the distribution of ranks of Selmer groups. URI: http://hdl.handle.net/123456789/5533 Files in this item: 1
Teza_Ilija_Vrecica.pdf ( 1.817Mb ) -
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 (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]