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Dissertations / Theses on the topic 'Multivariate distributions'

Author: Grafiati

Published: 4 June 2021

Last updated: 20 February 2023

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1

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 w

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2

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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3

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

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4

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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5

Fang, Hong-bin. "Some non-classical multivariate distributions." HKBU Institutional Repository, 1998. https://repository.hkbu.edu.hk/etd_ra/259.

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6

Goff, Matthew. "Multivariate discrete phase-type distributions." Online access for everyone, 2005. http://www.dissertations.wsu.edu/Dissertations/Spring2005/m%5Fgoff%5F032805.pdf.

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7

Adham, Samia Abbas Yahya. "Multivariate Gompertz and Gompertz type distributions." Thesis, Imperial College London, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.248450.

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8

Pivato, Marcus. "Analytical methods for multivariate stable probability distributions." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/NQ63698.pdf.

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9

Jama, Siphamandla. "An alternative model for multivariate stable distributions." Master's thesis, University of Cape Town, 2009. http://hdl.handle.net/11427/8959.

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Includes bibliographical references (leaves 52-55).<br>As the title, "An Alternative Model for Multivariate Stable Distributions", depicts, this thesis draws from the methodology of [J36] and derives an alternative to the sub-Gaussian alpha-stable distribution as another model for multivariate stable data without using the spectral measure as a dependence structure. From our investigation, firstly, we echo that the assumption of "Gaussianity" must be rejected, as a model for, particularly, high frequency financial data based on evidence from the Johannesburg Stock Exchange (JSE). Secondly, the

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10

Maa, Jen-Fue. "Simulation-based parameter estimation for multivariate distributions /." The Ohio State University, 1993. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487843688959692.

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11

Maddox, Wesley J. "Dependency Measures and Copulas for Multivariate Infinitely Divisible Distributions." Case Western Reserve University School of Graduate Studies / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=case1493912655994132.

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12

Sun, Yannan. "Extremal dependence of multivariate distributions and its applications." Pullman, Wash. : Washington State University, 2010. http://www.dissertations.wsu.edu/Dissertations/Spring2010/Y_SUN_041610.pdf.

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13

Lachos, Victor H., and Filidor V. Labra. "Multivariate skew-normal/independent distributions: properties and inference." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/97108.

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Liu (1996) discussed a class of robust normal/independent distributions which contains a group of thick-tailed cases. In this article, we develop a skewed version of these distributions in the multivariate setting, and we call them multivariate skew normal/independent distributions. We derive several useful properties for them. The main virtue of the members of this family is that they are easy to simulate and lend themselves to an EM-type algorithm for maximum likelihood estimation. For two multivariate models of practical interest, the EM-type algorithm has been discussed with emphasis on th

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14

Nævestad, Markus. "Multivariate Distributions Through Pair-Copula Construction: Theory and Applications." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2009. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-9846.

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<p>It is often very difficult, particularly in higher dimensions, to find a good multivariate model that describes both marginal behavior and dependence structure of data efficiently. The copula approach to multivariate models has been found to fit this purpose particularly well, and since it is a relatively new concept in statistical modeling, it is under frequent development. In this thesis we focus on the decomposition of a multivariate model into pairwise copulas rather than the usual multivariate copula approach. We account for the theory behind the decomposition of a multivariate mode

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15

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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16

Leydold, Josef. "A Rejection Technique for Sampling from Log-Concave Multivariate Distributions." Department of Statistics and Mathematics, Abt. f. Angewandte Statistik u. Datenverarbeitung, WU Vienna University of Economics and Business, 1998. http://epub.wu.ac.at/946/1/document.pdf.

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Different universal methods (also called automatic or black-box methods) have been suggested to sample from univariate log-concave distributions. The description of a suitable universal generator for multivariate distributions in arbitrary dimensions has not been published up to now. The new algorithm is based on the method of transformed density rejection. To construct a hat function for the rejection algorithm the multivariate density is tranformed by a proper transformation T into a concave function (in the case of log-concave density T(x) = log(x).) Then it is possible to construct a domin

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17

Gonzalez-Casanova, H. Pedro. "The recovery of smooth multivariate functions from discrete data distributions." Thesis, University of Oxford, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.236264.

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18

Wayne, Henry. "Towards a theory of multivariate interpolation using spaces of distributions." Thesis, University of Leicester, 1996. http://hdl.handle.net/2381/34574.

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The research contained in this thesis concerns the study of multivariate interpolation problems. Given a discrete set of possibly complex-valued data, indexed by a set of interpolation nodes in Euclidean space, it is desirable to generate a function which agrees with the data at the nodes. Within this general framework, this work pursues and generalizes one approach to the problem. Based on a variational theory, we construct a parameterised family of Hilbert spaces of tempered distributions, detail the necessary evolution of the interpolation problem, and provide a general error analysis. Some

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19

Akdemir, Deniz. "A Class of Multivariate Skew Distributions: Properties and Inferential Issues." Bowling Green, Ohio : Bowling Green State University, 2009. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=bgsu1237574643.

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20

Zheng, Shimin, J. M. Hardin, and A. K. Gupta. "The Inverse Problem of Multivariate and Matrix-Variate Skew Normal Distributions." Digital Commons @ East Tennessee State University, 2012. https://dc.etsu.edu/etsu-works/47.

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In this paper, we prove that the joint distribution of random vectors Z 1 and Z 2 and the distribution of Z 2 are skew normal provided that Z 1 is skew normally distributed and Z 2 conditioning on Z 1 is distributed as closed skew normal. Also, we extend the main results to the matrix variate case.

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21

Joly, Emilien. "Estimation robuste pour des distributions à queue lourde." Thesis, Université Paris-Saclay (ComUE), 2015. http://www.theses.fr/2015SACLS216/document.

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Nous nous intéressons à estimer la moyenne d'une variable aléatoire de loi à queue lourde. Nous adoptons une approche plus robuste que la moyenne empirique classique communément utilisée. L'objectif est de développer des inégalités de concentration de type sous-gaussien sur l'erreur d'estimation. En d'autres termes, nous cherchons à garantir une forte concentration sous une hypothèse plus faible que la bornitude : une variance finie. Deux estimateurs de la moyenne pour une loi à support réel sont invoqués et leurs résultats de concentration sont rappelés. Plusieurs adaptations en dimension sup

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22

Karawatzki, Roman, and Josef Leydold. "Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions." Department of Statistics and Mathematics, Abt. f. Angewandte Statistik u. Datenverarbeitung, WU Vienna University of Economics and Business, 2005. http://epub.wu.ac.at/294/1/document.pdf.

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Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that

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23

Chen, Youshi. "Estimation with multivariate extreme value distributions with applications to environmental data." Thesis, University of British Columbia, 1991. http://hdl.handle.net/2429/29795.

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Several parametric families of multivariate extreme value distributions (Hüsler and Reiss 1989, Tawn 1990, Joe 1990a, 1990b) have been proposed recently. Applications to multivariate extreme value data sets are needed to assess the adequacy of the known families in their fit to data. Different families are compared in their range of multivariate dependence and their ease of use for maximum likelihood estimation. Some useful conclusions have been made from experience with several environmental data sets.<br>Science, Faculty of<br>Statistics, Department of<br>Graduate

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24

Ayari, Samia. "Nonparametric estimation of the dependence function for multivariate extreme value distributions." Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4078.

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Dans cette thèse, nous abordons l'estimation non paramétrique de la fonction de dépendance des distributions multivariées à valeurs extrêmes. Dans une première partie, on adopte l’hypothèse classique stipulant que les variables aléatoires sont indépendantes et identiquement distribuées (i.i.d). Plusieurs estimateurs non paramétriques sont comparés pour une fonction de dépendance trivariée de type logistique dans deux différents cas. Dans le premier cas, on suppose que les fonctions marginales sont des distributions généralisées à valeurs extrêmes. La distribution marginale est remplacée par la

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25

Jairu, Desiderio N. "Distributions of some random volumes and their connection to multivariate analysis." Thesis, McGill University, 1987. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=63999.

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26

Karawatzki, Roman, Josef Leydold, and Klaus Pötzelberger. "Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions." Department of Statistics and Mathematics, WU Vienna University of Economics and Business, 2005. http://epub.wu.ac.at/1400/1/document.pdf.

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Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that

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27

Zhu, Hongxu. "AN R PACKAGE FOR FITTING DIRICHLET PROCESS MIXTURES OF MULTIVARIATE GAUSSIAN DISTRIBUTIONS." Case Western Reserve University School of Graduate Studies / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=case155752396390554.

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28

Wang, Sheng. "Regularized skewness parameter estimation for multivariate skew normal and skew t distributions." Diss., University of Iowa, 2019. https://ir.uiowa.edu/etd/6875.

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The skewed normal (SN) distribution introduced by Azzalini has opened a new era for analyzing skewed data. The idea behind it is that it incorporates a new parameter regulating shape and skewness on the symmetric Gaussian distribution. This idea was soon extended to other symmetric distributions such as the Student's t distribution, resulting in the invention of the skew t (ST) distribution. The multivariate versions of the two distributions, i.e. the multivariate skew normal (MSN) and multivariate skew t (MST) distributions, have received considerable attention because of their ability to t s

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29

Ley, Christophe. "Univariate and multivariate symmetry: statistical inference and distributional aspects." Doctoral thesis, Universite Libre de Bruxelles, 2010. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/210029.

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This thesis deals with several statistical and probabilistic aspects of symmetry and asymmetry, both in a univariate and multivariate context, and is divided into three distinct parts.<p><p>The first part, composed of Chapters 1, 2 and 3 of the thesis, solves two conjectures associated with multivariate skew-symmetric distributions. Since the introduction in 1985 by Adelchi Azzalini of the most famous representative of that class of distributions, namely the skew-normal distribution, it is well-known that, in the vicinity of symmetry, the Fisher information matrix is singular and the profile l

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30

潘成達 and Shing-Tat Poon. "Measuring the degree of dependence of lifetimes in some bivariate survival distributions." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1993. http://hub.hku.hk/bib/B31977443.

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31

Poon, Shing-Tat. "Measuring the degree of dependence of lifetimes in some bivariate survival distributions." [Hong Kong] : University of Hong Kong, 1993. http://sunzi.lib.hku.hk/hkuto/record.jsp?B13787421.

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32

Crawford, Jesse B. "Interpretation of eigenvalues in multivariate statistical analysis and Bartlett's test for Riesz distributions." [Bloomington, Ind.] : Indiana University, 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3331317.

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Thesis (Ph.D.)--Indiana University, Dept. of Mathematics, 2008.<br>Title from PDF t.p. (viewed on Jul 27, 2009). Source: Dissertation Abstracts International, Volume: 69-11, Section: B, page: 6890. Adviser: Steen A. Andersson.

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33

HUNTER, TINA D. "Gibbs Sampling and Expectation Maximization Methods for Estimation of Censored Values from Correlated Multivariate Distributions." University of Cincinnati / OhioLINK, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1212157899.

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34

Au, Charles. "Bayesian Hierarchical Models for Multivariate Continuous and Categorical Data." Thesis, The University of Sydney, 2019. https://hdl.handle.net/2123/21277.

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This thesis explores the use of the scale mixtures of normal (SMN) family of probability distributions as a data augmentation strategy in various Bayesian models for continuous and categorical data. The purpose is to facilitate efficient Bayesian computational methods, where Markov chain Monte Carlo (MCMC) algorithms have been a standard choice for handling sophisticated models. First, this thesis considers the use of the modified multivariate Student-t (Mod-t) distribution in seemingly unrelated regression (SUR) models. The Mod-t distribution allows for flexibly modelling the tails of its i

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35

Donner, Reik. "Advanced methods for analysing and modelling multivariate palaeoclimatic time series." Phd thesis, Universität Potsdam, 2006. http://opus.kobv.de/ubp/volltexte/2007/1256/.

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36

Krause, Dirk Verfasser], and Svetlozar T. [Akademischer Betreuer] [Račev. "Portfolio Analysis with Multivariate Normal Tempered Stable Processes and Distributions / Dirk Krause. Betreuer: S. T. Rachev." Karlsruhe : KIT-Bibliothek, 2011. http://d-nb.info/1025887433/34.

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37

Krause, Dirk [Verfasser], and Svetlozar T. [Akademischer Betreuer] Račev. "Portfolio Analysis with Multivariate Normal Tempered Stable Processes and Distributions / Dirk Krause. Betreuer: S. T. Rachev." Karlsruhe : KIT-Bibliothek, 2011. http://d-nb.info/1025887433/34.

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38

Adamski, Karien. "Generalised beta type II distributions - emanating from a sequential process." Thesis, University of Pretoria, 2013. http://hdl.handle.net/2263/40233.

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This study focuses on the development of a generalised multivariate beta type II distribution as well as the noncentral and bimatrix counterparts with positive domain. These models emanate from a sequential quality monitoring procedure with the normal and multivariate normal distributions as the underlying process distributions. Three different scenarios are considered, namely: 1. The variance is monitored from a normal process and the mean remains unchanged; 2. The above-mentioned scenario but the known mean also encounters a sustained shift; 3. The covariance structure of a multivari

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39

Leydold, Josef, and Wolfgang Hörmann. "A Sweep-Plane Algorithm for Generating Random Tuples in Simple Polytopes." Department of Statistics and Mathematics, Abt. f. Angewandte Statistik u. Datenverarbeitung, WU Vienna University of Economics and Business, 1997. http://epub.wu.ac.at/476/1/document.pdf.

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A sweep-plane algorithm by Lawrence for convex polytope computation is adapted to generate random tuples on simple polytopes. In our method an affine hyperplane is swept through the given polytope until a random fraction (sampled from a proper univariate distribution) of the volume of the polytope is covered. Then the intersection of the plane with the polytope is a simple polytope with smaller dimension. In the second part we apply this method to construct a black-box algorithm for log-concave and T-concave multivariate distributions by means of transformed density rejection. (author's abstra

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40

Haman, John T. "The energy goodness-of-fit test and E-M type estimator forasymmetric Laplace distributions." Bowling Green State University / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1524756256837676.

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41

Pant, Mohan Dev. "Simulating Univariate and Multivariate Burr Type III and Type XII Distributions Through the Method of L-Moments." OpenSIUC, 2011. https://opensiuc.lib.siu.edu/dissertations/401.

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The Burr families (Type III and Type XII) of distributions are traditionally used in the context of statistical modeling and for simulating non-normal distributions with moment-based parameters (e.g., Skew and Kurtosis). In educational and psychological studies, the Burr families of distributions can be used to simulate extremely asymmetrical and heavy-tailed non-normal distributions. Conventional moment-based estimators (i.e., the mean, variance, skew, and kurtosis) are traditionally used to characterize the distribution of a random variable or in the context of fitting data. However, conven

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42

Farias, Rafael Braz Azevedo. "Modelos multivariados binários com funções de ligação assimétricas." Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/45/45133/tde-15082012-233258/.

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Conjuntos de dados com respostas multivariadas aparecem frequentemente em pesquisas em que os dados são provenientes de questionários. Exemplos mais comuns são pesquisas de opinião, mais especificamente, pesquisas de marketing em que a preferência do consumidor em potencial é avaliado: pelo produto, marca, preço, praça, promoção e etc. Um tipo pesquisa de opinião que ganha grande destaque no Brasil de dois em dois anos são as pesquisas eleitorais de intenção de votos. Nós introduzimos nesta tese uma classe de modelos de regressão multivariados com funções de ligação assimétricas para o ajus

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43

Yuen, Hak-Keung. "Estimation of multivariate extreme value distributions by a kernel method with an application to non-Gaussian time series." Thesis, University of Surrey, 1988. http://epubs.surrey.ac.uk/843272/.

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In Chapter 1, we give a brief introduction to univariate extreme value theory. We also discuss the kernel method of density estimation and non-parametric regression analysis. Some methods of window-width choosing are also given. In Chapter 2, we develop a differentiable kernel estimator for the dependence function of a bivariate extreme value distribution. The estimator is applied to different sets of bivariate extreme value data. In Chapter 3, some existing methods for testing bivariate extreme pairs are discussed. Two new methods of testing independence against the alternative of a general b

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44

Dias, Raphael Antonio Prado. "Simulação estocástica de variáveis aleatórias Poisson correlacionadas: aplicação ao controle populacional do percevejo (Euschistus heros Fabricius) da soja (Glycine max L.)." Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/11/11134/tde-06052014-165527/.

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A simulação de dados que seguem distribuição de Poisson é essencial em muitas aplicações reais de várias áreas, tais como saúde, marketing, ciências agronômicas, entre outras em que os dados são contagens multivariadas. Métodos de simulação atuais sofrem de limitações computacionais e restrições à estrutura de correlação e, portanto, são raramente usados. Neste trabalho propôs-se uma modificação do método NORTA para gerar dados com distribuição Poisson multivariada a partir de uma distribuição normal multivariada com matriz de correlações e vetor de médias pré estabelecidos. Como as distribuiç

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45

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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46

MARCHANT, FUENTES Carolina Ivonne. "Essays on multivariate generalized Birnbaum-Saunders methods." Universidade Federal de Pernambuco, 2016. https://repositorio.ufpe.br/handle/123456789/18647.

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Submitted by Rafael Santana (rafael.silvasantana@ufpe.br) on 2017-04-26T17:07:37Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Carolina Marchant.pdf: 5792192 bytes, checksum: adbd82c79b286d2fe2470b7955e6a9ed (MD5)<br>Made available in DSpace on 2017-04-26T17:07:38Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Carolina Marchant.pdf: 5792192 bytes, checksum: adbd82c79b286d2fe2470b7955e6a9ed (MD5) Previous issue date: 2016-10-31<br>CAPES; BOLSA DO CHILE.<br>In the last decades, univa

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47

Mai, Jan-Frederik Verfasser], Rudi [Akademischer Betreuer] Zagst, Ralf [Akademischer Betreuer] [Korn, and Claudia [Akademischer Betreuer] Klüppelberg. "Multivariate exponential distributions with latent factor structure and related topics / Jan-Frederik Mai. Gutachter: Ralf Korn ; Claudia Klüppelberg. Betreuer: Rudi Zagst." München : Universitätsbibliothek der TU München, 2014. http://d-nb.info/1064383254/34.

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48

Frühwirth-Schnatter, Sylvia, and Rudolf Frühwirth. "Bayesian Inference in the Multinomial Logit Model." Austrian Statistical Society, 2012. http://epub.wu.ac.at/5629/1/186%2D751%2D1%2DSM.pdf.

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The multinomial logit model (MNL) possesses a latent variable representation in terms of random variables following a multivariate logistic distribution. Based on multivariate finite mixture approximations of the multivariate logistic distribution, various data-augmented Metropolis-Hastings algorithms are developed for a Bayesian inference of the MNL model.

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Goldbach, Johannes [Verfasser]. "A new approach to multivariate extreme value theory : f-implicit max-infinitely divisible distributions and f-implicit max-stable processes / Johannes Goldbach." Siegen : Universitätsbibliothek der Universität Siegen, 2016. http://d-nb.info/1114499404/34.

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Li, Zhonggai. "Objective Bayesian Analysis of Kullback-Liebler Divergence of two Multivariate Normal Distributions with Common Covariance Matrix and Star-shape Gaussian Graphical Model." Diss., Virginia Tech, 2008. http://hdl.handle.net/10919/28121.

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This dissertation consists of four independent but related parts, each in a Chapter. The first part is an introductory. It serves as the background introduction and offer preparations for later parts. The second part discusses two population multivariate normal distributions with common covariance matrix. The goal for this part is to derive objective/non-informative priors for the parameterizations and use these priors to build up constructive random posteriors of the Kullback-Liebler (KL) divergence of the two multivariate normal populations, which is proportional to the distance between the

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