Dissertations / Theses: 'Random variables'

1

Yan, Xiaosong. "Quantifications of random variables." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1996. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/NQ36218.pdf.

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Wanntorp, Henrik. "Summability Methods and Random Variables." Thesis, Uppsala University, Department of Mathematics, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-122047.

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3

Spencer, Steven Robert. "Renewal theory for uniform random variables." CSUSB ScholarWorks, 2002. https://scholarworks.lib.csusb.edu/etd-project/2248.

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This project will focus on finding formulas for E[N(t)] using one of the classical problems in the discipline first, and then extending the scope of the problem to include overall times greater than the time t in the original problem. The expected values in these cases will be found using the uniform and exponential distributions of random variables.

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4

Zarepour, Mahmoud. "Some topics on infinite variance random variables." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ28099.pdf.

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5

Jairu, Desiderio N. "Distributions of some random volumes of uniform and beta type-1 random variables." Thesis, City University London, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.283156.

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Venigella, Pavan Kumar. "Robust Mechanism synthesis with random and interval variables." Diss., Rolla, Mo. : University of Missouri-Rolla, 2007. http://scholarsmine.mst.edu/thesis/pdf/Venigella_09007dcc8046c1ff.pdf.

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Thesis (M.S.)--University of Missouri--Rolla, 2007.
Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed March 27, 2008) Includes bibliographical references (p. 86-89).

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7

Kharoufeh, Jeffrey P. "Density estimation for functions of correlated random variables." Ohio : Ohio University, 1997. http://www.ohiolink.edu/etd/view.cgi?ohiou1177097417.

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8

Njoroge, Moses M. "On jacobians connected with matrix variate random variables." Thesis, McGill University, 1988. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=61966.

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9

Nešlehová, Johana [Verfasser]. "Dependence of Non-Continuous Random Variables / Johana Nešlehová." Aachen : Shaker, 2004. http://d-nb.info/1181604567/34.

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Papathomas, Michail. "Non-parametric Bayesian procedures for binary random variables." Thesis, University of Sheffield, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.369897.

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Chen, Chu-ka. "Mosaics of dividing cells /." Hong Kong : University of Hong Kong, 1998. http://sunzi.lib.hku.hk/hkuto/record.jsp?B19235471.

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12

Hörmann, Wolfgang. "The transformed rejection method for generating Poisson random variables." Institut für Statistik und Mathematik, Abt. f. Angewandte Statistik u. Datenverarbeitung, WU Vienna University of Economics and Business, 1992. http://epub.wu.ac.at/352/1/document.pdf.

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The transformed rejection method, a combination of the inversion and the rejection method, which is used to generate non-uniform random numbers from a variety of continuous distributions can be applied to discrete distributions as well. For the Poisson distribution a short and simple algorithm is obtained which is well suited for large values of the Poisson parameter $\mu$, even when $\mu$ may vary from call to call. The average number of uniform deviates required is lower than for any of the known uniformly fast algorithms. Timings for a C implementation show that the algorithm needs only half of the code but is - for $\mu$ not too small - at least as fast as the current state-of-the-art algorithms. (author's abstract)
Series: Preprint Series / Department of Applied Statistics and Data Processing

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13

Horn, Wayne. "Laplace transforms of order statistics of Erlang random variables." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape10/PQDD_0014/MQ52571.pdf.

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Mutombo, Pierre Abraham Mulamba. "Two-phase behaviour in a sequence of random variables." Thesis, Stellenbosch : Stellenbosch University, 2007. http://hdl.handle.net/10019.1/19645.

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Thesis (MSc)--University of Stellenbosch, 2007.
ENGLISH ABSTRACT: Buying and selling in financial markets are driven by demand. The demand can be quantified by the imbalance in the number of shares QB and QS transacted by buyers and sellers respectively over a given time interval t. The demand in an interval t is given by (t) = QB − QS. The local noise intensity is given by = h|aiqi − haiqii|i where i = 1, . . . ,N labels the transactions in t, qi is the number of shares traded in transaction i, ai = ±1 denotes buyer- initiated and seller- initiated trades respectively and h· · · i is the local expectation value computed from all the transactions during the interval t. In a paper [1] based on data from the New York Stock Exchange Trade and Quote database during the period 1995-1996, Plerou, Gopikrishnan and Stanley [1] reported that the analysis of the probability distribution P( | ) of demand conditioned on the local noise intensity revealed the surprising existence of a critical threshold c. For < c, the most probable value of demand is roughly zero; they interpreted this as an equilibrium phase in which neither buying nor selling predominates. For > c two most probable values emerge that are symmetrical around zero demand, corresponding to excess demand and excess supply; they interpreted this as an out-of-equilibrium phase in which the market behaviour is buying for half of the time, and selling for the other half. It was suggested [1] that the two-phase behaviour indicates a link between the dynamics of a financial market with many interacting participants and the phenomenon of phase transitions that occurs in physical systems with many interacting units. This thesis reproduces the two-phase behaviour by means of experiments using sequences of random variables. We reproduce the two-phase behaviour based on correlated and uncorrelatd data. We use a Markov modulated Bernoulli process to model the transactions and investigate a simple interpretation of the two-phase behaviour. We sample data from heavy-tailed distributions and reproduce the two-phase behaviour. Our experiments show that the results presented in [1] do not provide evidence for the presence of complex phenomena in a trading market; the results are a consequence of the sampling method employed.
AFRIKAANSE OPSOMMING: Aankope en verkope in finansi¨ele markte word deur aanvraag gedryf. Aanvraag kan gekwantifiseer word in terme van die ongebalanseerdheid in die getal aandele QB en QB soos onderskeidelik verhandel deur kopers en verkopers in ’n gegewe tyd-interval t. Die aanvraag in ’n interval t word gegee deur (t) = QB −QS. Die lokale geraasintensiteit word gegee deur = h|aiqi − haiqii|i waar i = 1, . . . ,N die transaksies in t benoem, qi die getal aandele verhandel in transaksies verwys, en h· · · i op die lokale verwagte waarde dui, bereken van al die tansaksies tydens die interval t. In ’n referaat [1] wat op data van die New York Effektebeurs se Trade and Quote databasis in die periode tussen 1995 en 1996 geskoei was, het Plerou, Gopikrishnan en Stanley [1] gerapporteer dat ’n analise van die waarskynlikheidsverspreiding P( | ) van aanvraag gekondisioneer op die lokale geraasintensiteit , die verrassende bestaan van ’n kritieke drempelwaarde c na vore bring. Vir < c is die mees waarskynlike aanvraagwaarde nagenoeg nul; hulle het dit ge¨ınterpreteer as ’n ekwilibriumfase waartydens n`og aankope n`og verkope die oormag het. Vir > c is die twee mees waarskynlike aanvraagwaardes wat te voorskyn kom simmetries rondom nul aanvraag, wat oorenstem met ’n oormaat aanvraag en ’n oormaat aanbod; hulle het dit geinterpreteer as ’n buite-ewewigfase waartydens die markgedrag die helfte van die tyd koop en die anderhelfte verkoop. Daar is voorgestel [1] dat die tweefase gedrag op ’n verband tussen die dinamiek van ’n finansiele mark met baie deelnemende partye, en die verskynsel van fase-oorgange wat in fisieke sisteme met baie wisselwerkende eenhede voorkom, dui. Hierdie tesis reproduseer die tweefase gedrag deur middel van eksperimente wat gebruik maak van reekse van lukrake veranderlikes. Ons reproduseer die tweefase gedrag gebaseer op gekorreleerde en ongekorreleerde data. Ons gebruik ’n Markov-gemoduleerde Bernoulli proses om die transaksies te moduleer en ondersoek ’n eenvoudige interpretasie van die tweefase gedrag. Ons seem steekproefdata van “heavy-tailed” verspreidings en reproduseer die tweefase gedrag. Ons ekperimente wys dat die resultate in [1] voorgested is nie bewys lewer vir die teenwoordigheid van komplekse verskynsel in’n handelsmark nie; die resultate is as gevolg van die metode wat gebruik is vir die generering van die steekproefdata.

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15

Kasparavičiūtė, Aurelija. "Theorems of large deviations for the sums of a random number of independent random variables." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2014. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2014~D_20140121_101308-41106.

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The research object of this thesis is the sum of a random number of summands of independent identically distributed random variables with positive weights. Such sums appear as models, for example, in insurance, finance mathematics. Throughout the thesis, it is assumed that the random number of summands is independent of the summands, the summands satisfy S. N. Bernstein's condition, and the random number of summands together with weights satisfy some compatibility conditions. The aim of this dissertation is a normal approximation to a distribution of the sum of a random number of summands of independent identically distributed random variables with positive weights that takes into consideration large deviations in both the Cramer and the power Linnik zones.
Disertacinio darbo tyrimo objektas yra atsitiktinio dėmenų skaičiaus nepriklausomų vienodai pasiskirsčiusių atsitiktinių dydžių su teigiamais svoriniais koeficientais sumos, kurios kaip modelis sutinkamos, pavyzdžiui, finansų, draudos matematikose. Daromos prielaidos, kad atsitiktinis dėmenų skaičius yra nepriklausomas nuo sumos dėmenų, atsitiktiniai dėmenys tenkina apibendrintą S. N. Bernšteino sąlygą, o atsitiktinis dėmenų skaičius kartu su svoriais tenkina tam tikras suderinamumo sąlygas. Disertacijos tikslas yra standartizuotos (centruotos ir normuotos) minėtos atsitiktinės sumos skirstinio aproksimacija standartiniu normaliuoju dėsniu didžiųjų nuokrypių tiek Kramero, tiek ir laipsninėse Liniko zonose.

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16

Yu, Jihnhee. "Approaches to the multivariate random variables associated with stochastic processes." Texas A&M University, 2003. http://hdl.handle.net/1969.1/1209.

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Stochastic compartment models are widely used in modeling processes for biological populations. The residence time has been especially useful in describing the system dynamics in the models. The direct calculation of the distribution for the residence time of stochastic multi-compartment models is very complicated even with a relatively simple model and often impossible to calculate directly. This dissertation presents an analytical method to obtain the moment generating function for stochastic multi-compartment models and describe the distribution of the residence times, especially systems with nonexponential lifetime distributions. A common method for obtaining moments of the residence time is using the coefficient matrix, however it has a limitation in obtaining high order moments and moments for combined compartments in a system. In this dissertation, we first derive the bivariate moment generating function of the residence time distribution for stochastic two-compartment models with general lifetimes. It provides any order of moments and also enables us to approximate the density of the residence time using the saddlepoint approximation. The approximation method is applied to various situations including the approximation of the bivariate distribution of residence times in two-compartment models or approximations based on the truncated moment generating function. Special attention is given to the distribution of the residence time for multi-compartment semi-Markov models. The cofactor rule and the analytic approach to the two-compartment model facilitate the derivation of the moment generating function. The properties from the embedded Markov chain are also used to extend the application of the approach. This approach provides a complete specification of the residence time distribution based on the moment generating function and thus provides an easier calculation of high-order moments than the approach using the coefficient matrix. Applications to drug kinetics demonstrate the simplicity and usefulness of this approach.

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17

Bedbur, Stefan [Verfasser]. "Models of ordered random variables and exponential families / Stefan Bedbur." Aachen : Hochschulbibliothek der Rheinisch-Westfälischen Technischen Hochschule Aachen, 2012. http://d-nb.info/1021568015/34.

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18

Belu, Alexandru C. "Multivariate Measures of Dependence for Random Variables and Levy Processes." Case Western Reserve University School of Graduate Studies / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=case1333396376.

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19

Li, Xue. "A Novel Accurate Approximation Method of Lognormal Sum Random Variables." Wright State University / OhioLINK, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=wright1229358144.

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20

Phadke, Vidyadhar S. "Non-classical convergence results for sums of dependent random variables." Bowling Green, Ohio : Bowling Green State University, 2008. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=bgsu1224514478.

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21

Qeadan, Fares. "Bivariate distribution of n iid exponential random variables KPQ-EXP /." abstract and full text PDF (UNR users only), 2008. http://0-gateway.proquest.com.innopac.library.unr.edu/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:1456407.

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22

Fresen, Jill Winifred. "Random variables a CAI tutorial in statistics for distance education /." Thesis, Pretoria : [s.n.], 1996. http://upetd.up.ac.za/thesis/available/etd-10192001-124625.

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23

Brophy, Edmond M. "Prophet Inequalities for Multivariate Random Variables with Cost for Observations." Thesis, University of North Texas, 2019. https://digital.library.unt.edu/ark:/67531/metadc1538720/.

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In prophet problems, two players with different levels of information make decisions to optimize their return from an underlying optimal stopping problem. The player with more information is called the "prophet" while the player with less information is known as the "gambler." In this thesis, as in the majority of the literature on such problems, we assume that the prophet is omniscient, and the gambler does not know future outcomes when making his decisions. Certainly, the prophet will get a better return than the gambler. But how much better? The goal of a prophet problem is to find the least upper bound on the difference (or ratio) between the prophet's return, M, and the gambler's return, V. In this thesis, we present new prophet problems where we seek the least upper bound on M-V when there is a fixed cost per observations. Most prophet problems in the literature compare M and V when prophet and gambler buy (or sell) one asset. The new prophet problems presented in Chapters 3 and 4 treat a scenario where prophet and gambler optimize their return from selling two assets, when there is a fixed cost per observation. Sharp bounds for the problems on small time horizons are given; for the n-day problem, rough bounds and a description of the distributions for the random variables that maximize M-V are presented.

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Lo, Ambrose, and 羅彥博. "On some negative dependence structures and their applications." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2014. http://hdl.handle.net/10722/206695.

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Recently, the study of negative dependence structures has aroused considerable interest amongst researchers in actuarial science and quantitative risk management. This thesis centres on two extreme negative dependence structures in different dimensions - counter-monotonicity and mutual exclusivity, and develops their novel characterizations and applications to risk management. Bivariate random vectors are treated in the first part of the thesis, where the characterization of comonotonicity by the optimality of aggregate sums in convex order is extended to its bivariate antithesis, namely, counter-monotonicity. It is shown that two random variables are counter-monotonic if and only if their aggregate sum is minimal with respect to convex order. This defining property of counter-monotonicity is then exploited to identify a necessary and sufficient condition for merging counter-monotonic positions to be risk-reducing. In the second part, the notion of mutual exclusivity is introduced as a multi-dimensional generalization of counter-monotonicity. Various characterizations of mutually exclusive random vectors are presented, including their pairwise counter-monotonic behaviour, minimal convex sum property, and the characteristic function of their aggregate sums. These properties highlight the role of mutual exclusivity as the strongest negative dependence structure in a multi-dimensional setting. As an application, the practical problem of deriving general lower bounds on three common convex functionals of aggregate sums with arbitrary marginal distributions is considered. The sharpness of these lower bounds is characterized via the mutual exclusivity of the underlying random variables. Compared to existing bounds in the literature, the new lower bounds proposed enjoy the advantages of generality and simplicity.
published_or_final_version
Statistics and Actuarial Science
Doctoral
Doctor of Philosophy

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25

Kolesar, Michal. "Essays on Instrumental Variables." Thesis, Harvard University, 2013. http://dissertations.umi.com/gsas.harvard:10796.

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Sambale, Holger [Verfasser]. "Second order concentration for functions of independent random variables / Holger Sambale." Bielefeld : Universitätsbibliothek Bielefeld, 2016. http://d-nb.info/1084888173/34.

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Tilahun, Gelila. "Laws of large numbers for sequences and arrays of random variables." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp01/MQ29799.pdf.

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Tilahun, Gelila. "Laws of large numbers for sequences and arrays of random variables." Thesis, McGill University, 1996. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=27424.

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This thesis presents an up-to-date survey of results concerning laws of large numbers for sequences and arrays of random variables. We begin with Kolmogorov's pioneering result, the strong law of large numbers, and preceed through to Hu et al.'s, and Gut's recent result for weakly dominated random variables, for which we provide a simpler proof. We insist in particular on the techniques of proof of Etemadi and Jamison et al. Furthermore, analogues to the Marcinkiewicz-Zygmund theorem are given. This thesis illustrates the trade-off between the existence of higher moments and non i.i.d sequences and arrays of random variables to obtain the strong law of large numbers.

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Ivanchuk, M. A. "Separating of two normal distributed random variables by using their strewnfield." Thesis, БДМУ, 2020. http://dspace.bsmu.edu.ua:8080/xmlui/handle/123456789/18321.

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Li, Jia. "Investigation of empirical modeling of random vectors and its applications to hydrosystem problems /." View abstract or full-text, 2007. http://library.ust.hk/cgi/db/thesis.pl?CIVL%202007%20LI.

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Wu, Hao-cun. "Independent component analysis and its applications in finance." Click to view the E-thesis via HKUTO, 2007. http://sunzi.lib.hku.hk/HKUTO/record/B39559099.

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陳楚嘉 and Chu-ka Chen. "Mosaics of dividing cells." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1998. http://hub.hku.hk/bib/B31215038.

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JIMENEZ, MARCELO ROBERTO BAPTISTA PEREIRA LUIS. "CYCLIC RANDOM VARIABLES AND THEIR APPLICATION IN THE STUDY OF INTERFEROMETRIC NOISE." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2001. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=1981@1.

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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
O ruído interferométrico é um fator limitante cada vez mais importante nos sistemas óticos, principalmente nas ligações de longa distância em redes óticas transparentes. O presente trabalho analisa modelos para este tipo de ruído, dando um tratamento matemático novo para o modelo não-gaussiano. A teoria matemática é desenvolvida em detalhes e comrigor. O modelo gaussiano foi usado a fim de fazer previsões quanto aos valores de chão da taxa de erro de bits. Os dois modelos foram simulados em computador e comparados com os testes realizados em laboratório e os resultados são apresentados.
The interferometric noise is becoming a serious limiting factor in optical systems,notably on long distance connections in transparent optical networks. The present work analyzes models for this kind of noise, giving a new mathematical treatment to the non-gaussian model. The mathematical theory is developed in detail and rigorously. The gaussian model was used in order to make predictions relative to bit error rate floors. Both models were simulated in computer and compared with the tests made in laboratory and the results are presented.
EL ruido interferométrico es un factor limitante cada vez más importante en los sistemas ópticos, principalmente en las llamadas a larga distancia en redes ópticas transparentes. EL presente trabajo analiza modelos para este tipo de ruido, dando un nuevo tratamiento matemático para el modelo no gausiano. La teoría matemática es desarrollada en detalles y con rigor. EL modelo gausiano fue usado para efectuar previsiones de las cotas inferiores de la tasa de error de bits. Los dos modelos fueron simulados en computador y comparados con las priuebas de laboratorio y se presentan los resultados.

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Stewart, Jaimee E. "A Comparison of Methods for Generating Bivariate Non-normally Distributed Random Variables." UNF Digital Commons, 2009. http://digitalcommons.unf.edu/etd/235.

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Many distributions of multivariate data in the real world follow a non-normal model with distributions being skewed and/or heavy tailed. In studies in which multivariate non-normal distributions are needed, it is important for simulations of those variables to provide data that is close to the desired parameters while also being fast and easy to perform. Three algorithms for generating multivariate non-normal distributions are reviewed for accuracy, speed and simplicity. They are the Fleishman Power Method, the Fifth-Order Polynomial Transformation Method, and the Generalized Lambda Distribution Method. Simulations were run in order to compare the three methods by how well they generate bivariate distributions with the desired means, variances, skewness, kurtoses, and correlation, simplicity of the algorithms, and how quickly the desired distributions were calculated.

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Moldovan, Max. "Stochastic modelling of random variables with an application in financial risk management." Thesis, Queensland University of Technology, 2003. https://eprints.qut.edu.au/15796/1/Max_Moldovan_Thesis.pdf.

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The problem of determining whether or not a theoretical model is an accurate representation of an empirically observed phenomenon is one of the most challenging in the empirical scientific investigation. The following study explores the problem of stochastic model validation. Special attention is devoted to the unusual two-peaked shape of the empirically observed distributions of the conditional on realised volatility financial returns. The application of statistical hypothesis testing and simulation techniques leads to the conclusion that the conditional on realised volatility returns are distributed with a specific previously undocumented distribution. The probability density that represents this distribution is derived, characterised and applied for validation of the financial model.

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Moldovan, Max. "Stochastic Modelling of Random Variables with an Application in Financial Risk Management." Queensland University of Technology, 2003. http://eprints.qut.edu.au/15796/.

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The problem of determining whether or not a theoretical model is an accurate representation of an empirically observed phenomenon is one of the most challenging in the empirical scientific investigation. The following study explores the problem of stochastic model validation. Special attention is devoted to the unusual two-peaked shape of the empirically observed distributions of the conditional on realised volatility financial returns. The application of statistical hypothesis testing and simulation techniques leads to the conclusion that the conditional on realised volatility returns are distributed with a specific previously undocumented distribution. The probability density that represents this distribution is derived, characterised and applied for validation of the financial model.

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37

Kondapaneni, Rajesh. "A Study of the Delta-Normal Method of Measuring VaR." Link to electronic thesis, 2005. http://www.wpi.edu/Pubs/ETD/Available/etd-050905-104553/.

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Fielden, Thomas Robert. "Modeling Market and Regulatory Mechanisms for Pollution Abatement with Sharp and Random Variables." PDXScholar, 2011. https://pdxscholar.library.pdx.edu/open_access_etds/282.

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This dissertation is motivated by the problem of uncertainty and sensitivity in business- class models such as the carbon emission abatement policy model featured in this work. Uncertain model inputs are represented by numerical random variables and a computational methodology is developed to numerically compute business-class models as if sharp inputs were given. A new description for correlation of random variables is presented that arises spontaneously within a numerical model. Methods of numerically computing correlated random variables are implemented in software and represented. The major contribution of this work is a methodology for the numerical computation of models under uncertainty that expresses no preference for unlikelihood of model input combinations. The methodology presented here serves a sharp contrast to traditional Monte Carlo methods that implicitly equate likelihood of model input values with importance of results. The new methodology herein shifts the computational burden from likelihood of inputs to resolution of input space.

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39

吳浩存 and Hao-cun Wu. "Independent component analysis and its applications in finance." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2007. http://hub.hku.hk/bib/B39559099.

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Peng, Xiaoling. "Methods of variable selection and their applications in quantitative structure-property relationship (QSPR)." HKBU Institutional Repository, 2005. http://repository.hkbu.edu.hk/etd_ra/594.

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41

Karniychuk, Maryna. "Comparing Approximations for Risk Measures Related to Sums of Correlated Lognormal Random Variables." Master's thesis, Universitätsbibliothek Chemnitz, 2007. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200700024.

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In this thesis the performances of different approximations are compared for a standard actuarial and financial problem: the estimation of quantiles and conditional tail expectations of the final value of a series of discrete cash flows. To calculate the risk measures such as quantiles and Conditional Tail Expectations, one needs the distribution function of the final wealth. The final value of a series of discrete payments in the considered model is the sum of dependent lognormal random variables. Unfortunately, its distribution function cannot be determined analytically. Thus usually one has to use time-consuming Monte Carlo simulations. Computational time still remains a serious drawback of Monte Carlo simulations, thus several analytical techniques for approximating the distribution function of final wealth are proposed in the frame of this thesis. These are the widely used moment-matching approximations and innovative comonotonic approximations. Moment-matching methods approximate the unknown distribution function by a given one in such a way that some characteristics (in the present case the first two moments) coincide. The ideas of two well-known approximations are described briefly. Analytical formulas for valuing quantiles and Conditional Tail Expectations are derived for both approximations. Recently, a large group of scientists from Catholic University Leuven in Belgium has derived comonotonic upper and comonotonic lower bounds for sums of dependent lognormal random variables. These bounds are bounds in the terms of "convex order". In order to provide the theoretical background for comonotonic approximations several fundamental ordering concepts such as stochastic dominance, stop-loss and convex order and some important relations between them are introduced. The last two concepts are closely related. Both stochastic orders express which of two random variables is the "less dangerous/more attractive" one. The central idea of comonotonic upper bound approximation is to replace the original sum, presenting final wealth, by a new sum, for which the components have the same marginal distributions as the components in the original sum, but with "more dangerous/less attractive" dependence structure. The upper bound, or saying mathematically, convex largest sum is obtained when the components of the sum are the components of comonotonic random vector. Therefore, fundamental concepts of comonotonicity theory which are important for the derivation of convex bounds are introduced. The most wide-spread examples of comonotonicity which emerge in financial context are described. In addition to the upper bound a lower bound can be derived as well. This provides one with a measure of the reliability of the upper bound. The lower bound approach is based on the technique of conditioning. It is obtained by applying Jensen's inequality for conditional expectations to the original sum of dependent random variables. Two slightly different version of conditioning random variable are considered in the context of this thesis. They give rise to two different approaches which are referred to as comonotonic lower bound and comonotonic "maximal variance" lower bound approaches. Special attention is given to the class of distortion risk measures. It is shown that the quantile risk measure as well as Conditional Tail Expectation (under some additional conditions) belong to this class. It is proved that both risk measures being under consideration are additive for a sum of comonotonic random variables, i.e. quantile and Conditional Tail Expectation for a comonotonic upper and lower bounds can easily be obtained by summing the corresponding risk measures of the marginals involved. A special subclass of distortion risk measures which is referred to as class of concave distortion risk measures is also under consideration. It is shown that quantile risk measure is not a concave distortion risk measure while Conditional Tail Expectation (under some additional conditions) is a concave distortion risk measure. A theoretical justification for the fact that "concave" Conditional Tail Expectation preserves convex order relation between random variables is given. It is shown that this property does not necessarily hold for the quantile risk measure, as it is not a concave risk measure. Finally, the accuracy and efficiency of two moment-matching, comonotonic upper bound, comonotonic lower bound and "maximal variance" lower bound approximations are examined for a wide range of parameters by comparing with the results obtained by Monte Carlo simulation. It is justified by numerical results that, generally, in the current situation lower bound approach outperforms other methods. Moreover, the preservation of convex order relation between the convex bounds for the final wealth by Conditional Tail Expectation is demonstrated by numerical results. It is justified numerically that this property does not necessarily hold true for the quantile.

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42

Spencer, Neil. "DrSMC : a sequential Monte Carlo sampler for deterministic relationships on continuous random variables." Thesis, University of British Columbia, 2015. http://hdl.handle.net/2429/54647.

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Computing posterior distributions over variables linked by deterministic constraints is a recurrent problem in Bayesian analysis. Such problems can arise due to censoring, identifiability issues, or other considerations. It is well-known that standard implementations of Monte Carlo inference strategies break down in the presence of these deterministic relationships. Although several alternative Monte Carlo approaches have been recently developed, few are applicable to deterministic relationships on continuous random variables. In this thesis, I propose Deterministic relationship Sequential Monte Carlo (DrSMC), a new Monte Carlo method for continuous variables possessing deterministic constraints. My exposition focuses on developing a DrSMC algorithm for computing the posterior distribution of a continuous random vector given its sum. I derive optimal settings for this algorithm and compare its performance to that of alternative approaches in the literature.
Science, Faculty of
Statistics, Department of
Graduate

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43

Nguyen, Quang Huy. "Tail distribution of the sums of regularly varying random variables, computations and simulations." Thesis, Lyon 1, 2014. http://www.theses.fr/2014LYO10224.

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Cette thèse s'intéresse à l'utilisation de techniques numériques par approximation sous forme de séries et de techniques de simulation pour l'approximation de la queue de distribution de sommes de variables aléatoires à variations régulières. Le calcul de la probabilité que la somme soit plus grande qu'un seuil donné est important en gestion des risques. En particulier, ce calcul est utilisé pour définir le besoin en capital des sociétés d'assurances ou d'autres institutions financières. Le premier chapitre constitue l'introduction de la thèse. Il explique les principaux résultats et présente les outils mathématiques qui sont développés dans la thèse. Le second chapitre est basé sur le travail : ”Series expansions for the sum of the independent Pareto random variables”, article rédigé avec le Professeur Christian ROBERT, directeur de la thèse. Cet article est soumis à publication. Il propose un algorithme de calcul pour déterminer la queue de distribution d'une somme de variables aléatoires de type Pareto non nécessairement équidistribuées. Il propose une approximation sous forme de série de la fonction de survie de la somme. L'algorithme utilisé pour calculer l'approximation est simple, facile à implémenter, et offre de très bons résultats numériques. Le troisième chapitre de cette thèse est basée sur l'article : ”New efficient estimators in rare event simulation with heavy tails”, publié dans Journal of Computational and Applied Mathematics, et co-écrit avec le Professeur Christian ROBERT. Il s'intéresse à l'approximation par simulation de la probabilité que la somme de variables aléatoires indépendantes à variations régulières soit plus grande qu'un seuil élevé. Des estimateurs efficaces ont déjà été introduits dans la littérature associée à la simulation d'évènements rares. Nous proposons de nouvelles techniques de simulation qui sont plus efficaces que les méthodes précédemment proposées. Le quatrième chapitre poursuit l'analyse de la simulation d'évènements rares du type ”la somme est plus grande qu'un seuil”, mais cette fois-ci il s'intéresse à des situations où les variables aléatoires sont dépendantes. Il se focalise sur le cas où la dépendance est donnée par une copule archimédienne. Ce chapitre est basé sur l'article en relecture : ”Efficient simulation of tail probabilities of sums with heavy tailed random variables and Archimedean copulas”. Les équivalents asymptotiques de la probabilité de dépassement de seuil ne sont connus que dans des cas particuliers et ils fournissent en général des approximations très médiocres de la vraie valeur. Les techniques de simulation sont donc très appréciables pour obtenir rapidement des approximations précises. Nous proposons quatre estimateurs et quatre techniques de simulation associées. Nous montrons que les erreurs relatives sont asymptotiquement bornées pour presque tous les estimateurs. Les simulations montrent que certains estimateurs sont plus précis
This thesis aims to study computation and simulation methods to approximate tail distribution of the sums of regularly varying random variables. The paper proceeds as follows: The ﬁrst chapter provides the general introduction of the thesis. The second chapter is essentially constituted by the article ”Series expansions for the sum of the independent Pareto random variables” which was co-written with Professor Christian ROBERT, actually submitted for publication. It deals with the problem of estimating tail distribution of the sum of independent Pareto variables. This problem has been studied for a long time but a complete solution has not yet been found. In this section, we acquire an exact formula, a series expansions, for the distribution of the sum of independent Pareto of non-integer tail indices. Not only is this formula simple and easy to apply but it also gives better numerical results than most of existing methods.The third chapter rests on the article ”New eﬃcient estimators in rare event simulation with heavy tails”, co-written with Professor Christian ROBERT, currently published on ”Journal of Computational and Applied Mathematics 261, 39-47” in 2013. Practically, eﬃcient estimation for tail distribution of the sum of i.i.d. regularly varying random variables is one of widely researched problems in rare event simulation. In this context, Asmussen and Kroese’s estimator has performed better than other works. This part will introduce a new way to approach the sum. Our obtained estimator is more eﬃcient than Asmussen and Kroese’s estimator in the case of regularly varying tail. In other cases, combined with techniques of conditional Monte Carlo and importance sampling, our estimator is still better. In the fourth chapter, we continue to study the tail behavior of the sum of regularly varying variables, with additional assumption that the dependence follows an Archimedean copula or an Archimedean survival copula. This section hinges on the article ”Eﬃcient simulation of tail probabilities of sums with heavy tailed random variables and Archimedean copulas” which is under consideration for being published. Almost all previous studies on this problem used asymptotic approaches which are hard to control the errors. Therefore, techniques of simulation to calculate the tail probability of the sum are presented. Though some of our estimators have bounded relative errors while the others do not, all of them give favorable numerical performances for such a challenging problem

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44

Jiang, Xinxin. "Central limit theorems for exchangeable random variables when limits are mixtures of normals /." Thesis, Connect to Dissertations & Theses @ Tufts University, 2001.

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Thesis (Ph.D.)--Tufts University, 2001.
Adviser: Marjorie G. Hahn. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves44-46). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

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45

黃彥青. "PAIRWISE INDEPENDENT RANDOM VARIABLES." Thesis, 2001. http://ndltd.ncl.edu.tw/handle/47864460351789081997.

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碩士
國立中央大學
數學研究所
89
Abstract Pairwise independence is not enough for the central limit theorem to hold. In my thesis, some related results are mentioned. I also give some new version of conditions such that the central limit theorem would hold for pairwise independent sequences. Finally, I give an example to illustrate the results.

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46

Huang, Shih-Feng, and 黃士峯. "On the integrability of random variables and uniformly integrability of sequence of random variables." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/wtc9e7.

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47

Pincus, Bianca. "Combinatorics of geometrically distributed random variables." Thesis, 2012. http://hdl.handle.net/10539/11545.

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The aim of this thesis is to study various combinatorial problems relating to geometrically distributed random variables. In particular, we study sequences of geometric random variables with respect to the left-to-right maxima of the elements of the sequence. Traditionally, left-to-right maxima or records have been studied for permutations rather than sequences. For each of the parameters explained, we compute the mean and the variance both explicitly and asymptotically.

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48

Çagin, Tonguç. "Weighted sums of associated random variables." Doctoral thesis, 2015. http://hdl.handle.net/10316/26927.

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49

Moharana, Rajesh. "Information Measures for Truncated Random Variables." Thesis, 2019. http://ethesis.nitrkl.ac.in/10089/1/2019_PhD_RMoharana_514MA3001_Information.pdf.

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The concept of entropy plays a crucial role in information theory. Many authors obtained several properties of entropy and its various generalizations so far. The divergence measure was proposed to measure the inefficiency of taking an approximate distribution when the actual distribution is known. In reliability theory we often get random observations which are truncated in nature. The purpose of the thesis is to study properties of various information measures in truncated domain. First, we consider measures based on the probability density functions. These are Shannon’s entropy, weighted Shannon’s entropy, weighted generalized entropy, weighted Kullback-Leibler divergence and weighted generalized divergence. In these cases, we study various properties. Mainly, we obtain characterizations, inequalities, bounds, uncertainty orders and the effect of monotone transformations. Nonparametric classes based on monotonicity property of the uncertainty measures are introduced. The measures based on the density functions have some drawbacks. So, we consider weighted cumulative residual entropy, weighted cumulative entropy and their generalizations. These measures are based on the distribution function and the survival function. We obtain several characterizations. The importance of the characterization result is that it determines the distribution function uniquely under certain conditions. Always it is not an easy task to obtain a closed-form expression of the proposed measure. Therefore, we obtain various bounds and inequalities. The effect of monotone transformations is discussed. New uncertainty orders and classes of lifetime distributions are introduced. In addition, we propose estimators for the case of weighted extended cumulative residual entropy and weighted generalized cumulative entropy using empirical approach. Large sample property of the proposed estimators is studied. It is noticed that there are distributions such as power-Pareto and Govindarajulu which do not have closed-form distribution functions though these have closed-form quantile functions. Motivated by this, we consider quantile-based inaccuracy measure and its dynamic versions. Various results including characterization, effect of transformation and bounds are derived. Few examples are considered to illustrate the results.

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Lien, Meng-Ming, and 連萌敏. "UNUSUAL LAWS OF LARGE NUMBERS FOR RANDOM VARIABLES AND RANDOM ELEMENTS." Thesis, 1997. http://ndltd.ncl.edu.tw/handle/96013693244578260839.

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Dissertations / Theses on the topic 'Random variables'

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