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1
Al-Awadhi, Shafeeqah. "Elicitation of prior distributions for a multivariate normal distribution." Thesis, University of Aberdeen, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.387799.
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This thesis focuses on elicitation methods for quantifying an expert's subjective opinion about a multivariate normal distribution. Firstly, it is assumed that the expert's opinion can be adequately represented by a natural conjugate prior distribution (a normal inverse-Wishart distribution) and an elicitation method is developed in which the expert performs various assessment tasks that enable the hyperparameters of the distribution to be estimated. An example illustrating use of the method is given. There are some choices in the way hyperparameters are determined and empirical work underlies the choices made. The empirical work aimed to provide a basis for choosing between alternative assessment tasks that may be used in the elicitation method and to examine different ways of using the elicited assessments to estimate the hyperparameters of the prior distribution. In particular, we compare two methods for estimating a spread matrix. The method is implemented in an interactive computer program that questions the expert and forms the subjective distribution. In some practical situations, it may not be possible to accurately represent an expert's opinions by a natural conjugate prior distribution, especially as the conjugate prior description suffers from some restrictions in the manner it represents dependencies between the mean vector and the covariance matrix. As a more flexible alternative, non-conjugate prior distributions are considered in which independent prior distributions for the mean vector and spread matrix are employed. A method of eliciting a prior distribution for the mean when it is assumed to be a multivariate normal distribution is developed. The implementation of the method is given through a pilot study. The prior distribution for the variance is assumed to have one of two forms: either an inverse-Wishart distribution or a generalised inverse-Wishart distribution. An elicitation method is developed for each of these forms of prior distribution. An example illustrating the implementation of the methods is given. Finally, the elicitation methods for the conjugate and the non-conjugate prior distributions are studied and compared in depth through an experiment with subject-matter experts. In this experiment two assessment tasks are used: one is related to the distribution of a sample mean and the other to the distribution of an individual item. A comparison is made between the expert assessments for these two types of task and marked differences are observed.
2
Wang, Min. "Generalized stable distributions and free stable distributions." Thesis, Lille 1, 2019. http://www.theses.fr/2019LIL1I032/document.
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Cette thèse porte sur les lois stables réelles au sens large et comprend deux parties indépendantes. La première partie concerne les lois stables généralisées introduites par Schneider dans un contexte physique et étudiées ensuite par Pakes. Elles sont définies par une équation différentielle fractionnaire dont on caractérise ici l'existence et l'unicité des solutions densité à l'aide de deux paramètres positifs, l'un de stabilité et l'autre de biais. On montre ensuite diverses identités en loi pour les variables aléatoires sous-jacentes. On étudie le comportement asymptotique précis de la densité aux deux extrémités du support. Dans certains cas, on donne des représentations exactes de ces densités comme fonctions de Fox. Enfin, on résout entièrement les questions ouvertes autour de l'infinie divisibilité des lois stables généralisées. La seconde partie, plus longue, porte sur l'analyse classique des lois alpha-stables libres réelles. Introduites par Bercovici et Pata, ces lois ont ensuite étudiées par Biane, Demni et Hasebe-Kuznetsov sous divers points de vue. Nous montrons qu'elles sont classiquement infiniment divisibles pour alpha inférieur ou égal à 1 et qu'elles appartiennent à la classe de Thorin étendue pour alpha inférieur ou égal à 3/4. La mesure de Lévy est calculée explicitement pour alpha = 1 et ce calcul entraîne que les lois 1-stables libres n'appartiennent pas à la classe de Thorin, sauf dans le cas de la loi de Cauchy avec dérive. Dans le cas symétrique, nous montrons que les densités alpha-stables libres ne sont pas infiniment divisibles quand alpha supérieur à 1. Dans le cas de signe constant nous montrons que les densités stables libres ont une courbe en baleine, autrement dit que leurs dérivées successives ne s'annulent qu'une seule fois sur leurs supports, ce qui constitue un raffinement de l'unimodalité et fait écho à la courbe en cloche des densités stables classiques récemment montrée rigoureusement. Nous établissons enfin plusieurs propriétés précises des densités stables libres spectralement de signe constant, parmi lesquelles une analyse détaillée de la variable aléatoire de Kanter, des expansions asymptotiques complètes en zéro, ainsi que plusieurs propriétés intrinsèques des courbes en baleine. Nous montrons enfin une nouvelle identité en loi pour l'algèbre Beta-Gamma, diverses propriétés d'ordre stochastique et nous étudions le problème classique de Van Dantzig pour la loi semi-circulaire généralisée<br>This thesis deals with real stable laws in the broad sense and consists of two independent parts. The first part concerns the generalized stable laws introduced by Schneider in a physical context and then studied by Pakes. They are defined by a fractional differential equation, whose existence and uniqueness of the density solutions is here characterized via two positive parameters, a stability parameter and a bias parameter. We then show various identities in law for the underlying random variables. The precise asymptotic behaviour of the density at both ends of the support is investigated. In some cases, exact representations as Fox functions of these densities are given. Finally, we solve entirely the open questions on the infinite divisibility of the generalized stable laws. The second and longer part deals with the classical analysis of the free alpha-stable laws. Introduced by Bercovici and Pata, these laws were then studied by Biane, Demni and Hasebe-Kuznetsov, from various points of view. We show that they are classically infinitely divisible for alpha less than or equal to 1 and that they belong to the extended Thorin class extended for alpha less than or equal to 3/4. The Lévy measure is explicitly computed for alpha = 1, showing that free 1-stable distributions are not in the Thorin class except in the drifted Cauchy case. In the symmetric case we show that the free alpha-stable densities are not infinitely divisible when alpha larger than 1. In the one-sided case we prove, refining unimodality, that the densities are whale-shaped, that is their successive derivatives vanish exactly once on their support. This echoes the bell shape property of the classical stable densities recently rigorously shown. We also derive several fine properties of spectrally one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, and several intrinsic features of whale-shaped functions. Finally, we display a new identity in law for the Beta-Gamma algebra, various stochastic order properties, and we study the classical Van Danzig problem for the generalized semi-circular law
3
Vu, Tuan T. "Invariant distributions." Thesis, McGill University, 1986. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=74019.
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4
Watt, Graeme. "Parton distributions." Thesis, Durham University, 2004. http://etheses.dur.ac.uk/2813/.
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Parton distributions, α(χ,μ(^₂) are essential ingredients for almost all theoretical calculations at hadron colliders. They give the number densities of the colliding par- tons (quarks and gluons) inside their parent hadrons at a given momentum fraction χand scale μ(^₂). The scale dependence of the parton distributions is given by DGLAP evolution, while the X dependence must be determined from a global analysis of deep-inelastic scattering (DIS) and related hard-scattering data. In Part I we introduce ‘doubly-unintegrateď parton distributions, fa(x, z, k(^₂),μ(^₂)), which additionally depend on the splitting fraction z and the transverse momentum (k) associated with the last evolution step. We show how these distributions can be used to calculate cross sections for inclusive jet production in DIS and compare the predictions to data taken at the HERA ep collider. We then calculate the transverse momentum distributions of พ and z bosons at the Tevatron pp collider and of Standard Model Higgs bosons at the forthcoming LHC. In Part II we study diffractive DIS, which is characterised by a large rapidity gap between the slightly deflected proton and the products of the virtual photon dissociation. We perform a novel QCD analysis of recent HERA data and extract diffractive parton distributions. The results of this analysis are used to investigate the effect of absorptive corrections in inclusive DIS. These absorptive corrections are due to the recombination of partons within the proton and are found to enhance the size of the gluon distribution at small X. We discuss the problem that the gluon distribution decreases with decreasing X at low scales while the sea quark distribution increases with decreasing X, whereas Regge theory predicts that both should have the same small-X behaviour. Our study hints at the possible importance of power corrections at low scales of around 1 GeV.
5
Feng, Jingyu. "Modeling Distributions of Test Scores with Mixtures of Beta Distributions." Diss., CLICK HERE for online access, 2005. http://contentdm.lib.byu.edu/ETD/image/etd1068.pdf.
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6
Lamb, Robert. "Dynamic Loss Distributions." Thesis, Imperial College London, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.520974.
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Kimber, M. A. "Unintegrated parton distributions." Thesis, Durham University, 2001. http://etheses.dur.ac.uk/3848/.
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We develop the theory of parton distributions f(_a)(π, k(^t2), μ(^2), unintegrated with respect to transverse momentum k(_t), from a phenomenological standpoint. In particular, we demonstrate a convenient approximation in which the unintegrated functions are obtained by explicitly performing the last step of parton evolution in perturbative QCD, with single-scale functions a(π, Q(^2) as input. Results are presented in the context of DGLAP and combined BFKL-DGLAP evolution, but with angular ordering imposed in the last step of the evolution. We illustrate the application of these unintegrated distributions to predict cross sections for physical processes at lepton-hadron and hadron-hadron colliders. The use of partons with incoming transverse momentum, based on k(_t)-factorisation, is intended to replace phenomenological "smearing" in the perturbative region k(_t) > k(_o) (k(_o) ≈ 1 GeV), and enables the full kinematics of a process to be included even at leading order. We apply our framework to deep inelastic scattering and the fitting of F(_2)(π, Q(^2), to the transverse momentum spectra of prompt photons in hadroproduction and in photoproduction, and to the topical problem of bb production at HERA. Finally, we address the issue of parton-parton recombination (shadowing) at very low values of π, building on recent work by Kovchegov and others to make predictions for the likely magnitude of shadowing effects at the LHC.
8
REY, DAVID. "DISTRIBUTIONS AND IMMERSIONS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2007. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=11943@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO<br>COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>Os desafios de estudar formas levaram matemáticos a criar
abstrações, em particular através da geometria diferencial.
Porém, formas simples como cubos não se adequam a
ferramentas diferenciáveis. Este trabalho é uma tentativa
de usar avanços recentes da análise, no caso a teoria das
distribuições, para estender quantidades diferenciáveis a
objetos singulares. Como as distribuições generalizam as
funções e permitem derivações infinitas, substituição das
parametrizações de subvariedades clássicas por
distribuições poderia naturalmente generalizar as
subvariedades suaves. Isso nos leva a definir D-imersões.
Esse trabalho demonstra que essa formulação, de fato,
generaliza as imersões suaves. Extensões para outras
classes de subvariedades são discutidas através de exemplos
e casos particulares.<br>The challenge of studying shapes has led mathematicians
to create powerful abstract concepts, in particular
through Differential Geometry. However, differential
tools do not apply to simple shapes like cubes. This work
is an attempt to use modern advances of the Analysis,
namely Distribution Theory, to extend differential
quantities to singular objects. Distributions generalize
functions, while allowing infinite differentiation. The
substitution of classical immersions, which usually serve
as submanifold parameterizations, by distributions might
thus naturally generalize smooth immersion. This leads to
the concept of D-immersion. This work proves that this
formulation actually generalizes smooth immersions.
Extensions to non-smooth of immersions are discussed
through examples and specific cases.
9
Anabila, Moses A. "Skew Pareto distributions." abstract and full text PDF (free order & download 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:1453191.
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Abuhassan, Hassan. "Some transformed distributions /." Available to subscribers only, 2007. http://proquest.umi.com/pqdweb?did=1456289011&sid=9&Fmt=2&clientId=1509&RQT=309&VName=PQD.
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Barrowes, Benjamin E. "YSCAT backscatter distributions /." Diss., CLICK HERE for online access, 1999. http://contentdm.lib.byu.edu/ETD/image/etd2.pdf.
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Barrowes, Benjamin E. "YSCAT Backscatter Distributions." BYU ScholarsArchive, 2003. https://scholarsarchive.byu.edu/etd/67.
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YSCAT is a unique ultrawideband microwave scatterometer developed to investigate the sea surface under a variety of environmental and radar parameters. The YSCAT94 experiment consisted of a six month deployment on the WAVES research tower operated by the Canada Center for inland Waters (CCIW). Over 3500 hours of data were collected at 2Γ 3.05Γ 5.3Γ 10.02Γ and 14 GHz and at a variety of wind speeds, relative azimuth angles, and incidence angle.
A low wind speed "rolloff" of the normalized radar cross section (σ°) in YSCAT94 data is found and quantified. The rolloff wind speedΓ γΓ is estimated through regression estimation analysis using an Epanechnikov kernel. For YSCAT94 data, the rolloff is most noticeable at mid-range incidence angles with γ values ranging from 3 to 6 m/s.
In order to characterized YSCAT94 backscatter distributions, a second order polynomial in log space is developed as a model for the probability of the radar cross sectionΓρ(σ°). Following Gotwols and ThompsonΓρ(σ°) is found to adhere to a log-normal distribution for horizontal polarization and a generalized log-normal distribution for vertical polarization. If ρ(α|σ°) is assumed to be Rayleigh distributed, the instantaneous amplitude distribution ρ(α) is found to be the integral of a Rayleigh/generalized log-normal distribution.
A robust algorithm is developed to fit this probability density function to YSCAT94 backscatter distributions. The mean and variance of the generalized log-normal distribution are derived to facilitate this algorithm. Over 2700 distinct data cases sorted according to five different frequencies, horizontal and vertical polarizations, upwind and downwind, eight different incidence angles Γ1-10 m/s wind speeds, and 0.1-0.38 mean wave slope are considered. Definite trends are recognizable in the fitted parameters a1Γ a2Γ and C of the Rayleigh/generalized log-normal distribution when sorted according to wind speed and mean wave slope.
At mid-range incidence angles, the Rayleigh/generalized log-normal distribution is found to adequately characterize both low and high amplitude portions of YSCAT94 backscatter distributions. However, at higher incidence angels (50°and 60°) the more general Weibull/generalized log-normal distributions is found to better characterized the low amplitude portion of the backscatter distributions.
13
Liu, Liang. "Reconstructing posterior distributions of a species phylogeny using estimated gene tree distributions." Columbus, Ohio : Ohio State University, 2006. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1155754980.
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Bavouzet, Nicolas. "Les galaxies infrarouges : distribution spatiale, contributions au fond extragalactique et distributions spectrales d'énergie." Phd thesis, Université Paris Sud - Paris XI, 2008. http://tel.archives-ouvertes.fr/tel-00363975.
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Si la formation des grandes structures de l'Univers est plutot bien comprise, celle des galaxies ainsi que leur évolution l'est beaucoup moins. On s'intéresse en particulier aux mécanismes de mise en route de la formation stellaire dans les galaxies. L'étude des galaxies lumineuses en infrarouge constitue une des approches pour répondre à ces questions. Le travail effectué au cours de cette thèse repose essentiellement sur l'analyse de données infrarouges provenant du satellite Spitzer.<br /><br />La première partie de ce travail porte sur l'étude de la<br />distribution spatiale des galaxies infrarouges. Nous avons introduit une nouvelle méthode pour mesurer la fonction de corrélation angulaire des galaxies. Cette méthode a été validée sur des simulations et des données. Nous avons également montré comment les effets de corrélation spatiale pouvaient biaiser les mesures de flux moyen réalisées par la méthode d'empilement. De plus, la<br />fonction de corrélation angulaire mesurée pour les sources sélectionnées à 3.6 microns et 24 microns montre un excès de corrélation aux petites échelles angulaires. Ceci pourrait être lié à l'interaction des galaxies à l'intérieur d'un meme halo de matière noire qui favoriserait alors les mécanismes d'émission infrarouge.<br /><br />Dans un second temps, nous nous sommes attachés à mieux caractériser le fond diffus infrarouge (CIB) en déterminant la contribution à ce fond des sources détectées à 3.6 microns et en la comparant à celle des sources sélectionnées à 24 microns. Nous avons également estimé la contribution au CIB à 3.6 et 24 microns des sources sélectionnées à 3.6 microns en fonction de leur taux de formation stellaire spécifique.<br /><br />Enfin, nous avons étudié les distributions spectrales d'énergie d'un grand nombre de galaxies situées entre z=0 et z=2 : nous avons montré d'une part que les luminosités à 8 et 24 microns étaient de bons traceurs de la luminosité totale infrarouge (et donc du taux de formation stellaire) et, d'autre part, que les propriétés de ces galaxies ne semblaient pas évoluer entre entre z=0 et z=1. Nous avons également étudié de façon détaillée le spectre infrarouge de 17 galaxies sélectionnées à 70 microns et nous avons montré que la luminosité relative des PAHs diminuait lorsque le champ de rayonnement<br />augmentait.
15
Godin, Antoine. "Deciphering synaptic receptor distributions, clustering and stoichiometry using spatial intensity distribution analysis (SpIDA)." Thesis, McGill University, 2011. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=96774.
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Measuring protein interactions in subcellular compartments is key to understanding cell signalling mechanisms, but quantitative analysis of these interactions in situ has remained a major challenge. This thesis presents a novel analysis technique, spatial intensity distribution analysis (SpIDA), which may be applied to images obtained using fluorescence microscopy. SpIDA measures fluorescent particle densities and oligomerization states within individual images. The method is based on fitting intensity histograms from single images with super-Poissonian distributions to obtain density maps of fluorescent molecules and their quantal brightness. Since distributions are acquired spatially rather than temporally, this analysis may be applied to both live and chemically fixed cells and tissue. The technique does not rely on spatial correlations, freeing it from biases due to subcellular compartmentalization and heterogeneity within tissue samples. First, we validated the analysis technique evaluating its limits and demonstrating how it can be used to obtain useful information from complex biological samples. Analysis of simulations and heterodimeric GABAB receptors in spinal cord samples shows that the approach yields accurate estimates over a broad range of densities. SpIDA is applicable to sampling within subcell areas and reveals the presence of monomers and multimers with single dye labeling. We show that the substance P receptor (NK-1r) almost exclusively forms homodimers on the membrane and is primarily monomeric in the cytoplasm of dorsal horn neurons. Triggering receptor internalization caused a measurable decrease in homodimer density on the membrane surface. Finally, using GFP-tagged receptor subunits, we show that SpIDA can resolve dynamic changes in receptor oligomerization in live cells and is applicable to detection of high order oligomerization states. We then compared SpIDA results with those obtained from fluorescence lifetime imaging, and used it to extract information on receptor tyrosine kinase (RTK) dimerization at the cell membrane in response to GPCR activation. We show that RTK dimerization can be used as an index of activation or transactivation and then characterize the level of transactivation of many RTK-GPCR pairs, with cell cultures and primary neuron cultures with endogenous levels of RTKs and GPCRs. Dose-response curves were obtained from which pharmalogical parameters can be compared for each GPCR studied. Our data demonstrates that by allowing for time and space quantification of heterogenous oligomeric states, SpIDA enables systematic quantitative mechanistic studies not only of RTK transactivation at the cell membrane, but also of other cell signaling processes involving changes in protein oligomerization, trafficking and activity in different subcellular localizations. Finally, we studied the changes in number of synaptic sites in the neurons of the dorsal horn of the spinal cord of rats after a peripheral nerve injury (PNI), which consists of our model for chronic pain. We show that, after the PNI, there is a general decrease in synaptic sites together with a scaling or increasing of some of the GABAA receptor subunits. This scaling of the GABAA receptors at the postsynaptic sites was replicated by incubating the histological sections in a brain derivative nerve factor. Furthermore, we use SpIDA to obtain stoichiometry information for the GABAA receptor subunits directly at the postsynaptic sites. In short, we observe a switch from receptors containing two alpha1 to receptors containing two alpha2 and alpha3. This general change in subunits will have a direct effect on the cell as it will have different effects on the cell membrane conductance in response to GABA. As demonstrated, the advantages and greater versatility of SpIDA over current techniques opens the door to a new level of quantification for studies of protein interactions in native tissue using standard fluorescence microscopy.<br>Measuring protein interactions in subcellular compartments is key to understanding cell signalling mechanisms, but quantitative analysis of these interactions in situ has remained a major challenge. This thesis presents a novel analysis technique, spatial intensity distribution analysis (SpIDA), which may be applied to images obtained using fluorescence microscopy. SpIDA measures fluorescent particle densities and oligomerization states within individual images. The method is based on fitting intensity histograms from single images with super-Poissonian distributions to obtain density maps of fluorescent molecules and their quantal brightness. Since distributions are acquired spatially rather than temporally, this analysis may be applied to both live and chemically fixed cells and tissue. The technique does not rely on spatial correlations, freeing it from biases due to subcellular compartmentalization and heterogeneity within tissue samples. First, we validated the analysis technique evaluating its limits and demonstrating how it can be used to obtain useful information from complex biological samples. Analysis of simulations and heterodimeric GABAB receptors in spinal cord samples shows that the approach yields accurate estimates over a broad range of densities. SpIDA is applicable to sampling within subcell areas and reveals the presence of monomers and multimers with single dye labeling. We show that the substance P receptor (NK-1r) almost exclusively forms homodimers on the membrane and is primarily monomeric in the cytoplasm of dorsal horn neurons. Triggering receptor internalization caused a measurable decrease in homodimer density on the membrane surface. Finally, using GFP-tagged receptor subunits, we show that SpIDA can resolve dynamic changes in receptor oligomerization in live cells and is applicable to detection of high order oligomerization states. We then compared SpIDA results with those obtained from fluorescence lifetime imaging, and used it to extract information on receptor tyrosine kinase (RTK) dimerization at the cell membrane in response to GPCR activation. We show that RTK dimerization can be used as an index of activation or transactivation and then characterize the level of transactivation of many RTK-GPCR pairs, with cell cultures and primary neuron cultures with endogenous levels of RTKs and GPCRs. Dose-response curves were obtained from which pharmalogical parameters can be compared for each GPCR studied. Our data demonstrates that by allowing for time and space quantification of heterogenous oligomeric states, SpIDA enables systematic quantitative mechanistic studies not only of RTK transactivation at the cell membrane, but also of other cell signaling processes involving changes in protein oligomerization, trafficking and activity in different subcellular localizations. Finally, we studied the changes in number of synaptic sites in the neurons of the dorsal horn of the spinal cord of rats after a peripheral nerve injury (PNI), which consists of our model for chronic pain. We show that, after the PNI, there is a general decrease in synaptic sites together with a scaling or increasing of some of the GABAA receptor subunits. This scaling of the GABAA receptors at the postsynaptic sites was replicated by incubating the histological sections in a brain derivative nerve factor. Furthermore, we use SpIDA to obtain stoichiometry information for the GABAA receptor subunits directly at the postsynaptic sites. In short, we observe a switch from receptors containing two alpha1 to receptors containing two alpha2 and alpha3. This general change in subunits will have a direct effect on the cell as it will have different effects on the cell membrane conductance in response to GABA. As demonstrated, the advantages and greater versatility of SpIDA over current techniques opens the door to a new level of quantification for studies of protein interactions in native tissue using standard fluorescence microscopy.
16
Hohn, Jennifer Lynn. "Generalized Probabilistic Bowling Distributions." TopSCHOLAR®, 2009. http://digitalcommons.wku.edu/theses/82.
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Have you ever wondered if you are better than the average bowler? If so, there are a variety of ways to compute the average score of a bowling game, including methods that account for a bowler’s skill level. In this thesis, we discuss several different ways to generate bowling scores randomly. For each distribution, we give results for the expected value and standard deviation of each frame's score, the expected value of the game’s final score, and the correlation coefficient between the score of the first and second roll of a single frame. Furthermore, we shall generalize the results in each distribution for an frame game on pins. Additionally, we shall generalize the number of possible games when bowling frames on pins. Then, we shall derive the frequency distribution of each frame’s scores and the arithmetic mean for frames on pins. Finally, to summarize the variety of distributions, we shall make tables that display the results obtained from each distribution used to model a particular bowler’s score. We evaluate the special case when bowling 10 frames on 10 pins, which represents a standard bowling game.
17
Dai, Tao. "On multivariate unimodal distributions." Thesis, University of British Columbia, 1989. http://hdl.handle.net/2429/27411.
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In this thesis, Kanter's representation of multivariate unimodal distributions is shown equivalent to the usual mixture of uniform distributions on symmetric, compact and convex sets. Kanter's idea is utilized in several contexts by viewing multivariate distributions as mixtures of uniform distributions on sets of various shapes. This provides a unifying viewpoint of what is in the literature and gives some important new classes of multivariate unimodal distributions. The closure properties of these new classes under convolution, marginality and weak convergence, etc. and their relationships with other notions of multivariate unimodality are discussed. Some interesting examples and their 2- or 3-dimensional pictures are presented.<br>Science, Faculty of<br>Statistics, Department of<br>Graduate
18
Kim, Young Do. "Return distributions and applications." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2007. http://wwwlib.umi.com/cr/ucsd/fullcit?p3266772.
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Thesis (Ph. D.)--University of California, San Diego, 2007.<br>Title from first page of PDF file (viewed August 7, 2007). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references.
19
Viraswami, Kalyanee. "On multivariate gamma distributions." Thesis, McGill University, 1991. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=60513.
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This thesis is an exposition of the various forms of Multivariate Gamma Distributions available in the literature. We begin by discussing the univariate gamma distributions and some of their most important properties. These distributions are then extended to the vector variate and matrix variate cases. Derivations, applications and properties are given for gamma distributions in these two categories. Further generalizations associated with several matrix variate gamma variables are also included.
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PAIXÃO, Ana Carla Percontini da. "New Extended Lifetime Distributions." Universidade Federal de Pernambuco, 2014. https://repositorio.ufpe.br/handle/123456789/12240.
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Previous issue date: 2014<br>Este trabalho está dividido em quatro capítulos independentes. Nos Capítulos 2 e 3 propomos
extensões para a distribuição Weibull. A primeira delas, com cinco parâmetros, é uma
composição das distribuições beta e Weibull Poisson. Essa nova distribuição tem como submodelos
algumas importantes distribuições descritas na literatura e outras ainda não discutidas
tais como: bata exponencial Poisson, Weibull Poisson exponencializada, Rayleigh Poisson exponencializada,
beta Weibull, Weibull, exponencial, entre outras. Obtemos algumas propriedades
matemáticas tais como momentos ordinários e incompletos, estatísticas de ordem e seus momentos
e entropia de Rényi. Usamos o método da máxima verossimilhança para obter estimativas
dos parâmetros. A potencialidade desse novo modelo é mostrada por meio de um conjunto de
dados reais. A segunda extensão, com quatro parâmetros, é uma composição das distribuições
Poisson generalizada e Weibull, tendo a Poisson generalizada exponencial, a Rayleigh Poisson,
Weibull Poisson e Weibull como alguns de seus sub-modelos. Várias propriedades matemáticas
foram investigadas, incluíndo expressões explícitas para os momentos ordinários e incompletos,
desvios médios, função quantílica, curvas de Bonferroni e Lorentz, con abilidade e as entropias
de Rényi e Shannon. Estatísticas de ordem e seus momentos são investigados. A estimativa de
parâmetros é feita pelo método da máxima verossimilhança e é obtida a matriz de informação
obsevada. Uma aplicação a um conjunto de dados reais mostra a utilidade do novo modelo. Nos
dois últimos capítulos propomos duas novas classes de distribuições. No Capítulo 4 apresentamos
a família G- Binomial Negativa com dois parâmetros extras. Essa nova família inclui como caso
especial um modelo bastante popular, a Weibull binomial negativa, discutida por Rodrigues et
al.(Advances and Applications in Statistics 22 (2011), 25-55.) Algumas propriedades matemáticas
da nova classe são estudadas, incluindo momentos e função geradora. O método de máxima
verossimilhança é utilizado para obter estimativas dos parâmetros. A utilidade da nova classe
é mostrada através de um exemplo com conjuntos de dados reais. No Capítulo 5 apresentamos
a classe Zeta-G com um parâmetro extra e algumas nova distribuições desta classe. Obtemos
expressões explícitas para a função quantílica, momentos ordinários e incompletos, dois tipos de
entropia, con abilidade e momentos das estatísticas de ordem. Usamos o método da máxima
verossimilhança para estimar os parâmetros e a utilidade da nova classe é exempli cada com um
conjunto de dados reais.
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Hamdi, Fatma. "Apprentissage en distributions déséquilibrées." Paris 13, 2012. http://scbd-sto.univ-paris13.fr/intranet/edgalilee_th_2012_hamdi.pdf.
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Le travail de recherche présenté dans cette thèse concerne le développement d’approches à base d’apprentissage artificiel pour le traitement des bases de données déséquilibrées. Afin d’apporter des éléments de réponse pour cette problématique, nous proposons différentes contributions. Une première méthode qui agit au niveau des données d’apprentissage SNCR, c’est une technique d’échantillonnage structurel adaptatif qui permet de rééquilibrer les données par sous-échantillonnage de la classe majoritaire. La méthode proposée est guidée par la structure topologique des données et leur distribution. La seconde contribution proposée dans cette thèse aborde le problème de l’apprentissage à partir d’une seule classe, c’est un moyen permettant de contourner le problème de classes déséquilibrées à un problème de détection de nouveauté. Le modèle RS-NDF est basée sur un ensemble de filtres adaptatif. Chaque filtre est conçu dans un espace de description dont les composantes et la dimension sont choisies aléatoirement. Nous avons proposé en outre une amélioration de la qualité de RS-NDF par une extension plus économe SRS-NDF, permettant de réduire le nombre de modèles participant à la prise de décision. L’objectif est de choisir parmi cet ensemble de filtres, le sous ensemble qui permet d’atteindre les meilleurs performances. Enfin nous avons proposé une adaptation de l’approche RS-NDF au problème de la détection de la dérive de concept. Les résultats obtenus sur la validation des approches traités an cette étude sont encourageants et prometteurs<br>The research work exposed in this thesis concerns the development of approaches for processing and modeling unbalanced databases. In order to afford solutions to this problem, we propose different contributions. A first proposition acting at the learning data level SNCR, it is a technique of adaptive structural sampling that allowing data rebalancing by sub-sampling of the ma jority class. The proposed method is guided by the topological structure of the data and their distribution. The second proposed approach in this thesis discuss the problem of one class learning, it is a way allowing to bypass the problem of unbalanced classes to a novelty detection problem. The model RS-NDF is based on a set of adaptive filters. Every filter is conceived in a description subspace which the components and the dimension are randomly chosen. Besides, we propose an improvement of the quality of RS-NFD by an extension SRS-NDF allowing to reduce the number of models participating in the decision. The goal is to choose between those filters the sub-set which allows to reach the best performances. Finally, we propose an adaptation of the RS-NDF approach to the concept drift detection problem. The results obtained using the proposed approaches are encouraging and promising
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Acharya, Tanka Prasad Somers Greg Lynn. "Prediction of distribution for total height and crown ratio using normal versus other distributions." Auburn, Ala., 2006. http://repo.lib.auburn.edu/2006%20Fall/Theses/ACHARYA_TANKA_3.pdf.
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Yoo, Jiyoun. "Modeling Compressive Stress Distributions at the Interface between a Pallet Deck and Distribution Packaging." Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/39939.
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Three components, a pallet, packaging, and material handling equipment, of the unit load portion of the supply chain are physically and mechanically interacting during product storage and shipping. Understanding the interactions between two primary components, a pallet and packaging, in a unit load is a key step towards supply chain cost reduction and workplace safety improvement. Designing a unit load without considering physical and mechanical interactions, between those two components, can result in human injury or death caused from a unsafe workplace environment and increased supply chain operating costs, due to product damage, high packaging cost, disposal expense, and waste of natural resources.
This research is directed towards developing predictive models of the compressive stress distributions using the principle of the beam on an elastic foundation and experimentally quantifying the compressive stress distributions. The overall objective of this study is to develop a model that predicts compressive stress distributions at the interface between a pallet deck and packaging as a function of: pallet deck stiffness, packaging stiffness, and pallet joint fixity. The developed models were validated by comparison to the results of physical testing of the unit load section. Design variables required for modeling included Modulus of Elasticity (MOE) of pallet deckboards, Rotation Modulus (RM) for nailed joints, and packaging stiffness.
Predictive models of the compressive stress distributions were non-uniformly distributed across the interface between pallet deckboards and packaging. Maximum compressive stresses were observed at the deckboard ends over stringer segments. All predictive compressive stress
distributions were influenced by pallet deck stiffness, packaging stiffness, and joint fixity. The less the joint fixity the greater the pallet deck deflection. The stiffer deckboards are more sensitive to joint fixity. For predictive compressive stress distribution models, the measure of the stress concentrations was the Compressive Stress Intensity Factor (SIF), which was the ratio of the estimated maximum compressive stress to the applied stress. Less stiff pallets and stiffer packaging resulted in greater SIF for all end condition models. SIF was reduced by stiffer joint, stiffer pallet deck and more flexible packaging. The stiffer the pallet deck and pallet joint the greater the effective bearing area. The lower stiffness packaging resulted in the greater effective bearing area with all three packages. The predicted effective bearing area was more influenced by pallet deck stiffness than the packaging stiffness.
The developed prediction models were validated by comparison to experimental results. All prediction models fell within 95% confidence bounds except the 3/8-inch deck with free ends and 3/4-inch deck with fixed ends. The difference between predicted and measured results was due to a limitation in pressure sensor range and test specimen construction for the free end model and fixed end model, respectively.
The results show effects of pallet deck stiffness and packaging stiffness on SIFs with percentage changes ranging from 2 to 26% (absolute value of change) for all three end conditions. The sensitivity study concluded that changing both pallet deck stiffness and packaging stiffness more significantly influenced the SIFs than bearing areas.<br>Ph. D.
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Andronikof, Emmanuel. "Microlocalisation tempérée des distributions et des fonctions holomorphes." Paris 13, 1987. http://www.theses.fr/1987PA132011.
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1ere partie : on introduit la notion de système normal d'équations microdifférentielles à l'aide de filtrations de cohen-macaulay pour généraliser la notion classique de matrice (carrée) normale. Le résultat essentiel est alors que cette notion est stable par restriction non caractéristique, en particulier une telle restriction d'un système présente par une matrice normale est un module localement libre. Cela n'est plus vrai en général quand la matrice n'est pas normale : c'est ce qu'on indique dans l'appendice ou l'on construit un contre-exemple après avoir caracterisé les systemes déterminés en termes de dimension homologique. 2eme partie : sur une varièté analytique réelle, on definit la microlocalisation temperee des faisceaux de distributions th(f) de kashiwara et on donne une version microlocale du foncteur th et de ses operations. Sur une variete complexe, on obtient la version microlocale du foncteur rh de kashiwara avec ses opérations, et on construit les analogues tempérés des faisceaux de microfonctions et d'opérateurs microlocaux de sato, kashiwara et kawai. On donne également une version microlocale de la correspondance de riemann-hilbert
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Schluter, Christian. "Topics in distributional analysis : the importance of intermediate institutions for income distributions, inequality and intra-distributional mobility." Thesis, London School of Economics and Political Science (University of London), 1999. http://etheses.lse.ac.uk/1487/.
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The unifying theme of this dissertation is the importance of intermediate institutions for income distributions, inequality and intra-distributional mobility. First, we analyse the effects of informational problems in a general equilibrium model with dynamically evolving wealth distributions. Poor agents need to borrow funds but a non-commitment problem on the capital market leads to persistent inequality. The next important institution to be examined is the tax-benefit system. The third chapter investigates the relative performance of alternative unemployment benefit regimes in a search-theoretic general equilibrium model of the labour market. Policy objectives such as the reduction in inequality or the alleviation of poverty are considered and the incentive problems are examined. Prior to the empirical analysis, the fourth chapter develops the large sample distribution of a number of inequality and mobility indices. Moreover, the relative performance of these (asymptotic) approximations and various bootstrap estimators is examined. The data is described in chapter five. The sixth chapter analyses the distributional consequences of the German tax-benefit system using the German Socio-Economic Panel. Two dimensions income dynamics are investigated by distinguishing between shape dynamics and intra-distributional mobility. The complementarity between various tools such as non-parametric stochastic kernel density estimates and transition matrices is explored. As the transition probabilities are found to be time-varying, several statistical models of income mobility are estimated (and a new mover-stayer model is proposed) in the last chapter. In order to give an economic explanation of the observed mobility patterns various duration models (with duration dependent hazards and unobserved heterogeneity) are estimated.
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Hodis, Flaviu-Adrian. "Simulating univariate and multivariate nonnormal distributions based on a system of power method distributions /." Available to subscribers only, 2008. http://proquest.umi.com/pqdweb?did=1594480491&sid=3&Fmt=2&clientId=1509&RQT=309&VName=PQD.
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Thesis (Ph.D.)--Southern Illinois University Carbondale, 2008.<br>"Department of Educational Psychology and Special Education." Includes bibliographical references (p. 132-138). Also available online.
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Dominicy, Yves. "Quantile-based inference and estimation of heavy-tailed distributions." Doctoral thesis, Universite Libre de Bruxelles, 2014. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209311.
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This thesis is divided in four chapters. The two first chapters introduce a parametric quantile-based estimation method of univariate heavy-tailed distributions and elliptical distributions, respectively. If one is interested in estimating the tail index without imposing a parametric form for the entire distribution function, but only on the tail behaviour, we propose a multivariate Hill estimator for elliptical distributions in chapter three. In the first three chapters we assume an independent and identically distributed setting, and so as a first step to a dependent setting, using quantiles, we prove in the last chapter the asymptotic normality of marginal sample quantiles for stationary processes under the S-mixing condition.<p><p><p>The first chapter introduces a quantile- and simulation-based estimation method, which we call the Method of Simulated Quantiles, or simply MSQ. Since it is based on quantiles, it is a moment-free approach. And since it is based on simulations, we do not need closed form expressions of any function that represents the probability law of the process. Thus, it is useful in case the probability density functions has no closed form or/and moments do not exist. It is based on a vector of functions of quantiles. The principle consists in matching functions of theoretical quantiles, which depend on the parameters of the assumed probability law, with those of empirical quantiles, which depend on the data. Since the theoretical functions of quantiles may not have a closed form expression, we rely on simulations.<p><p><p>The second chapter deals with the estimation of the parameters of elliptical distributions by means of a multivariate extension of MSQ. In this chapter we propose inference for vast dimensional elliptical distributions. Estimation is based on quantiles, which always exist regardless of the thickness of the tails, and testing is based on the geometry of the elliptical family. The multivariate extension of MSQ faces the difficulty of constructing a function of quantiles that is informative about the covariation parameters. We show that the interquartile range of a projection of pairwise random variables onto the 45 degree line is very informative about the covariation.<p><p><p>The third chapter consists in constructing a multivariate tail index estimator. In the univariate case, the most popular estimator for the tail exponent is the Hill estimator introduced by Bruce Hill in 1975. The aim of this chapter is to propose an estimator of the tail index in a multivariate context; more precisely, in the case of regularly varying elliptical distributions. Since, for univariate random variables, our estimator boils down to the Hill estimator, we name it after Bruce Hill. Our estimator is based on the distance between an elliptical probability contour and the exceedance observations. <p><p><p>Finally, the fourth chapter investigates the asymptotic behaviour of the marginal sample quantiles for p-dimensional stationary processes and we obtain the asymptotic normality of the empirical quantile vector. We assume that the processes are S-mixing, a recently introduced and widely applicable notion of dependence. A remarkable property of S-mixing is the fact that it doesn't require any higher order moment assumptions to be verified. Since we are interested in quantiles and processes that are probably heavy-tailed, this is of particular interest.<p><br>Doctorat en Sciences économiques et de gestion<br>info:eu-repo/semantics/nonPublished
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Crippa, Davide. "q-distributions and random graphs /." [S.l.] : [s.n.], 1994. http://e-collection.ethbib.ethz.ch/show?type=diss&nr=10923.
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Kiptily, Dmitri V. "Properties of generalised parton distributions." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=974204722.
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Liu, Xinyu. "Schwartz analysis and intertwining distributions." Thesis, University of British Columbia, 2017. http://hdl.handle.net/2429/64171.
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In this dissertation, we combine the work of A. Aizenbud and D. Gourevitch on Schwartz functions on Nash manifolds, and the work of F. du Cloux on Schwartz inductions, to develop a toolbox of Schwartz analysis. We then use these tools to study the intertwining operators between parabolic inductions, and study the behavior of intertwining distributions on certain open subsets. Finally we use our results to give new proof of results in Bruhat's thesis on irreducibilities of degenerate principal series and minimal principal series.<br>Science, Faculty of<br>Mathematics, Department of<br>Graduate
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Bailey, Lee. "Spatial distributions within electroactive films." Thesis, University of Leicester, 2000. http://hdl.handle.net/2381/30039.
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The spatial distributions of polymer and solvent within three types of polymer modified electrode have been investigated using the technique of neutron reflectivity. The overall aim of this research is to rationalise electrochemical behaviour of polymer modified electrodes through improved knowledge of in situ electrode structure. This may in future assist the design and fabrication of modified electrodes of desired function. The permselectivity failure of poly(vinylferrocene) has been investigated, with neutron reflectivity used to quantify mobile species transfer which occurs during redox cycling. Short time scale reflectivity studies have allowed observation of the changes in spatial distributions which occur within such films during oxidation and reduction, and rate limiting species transfers have been proposed. The reaction of the polymer poly(pentafluorophenyl 3-(pyrrol-1-yl)propanoate) with the nucleophile ferrocene-ethylamine has been followed and in particular the progress of the nucleophile's reaction front through the polymer monitored as a function of time. Definite regional differences in polymer structure were identified, which had an effect on the reaction progress. Studies on a third system looked at the effect of electrolyte pH on the structure of poly(orthotoluidine) films. At high acid concentration films are observed to swell, with the species responsible dependent on acid concentration. Neutron reflectivity was also used to rationalise combined EQCM/PBD responses obtained during redox cycling of poly(orthotoluidine).
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Quinn, Timothy Trent. "Spatial tessellations of wildlife distributions." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp04/mq22660.pdf.
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Finke, Jorge. "Stable emergent ideal free distributions." Columbus, Ohio : Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1172757689.
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Sitton, David E. R. "Generating random absolutely continuous distributions." Diss., Georgia Institute of Technology, 2001. http://hdl.handle.net/1853/28989.
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Reiss, Arie. "Pricing options on real distributions." Thesis, Imperial College London, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.272108.
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Titchener, J. B. "Continuum models for dislocation distributions." Thesis, University of Oxford, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.382704.
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Elfadaly, Fadlalla Ghaly Hassan Mohamed. "Elicitation of subjective probability distributions." Thesis, Open University, 2012. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.578703.
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To incorporate expert opinion into a Bayesian analysis, it must be quantified as a prior distribution through an elicitation process that asks the expert meaningful questions whose answers determine this distribution. The aim of this thesis is to fill some gaps in the available techniques for eliciting prior distributions for Generalized Linear Models (GLMs) and multinomial models. A general method for quantifying opinion about GLMs was developed in Garthwaite and Al- Awadhi (2006). They model the relationship between each continuous predictor and the dependant variable as a piecewise-linear function with a regression coefficient at each of its dividing points. How- ever, coefficients were assumed a priori independent if associated with different predictors. We relax this simplifying assumption and propose three new methods for eliciting positive-definite variance- covariance matrices of a multivariate normal prior distribution. In addition, we extend the method of Garthwaite and Dickey (1988) for eliciting an inverse chi-squared conjugate prior for the error variance in normal linear models. We also propose a novel method for eliciting a lognormal prior distribution for the scale parameter of a gamma GLM. For multinomial models, novel methods are proposed that quantify expert opinion about a conjugate Dirichlet distribution and, additionally, about three more general and flexible prior distributions. First, an elicitation method is proposed for the generalized Dirichlet distribution that was introduced by Connor and Mosimann (1969). Second, a method is developed for eliciting the Gaussian copula as a multivariate distribution with marginal beta priors. Third, a further novel method is constructed that quantifies expert opinion about the most flexible alternate prior, the logistic normal distribution (Aitchison, 1986). This third method is extended to the case of multinomial models with explanatory covariates.
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Pötzelberger, Klaus. "The Quantization Dimension of Distributions." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 1999. http://epub.wu.ac.at/1430/1/document.pdf.
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We show that the asymptotic behavior of the quantization error allows the definition of dimensions for probability distributions, the upper and the lower quantization dimension. These concepts fit into standard geometric measure theory, as the upper quantization dimension is always between the packing and the upper box-counting dimension, whereas the lower quantization dimension is between the Hausdorff and the lower box-counting dimension. (author's abstract)<br>Series: Forschungsberichte / Institut für Statistik
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Arrell, Katherine E. "Predicting glacier accumulation area distributions." Thesis, Durham University, 2005. http://etheses.dur.ac.uk/2805/.
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A mass balance model based on energy balance at the terrain surface was developed and used to predict glacier accumulation areas in the Jotunheimen, Norway. Spatially distributed melt modelling used local climate and energy balance surfaces to drive predictions, derived from regional climate and topographic data. Predictions had a temporal resolution of 1 month and a spatial resolution of 100 m, which were able to simulate observed glacier accumulation area distributions. Data were stored and manipulated within a GIS and spatial trends and patterns within the data were explored. These trends guided the design of a suite of geomorphologically and climatologically significant variables which were used to simulate the observed spatial organisation of climatic variables, specifically temperature, precipitation and wind speed and direction. DEM quality was found as a critical factor in minimising error propagation. A new method of removing spatially and spectrally organised DEM error is presented using a fast Fourier transformation. This was successfully employed to remove error within the DEM minimising error propagation into model predictions. With no parameter fitting the modeled spatial distribution of snowcover showed good agreement with observed distributions. Topographic maps and a Landsat ETM+ image are used to validate the predictions and identify areas of over or under prediction. Topographically constrained glaciers are most effectively simulated, where aspect, gradient and altitude impose dominant controls on accumulation. Reflections on the causes of over or under prediction are presented and future research directions to address these are outlined. Sensitivity of snow accumulation to climatic and radiative variables was assessed. Results showed the mass balance of accumulation areas is most sensitive to air temperature and cloud cover parameterisations. The model was applied to reconstruct snow accumulation at the last glacial maximum and under IPCC warming scenarios to assess the sensitivity of melt to changing environmental conditions, which showed pronounced sensitivity to summer temperatures Low data requirements: regional climate and elevation data identify the model as a powerful tool for predicting the onset, duration and rate of melt for any geographical area.
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Moala, Fernando Antonio. "Elicitation of multivariate prior distributions." Thesis, University of Sheffield, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.434948.
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Lynch, Andrew Graeme. "Covariate models for size distributions." Thesis, University of Sheffield, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.251220.
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Loder, Natasha. "Insect species-body size distributions." Thesis, University of Sheffield, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.284385.
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Strain, J. A. "Angular distributions of sputtered particles." Thesis, University of Sussex, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.375144.
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Cleaver, James Arnold Stafford. "Velocity distributions in conical hoppers." Thesis, University of Cambridge, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.387131.
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Xie, Ning Ph D. Massachusetts Institute of Technology. "Testing k-wise independent distributions." Thesis, Massachusetts Institute of Technology, 2012. http://hdl.handle.net/1721.1/78457.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 119-123).<br>A probability distribution over {0, 1}' is k-wise independent if its restriction to any k coordinates is uniform. More generally, a discrete distribution D over E1 x ... x E, is called (non-uniform) k-wise independent if for any subset of k indices {ii, . . . , ik} and for any zi E Ei 1, .. , Zk E Eik , PrX~D [Xi 1 - - -Xi, = Z1 .. z] = PrX-D[Xi 1 = zi] ... PrX~D [Xik = Zk]. k-wise independent distributions look random "locally" to an observer of only k coordinates, even though they may be far from random "globally". Because of this key feature, k-wise independent distributions are important concepts in probability, complexity, and algorithm design. In this thesis, we study the problem of testing (non-uniform) k-wise independent distributions over product spaces. For the problem of distinguishing k-wise independent distributions supported on the Boolean cube from those that are 6-far in statistical distance from any k-wise independent distribution, we upper bound the number of required samples by O(nk/6 2 ) and lower bound it by Q (n 2 /6) (these bounds hold for constant k, and essentially the same bounds hold for general k). To achieve these bounds, we use novel Fourier analysis techniques to relate a distribution's statistical distance from k-wise independence to its biases, a measure of the parity imbalance it induces on a set of variables. The relationships we derive are tighter than previously known, and may be of independent interest. We then generalize our results to distributions over larger domains. For the uniform case we show an upper bound on the distance between a distribution D from k-wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. For the non-uniform case, we give a new characterization of distributions being k-wise independent and further show that such a characterization is robust based on our results for the uniform case. Our results yield natural testing algorithms for k-wise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.<br>by Ning Xie.<br>Ph.D.
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Atkins, Henry Hornblower. "Background maintenance utilizing common distributions." Thesis, Massachusetts Institute of Technology, 2003. http://hdl.handle.net/1721.1/87372.
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Thesis (M.Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.<br>Includes bibliographical references (leaves 28-29).<br>by Henry Hornblower Atkins, III.<br>M.Eng.
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Valiant, Paul (Paul Andrew). "Testing symmetric properties of distributions." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/44717.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.<br>Includes bibliographical references (p. 65-66).<br>We introduce the notion of a Canonical Tester for a class of properties on distributions, that is, a tester strong and general enough that "a distribution property in the class is testable if and only if the Canonical Tester tests it". We construct a Canonical Tester for the class of symmetric properties of one or two distributions, satisfying a certain weak continuity condition. Analyzing the performance of the Canonical Tester on specific properties resolves several open problems, establishing lower bounds that match known upper bounds: we show that distinguishing between entropy < a or > p on distributions over [n] requires nc/P-O(1) samples, and distinguishing whether a pair of distributions has statistical distance < a or > 0 requires n1-o(1) samples. Our techniques also resolve a conjecture about a property that our Canonical Tester does not apply to: distinguishing identical distributions from those with statistical distance > 0 requires Q(n2/3) samples.<br>by Paul Valiant.<br>Ph.D.
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Korte, Robert A. "Inference in Power Series Distributions." Kent State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=kent1352937611.
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Doran, Gary Brian Jr. "Multiple-Instance Learning from Distributions." Case Western Reserve University School of Graduate Studies / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=case1417736923.
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Fang, Hong-bin. "Some non-classical multivariate distributions." HKBU Institutional Repository, 1998. https://repository.hkbu.edu.hk/etd_ra/259.
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