DOCSLIB.ORG

Useful Properties of the Multivariate Normal*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Useful Properties of the Multivariate Normal* USEFUL PROPERTIES OF THE MULTIVARIATE NORMAL* 3.1. Conditionals and marginals For Bayesian analysis it is very useful to understand how to write joint, marginal, and conditional distributions for the multivariate normal. Given a vector x IR p the multivariate normal density is ∈ 1 1 − f(x) = exp (x µ)T Σ 1(x µ) . (2 π)p/ 2 Σ 1/2 −2 − − | | Now split the vector into two parts x µ q 1 x = 1 , µ = 1 , of size , x µ (p ×q) 1 2 2 − × and Σ Σ q q q (p q) Σ = 11 12 , of size . Σ Σ (p ×q) q (p ×q) −(p q) 21 22 − × − × − We now state the joint and marginal distributions x N( µ , Σ ), x N( µ , Σ ), x N( µ, Σ) , 1 ∼ 1 11 2 ∼ 2 22 ∼ and the conditional density − − x x N µ + Σ Σ 1(x µ ), Σ Σ Σ 1Σ . 1 | 2 ∼ 1 12 22 2 − 2 11 − 12 22 21 The same idea holds for other sizes of partitions. 3.2. Conjugate priors 3.2.1. Univariate normals 3.2.1.1. Fixed variance, random mean. We consider the parameter σ2 fixed so we are interested in the conjugate prior for µ: 1 1 π(µ µ , σ 2) exp (µ µ )2 , | 0 ∝ σ −2σ2 − 0 0 0 2 where µ0 and σ are hyper-parameters for the prior distribution (when we don’t 2 have informative prior knowledge we typically consider µ0 = 0 and σ large). 13 14 S. MUKHERJEE, PROBABILISTIC MACHINE LEARNING The posterior distribution for x1, ..., x n with a univariate normal likelihood and the above prior will be 2 2 −1 σ0 σ 1 n Post( µ x , ..., x ) N 2 x¯ + 2 µ , + . 1 n σ 2 σ 2 0 2 2 | ∼ ! + σ0 + σ0 σ0 σ " n n 3.2.1.2. Fixed mean, random variance. We will formulate this setting with two parameterizations of the scale parameter: (1) the variance σ2, (2) the precision 1 τ = σ2 . The two conjugate distributions are the Gamma and the inverse Gamma (really they are the same distribution, just reparameterized) α α β − − − β − IG( α, β ) : f(σ2) = (σ2) α 1 exp( β(σ2) 1), Ga( α, β ) : f(τ) = τ α 1 exp( βτ ). Γ( α) − Γ( α) − The posterior distribution of σ2 is n 1 σ2 x , ..., x IG α + , β + (x µ)2 . | 1 n ∼ 2 2 i − The posterior distribution of τ is not surprisinglyX n 1 τ x , ..., x Ga α + , β + (x µ)2 . | 1 n ∼ 2 2 i − 3.2.1.3. Random mean, random variance. We nowX put the previous priors together in what is called a Bayesian hierarchical model: iid −1 xi µ, τ N( µ, (τ) ) | ∼ − µ τ N( µ , (κ τ) 1) | ∼ 0 0 τ Ga( α, β ). ∼ For the above likelihood and priors the posterior distribution for the mean and precision is µ0κ0 + nx¯ −1 µ τ, x 1, ..., x n N , (τ(n + κ0)) | ∼ n + κ0 n 1 n (¯x x )2 τ x , ..., x Ga α + , β + (x x¯)2 + − i . | 1 n ∼ 2 2 i − n + 1 2 X 3.2.2. Multivariate normal Given a vector x IR p the multivariate normal density is ∈ 1 1 − f(x) = exp (x µ)T Σ 1(x µ) . (2 π)p/ 2 Σ 1/2 −2 − − | | We will work with the precision matrix instead of the covariance and we will consider the following Bayesian hierarchical model: iid −1 xi µ, Λ N( µ, (Λ) ) | ∼ − µ Λ N( µ , (κ Λ) 1) | ∼ 0 0 Λ Wi(Λ , n ), ∼ 0 0 the precision matrix is modeled using the Wishart distribution Λ (n−d−1) /2 exp( .5tr(Λ V −1)) f(Λ; V, n ) = | | − . 2nd/ 2 V n/ 2Γ (n/ 2) | | d USEFUL PROPERTIES OF THE MULTIVARIATE NORMAL* 15 For the above likelihood and priors the posterior distribution for the mean and precision is µ0κ0 + nx¯ −1 µ Λ, x 1, ..., x n N , (Λ( n + κ0)) | ∼ n + κ0 n 1 κ Λ x , ..., x Wi n + , Λ + Σ¯ + 0 (¯x µ )(¯x µ )T . | 1 n ∼ 0 2 0 2 κ + n − 0 − 0 0 LECTURE 4 A Bayesian approach to linear regression The main motivations behind a Bayesian formalism for inference are a coherent approach to modeling uncertainty as well as an axiomatic framework for inference. We will reformulate multivariate linear regression from a Bayesian formulation in this section. Bayesian inference involves thinking in terms of probability distributions and conditional distributions. One important idea is that of a conjugate prior. Another tool we will use extensively in this class is the multivariate normal distribution and its properties. 4.1. Conjugate priors Given a likelihood function p(x θ) and a prior π(θ) on can write the posterior as | p(x θ)π(θ) p(x, θ ) p(θ x) = | ′ ′ ′ = , | ′ p(x θ )π(θ ) d θ p(x) θ | R where p(x) is the marginal density for the data, p(x, θ ) is the joint density of the data and the parameter θ. The idea of a prior and likelihood being conjugate is that the prior and the posterior densities belong to the same family. We now state some examples to illustrate this idea. Beta, Binomial: Consider the Binomial likelihood with n (the number of trials) fixed n − f(x p, n ) = px(1 p)n x, | x − the parameter of interest (the probability of a success) is p [0 , 1]. A natural prior distribution for p is the Beta distribution which has density∈ Γ( α + β) − − π(p; α, β ) = pα 1(1 p)β 1, p (0 , 1) and α, β > 0, Γ( α)Γ( β) − ∈ 17 18 S. MUKHERJEE, PROBABILISTIC MACHINE LEARNING Γ( α+β) where Γ( α)Γ( β) is a normalization constant. Given the prior and the likelihood densities the posterior density modulo normalizing constants will take the form n Γ( α + β) − − − f(p x) px(1 p)n x pα 1(1 p)β 1, | ∝ x × Γ( α)Γ( β) − × − − − − px+α 1(1 p)n x+β 1, ∝ − which means that the posterior distribution of p is also a Beta with p x Beta( α + x, β + n x). | ∼ − Normal, Normal: Given a normal distribution with unknown mean the den- sity for the likelihood is 1 f(x θ, σ 2) exp (x θ)2 , | ∝ −2σ2 − and one can specify a normal prior 1 π(θ; θ , τ 2) exp (θ θ )2 , 0 0 ∝ −2τ 2 − 0 0 with hyper-parameters θ0 and τ0. The resulting posterior distribution will have the following density function 1 1 f(θ x) exp (x θ)2 exp (θ θ )2 , | ∝ −2σ2 − × −2τ 2 − 0 0 which after completing squares and reordering can be written as θ0 x 2 + 2 2 τ0 σ 2 1 θ x N( θ1, τ 1 ), θ 1 = 1 1 , τ 1 = 1 1 . | ∼ 2 + 2 2 + 2 τ0 σ τ0 σ 4.2. Bayesian linear regression We start with the likelihood as n 1 y βT x 2 f(Y X, β, σ 2) = exp k i − ik . | √ − 2σ2 i=1 σ 2π Y and the prior as 1 π(β) exp βT β . ∝ −2τ 2 0 The density of the posterior is n 1 y βT x 2 1 1 Post( β D) exp k i − ik exp βT β . | ∝ √ − 2σ2 × (2 π)p/ 2γ1/2 −2τ 2 #i=1 σ 2π $ 0 Y With a good bit of manipulation the above can be rewritten as a multivariate normal distribution β Y, X, σ 2 N (µ , Σ ) | ∼ p 1 1 with −2 −2 T −1 −2 T Σ1 = ( τ0 Ip + σ X X) , µ 1 = σ Σ1 X Y. Note the similarities of the above distribution to the MAP estimator. Relate the mean of the above estimator to the MAP estimator. LECTURE 4. A BAYESIAN APPROACH TO LINEAR REGRESSION 19 Predictive distribution: Given data D = (x , y ) n and and a new value x∗ { i i }i=1 one would like to estimate y∗. This can be done using the posterior and is called the posterior predictive distribution 2 2 2 2 2 f(y∗ D, x ∗, σ , τ 0 ) = f(y∗ x∗, β, σ )f(β Y, X, σ , τ 0 ) d β, | p | | ZIR where with some manipulation 2 2 2 y∗ D, x ∗, σ , τ N( µ∗, σ ∗), | 0 ∼ where 1 T 2 2 T µ∗ = Σ X Y x ∗, σ ∗ = σ + x∗ Σ x∗. σ2 1 1 .
Load more

Recommended publications
  • The Exponential Family 1 Definition
    The Exponential Family David M. Blei Columbia University November 9, 2016 The exponential family is a class of densities (Brown, 1986). It encompasses many familiar forms of likelihoods, such as the Gaussian, Poisson, multinomial, and Bernoulli. It also encompasses their conjugate priors, such as the Gamma, Dirichlet, and beta. 1 Definition A probability density in the exponential family has this form p.x / h.x/ exp >t.x/ a./ ; (1) j D f g where is the natural parameter; t.x/ are sufficient statistics; h.x/ is the “base measure;” a./ is the log normalizer. Examples of exponential family distributions include Gaussian, gamma, Poisson, Bernoulli, multinomial, Markov models. Examples of distributions that are not in this family include student-t, mixtures, and hidden Markov models. (We are considering these families as distributions of data. The latent variables are implicitly marginalized out.) The statistic t.x/ is called sufficient because the probability as a function of only depends on x through t.x/. The exponential family has fundamental connections to the world of graphical models (Wainwright and Jordan, 2008). For our purposes, we’ll use exponential 1 families as components in directed graphical models, e.g., in the mixtures of Gaussians. The log normalizer ensures that the density integrates to 1, Z a./ log h.x/ exp >t.x/ d.x/ (2) D f g This is the negative logarithm of the normalizing constant. The function h.x/ can be a source of confusion. One way to interpret h.x/ is the (unnormalized) distribution of x when 0. It might involve statistics of x that D are not in t.x/, i.e., that do not vary with the natural parameter.
    [Show full text]
  • Machine Learning Conjugate Priors and Monte Carlo Methods
    Hierarchical Bayes for Non-IID Data Conjugate Priors Monte Carlo Methods CPSC 540: Machine Learning Conjugate Priors and Monte Carlo Methods Mark Schmidt University of British Columbia Winter 2016 Hierarchical Bayes for Non-IID Data Conjugate Priors Monte Carlo Methods Admin Nothing exciting? We discussed empirical Bayes, where you optimize prior using marginal likelihood, Z argmax p(xjα; β) = argmax p(xjθ)p(θjα; β)dθ: α,β α,β θ Can be used to optimize λj, polynomial degree, RBF σi, polynomial vs. RBF, etc. We also considered hierarchical Bayes, where you put a prior on the prior, p(xjα; β)p(α; βjγ) p(α; βjx; γ) = : p(xjγ) But is the hyper-prior really needed? Hierarchical Bayes for Non-IID Data Conjugate Priors Monte Carlo Methods Last Time: Bayesian Statistics In Bayesian statistics we work with posterior over parameters, p(xjθ)p(θjα; β) p(θjx; α; β) = : p(xjα; β) We also considered hierarchical Bayes, where you put a prior on the prior, p(xjα; β)p(α; βjγ) p(α; βjx; γ) = : p(xjγ) But is the hyper-prior really needed? Hierarchical Bayes for Non-IID Data Conjugate Priors Monte Carlo Methods Last Time: Bayesian Statistics In Bayesian statistics we work with posterior over parameters, p(xjθ)p(θjα; β) p(θjx; α; β) = : p(xjα; β) We discussed empirical Bayes, where you optimize prior using marginal likelihood, Z argmax p(xjα; β) = argmax p(xjθ)p(θjα; β)dθ: α,β α,β θ Can be used to optimize λj, polynomial degree, RBF σi, polynomial vs.
    [Show full text]
  • A Skew Extension of the T-Distribution, with Applications
    J. R. Statist. Soc. B (2003) 65, Part 1, pp. 159–174 A skew extension of the t-distribution, with applications M. C. Jones The Open University, Milton Keynes, UK and M. J. Faddy University of Birmingham, UK [Received March 2000. Final revision July 2002] Summary. A tractable skew t-distribution on the real line is proposed.This includes as a special case the symmetric t-distribution, and otherwise provides skew extensions thereof.The distribu- tion is potentially useful both for modelling data and in robustness studies. Properties of the new distribution are presented. Likelihood inference for the parameters of this skew t-distribution is developed. Application is made to two data modelling examples. Keywords: Beta distribution; Likelihood inference; Robustness; Skewness; Student’s t-distribution 1. Introduction Student’s t-distribution occurs frequently in statistics. Its usual derivation and use is as the sam- pling distribution of certain test statistics under normality, but increasingly the t-distribution is being used in both frequentist and Bayesian statistics as a heavy-tailed alternative to the nor- mal distribution when robustness to possible outliers is a concern. See Lange et al. (1989) and Gelman et al. (1995) and references therein. It will often be useful to consider a further alternative to the normal or t-distribution which is both heavy tailed and skew. To this end, we propose a family of distributions which includes the symmetric t-distributions as special cases, and also includes extensions of the t-distribution, still taking values on the whole real line, with non-zero skewness. Let a>0 and b>0be parameters.
    [Show full text]
  • On the Scale Parameter of Exponential Distribution
    Review of the Air Force Academy No.2 (34)/2017 ON THE SCALE PARAMETER OF EXPONENTIAL DISTRIBUTION Anca Ileana LUPAŞ Military Technical Academy, Bucharest, Romania ([email protected]) DOI: 10.19062/1842-9238.2017.15.2.16 Abstract: Exponential distribution is one of the widely used continuous distributions in various fields for statistical applications. In this paper we study the exact and asymptotical distribution of the scale parameter for this distribution. We will also define the confidence intervals for the studied parameter as well as the fixed length confidence intervals. 1. INTRODUCTION Exponential distribution is used in various statistical applications. Therefore, we often encounter exponential distribution in applications such as: life tables, reliability studies, extreme values analysis and others. In the following paper, we focus our attention on the exact and asymptotical repartition of the exponential distribution scale parameter estimator. 2. SCALE PARAMETER ESTIMATOR OF THE EXPONENTIAL DISTRIBUTION We will consider the random variable X with the following cumulative distribution function: x F(x ; ) 1 e ( x 0 , 0) (1) where is an unknown scale parameter Using the relationships between MXXX( ) ; 22( ) ; ( ) , we obtain ()X a theoretical variation coefficient 1. This is a useful indicator, especially if MX() you have observational data which seems to be exponential and with variation coefficient of the selection closed to 1. If we consider x12, x ,... xn as a part of a population that follows an exponential distribution, then by using the maximum likelihood estimation method we obtain the following estimate n ˆ 1 xi (2) n i1 119 On the Scale Parameter of Exponential Distribution Since M ˆ , it follows that ˆ is an unbiased estimator for .
    [Show full text]
  • On a Problem Connected with Beta and Gamma Distributions by R
    ON A PROBLEM CONNECTED WITH BETA AND GAMMA DISTRIBUTIONS BY R. G. LAHA(i) 1. Introduction. The random variable X is said to have a Gamma distribution G(x;0,a)if du for x > 0, (1.1) P(X = x) = G(x;0,a) = JoT(a)" 0 for x ^ 0, where 0 > 0, a > 0. Let X and Y be two independently and identically distributed random variables each having a Gamma distribution of the form (1.1). Then it is well known [1, pp. 243-244], that the random variable W = X¡iX + Y) has a Beta distribution Biw ; a, a) given by 0 for w = 0, (1.2) PiW^w) = Biw;x,x)=\ ) u"-1il-u)'-1du for0<w<l, Ío T(a)r(a) 1 for w > 1. Now we can state the converse problem as follows : Let X and Y be two independently and identically distributed random variables having a common distribution function Fix). Suppose that W = Xj{X + Y) has a Beta distribution of the form (1.2). Then the question is whether £(x) is necessarily a Gamma distribution of the form (1.1). This problem was posed by Mauldon in [9]. He also showed that the converse problem is not true in general and constructed an example of a non-Gamma distribution with this property using the solution of an integral equation which was studied by Goodspeed in [2]. In the present paper we carry out a systematic investigation of this problem. In §2, we derive some general properties possessed by this class of distribution laws Fix).
    [Show full text]
  • Polynomial Singular Value Decompositions of a Family of Source-Channel Models
    Polynomial Singular Value Decompositions of a Family of Source-Channel Models The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Makur, Anuran and Lizhong Zheng. "Polynomial Singular Value Decompositions of a Family of Source-Channel Models." IEEE Transactions on Information Theory 63, 12 (December 2017): 7716 - 7728. © 2017 IEEE As Published http://dx.doi.org/10.1109/tit.2017.2760626 Publisher Institute of Electrical and Electronics Engineers (IEEE) Version Author's final manuscript Citable link https://hdl.handle.net/1721.1/131019 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ IEEE TRANSACTIONS ON INFORMATION THEORY 1 Polynomial Singular Value Decompositions of a Family of Source-Channel Models Anuran Makur, Student Member, IEEE, and Lizhong Zheng, Fellow, IEEE Abstract—In this paper, we show that the conditional expec- interested in a particular subclass of one-parameter exponential tation operators corresponding to a family of source-channel families known as natural exponential families with quadratic models, defined by natural exponential families with quadratic variance functions (NEFQVF). So, we define natural exponen- variance functions and their conjugate priors, have orthonor- mal polynomials as singular vectors. These models include the tial families next. Gaussian channel with Gaussian source, the Poisson channel Definition 1 (Natural Exponential Family). Given a measur- with gamma source, and the binomial channel with beta source. To derive the singular vectors of these models, we prove and able space (Y; B(Y)) with a σ-finite measure µ, where Y ⊆ R employ the equivalent condition that their conditional moments and B(Y) denotes the Borel σ-algebra on Y, the parametrized are strictly degree preserving polynomials.
    [Show full text]
  • A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess
    S S symmetry Article A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess Guillermo Martínez-Flórez 1, Víctor Leiva 2,* , Emilio Gómez-Déniz 3 and Carolina Marchant 4 1 Departamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Montería 14014, Colombia; [email protected] 2 Escuela de Ingeniería Industrial, Pontificia Universidad Católica de Valparaíso, 2362807 Valparaíso, Chile 3 Facultad de Economía, Empresa y Turismo, Universidad de Las Palmas de Gran Canaria and TIDES Institute, 35001 Canarias, Spain; [email protected] 4 Facultad de Ciencias Básicas, Universidad Católica del Maule, 3466706 Talca, Chile; [email protected] * Correspondence: [email protected] or [email protected] Received: 30 June 2020; Accepted: 19 August 2020; Published: 1 September 2020 Abstract: In this paper, we consider skew-normal distributions for constructing new a distribution which allows us to model proportions and rates with zero/one inflation as an alternative to the inflated beta distributions. The new distribution is a mixture between a Bernoulli distribution for explaining the zero/one excess and a censored skew-normal distribution for the continuous variable. The maximum likelihood method is used for parameter estimation. Observed and expected Fisher information matrices are derived to conduct likelihood-based inference in this new type skew-normal distribution. Given the flexibility of the new distributions, we are able to show, in real data scenarios, the good performance of our proposal. Keywords: beta distribution; centered skew-normal distribution; maximum-likelihood methods; Monte Carlo simulations; proportions; R software; rates; zero/one inflated data 1.
    [Show full text]
  • Generalized Inferences for the Common Scale Parameter of Several Pareto Populations∗
    InternationalInternational JournalJournal of of Statistics Statistics and and Management Management System System Vol.Vol. 4 No. 12 (January-June,(July-December, 2019) 2019) International Journal of Statistics and Management System, 2010, Vol. 5, No. 1–2, pp. 118–126. c 2010 Serials Publications Generalized inferences for the common scale parameter of several Pareto populations∗ Sumith Gunasekera†and Malwane M. A. Ananda‡ Received: 13th August 2018 Revised: 24th December 2018 Accepted: 10th March 2019 Abstract A problem of interest in this article is statistical inferences concerning the com- mon scale parameter of several Pareto distributions. Using the generalized p-value approach, exact confidence intervals and the exact tests for testing the common scale parameter are given. Examples are given in order to illustrate our procedures. A limited simulation study is given to demonstrate the performance of the proposed procedures. 1 Introduction In this paper, we consider k (k ≥ 2) independent Pareto distributions with an unknown common scale parameter θ (sometimes referred to as the “location pa- rameter” and also as the “truncation parameter”) and unknown possibly unequal shape parameters αi’s (i = 1, 2, ..., k). Using the generalized variable approach (Tsui and Weer- ahandi [8]), we construct an exact test for testing θ. Furthermore, using the generalized confidence interval (Weerahandi [11]), we construct an exact confidence interval for θ as well. A limited simulation study was carried out to compare the performance of these gen- eralized procedures with the approximate procedures based on the large sample method as well as with the other test procedures based on the combination of p-values.
    [Show full text]
  • Lecture 2 — September 24 2.1 Recap 2.2 Exponential Families
    STATS 300A: Theory of Statistics Fall 2015 Lecture 2 | September 24 Lecturer: Lester Mackey Scribe: Stephen Bates and Andy Tsao 2.1 Recap Last time, we set out on a quest to develop optimal inference procedures and, along the way, encountered an important pair of assertions: not all data is relevant, and irrelevant data can only increase risk and hence impair performance. This led us to introduce a notion of lossless data compression (sufficiency): T is sufficient for P with X ∼ Pθ 2 P if X j T (X) is independent of θ. How far can we take this idea? At what point does compression impair performance? These are questions of optimal data reduction. While we will develop general answers to these questions in this lecture and the next, we can often say much more in the context of specific modeling choices. With this in mind, let's consider an especially important class of models known as the exponential family models. 2.2 Exponential Families Definition 1. The model fPθ : θ 2 Ωg forms an s-dimensional exponential family if each Pθ has density of the form: s ! X p(x; θ) = exp ηi(θ)Ti(x) − B(θ) h(x) i=1 • ηi(θ) 2 R are called the natural parameters. • Ti(x) 2 R are its sufficient statistics, which follows from NFFC. • B(θ) is the log-partition function because it is the logarithm of a normalization factor: s ! ! Z X B(θ) = log exp ηi(θ)Ti(x) h(x)dµ(x) 2 R i=1 • h(x) 2 R: base measure.
    [Show full text]
  • Procedures for Estimation of Weibull Parameters James W
    United States Department of Agriculture Procedures for Estimation of Weibull Parameters James W. Evans David E. Kretschmann David W. Green Forest Forest Products General Technical Report February Service Laboratory FPL–GTR–264 2019 Abstract Contents The primary purpose of this publication is to provide an 1 Introduction .................................................................. 1 overview of the information in the statistical literature on 2 Background .................................................................. 1 the different methods developed for fitting a Weibull distribution to an uncensored set of data and on any 3 Estimation Procedures .................................................. 1 comparisons between methods that have been studied in the 4 Historical Comparisons of Individual statistics literature. This should help the person using a Estimator Types ........................................................ 8 Weibull distribution to represent a data set realize some advantages and disadvantages of some basic methods. It 5 Other Methods of Estimating Parameters of should also help both in evaluating other studies using the Weibull Distribution .......................................... 11 different methods of Weibull parameter estimation and in 6 Discussion .................................................................. 12 discussions on American Society for Testing and Materials Standard D5457, which appears to allow a choice for the 7 Conclusion ................................................................
    [Show full text]
  • Volatility Modeling Using the Student's T Distribution
    Volatility Modeling Using the Student’s t Distribution Maria S. Heracleous Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Economics Aris Spanos, Chair Richard Ashley Raman Kumar Anya McGuirk Dennis Yang August 29, 2003 Blacksburg, Virginia Keywords: Student’s t Distribution, Multivariate GARCH, VAR, Exchange Rates Copyright 2003, Maria S. Heracleous Volatility Modeling Using the Student’s t Distribution Maria S. Heracleous (ABSTRACT) Over the last twenty years or so the Dynamic Volatility literature has produced a wealth of uni- variateandmultivariateGARCHtypemodels.Whiletheunivariatemodelshavebeenrelatively successful in empirical studies, they suffer from a number of weaknesses, such as unverifiable param- eter restrictions, existence of moment conditions and the retention of Normality. These problems are naturally more acute in the multivariate GARCH type models, which in addition have the problem of overparameterization. This dissertation uses the Student’s t distribution and follows the Probabilistic Reduction (PR) methodology to modify and extend the univariate and multivariate volatility models viewed as alternative to the GARCH models. Its most important advantage is that it gives rise to internally consistent statistical models that do not require ad hoc parameter restrictions unlike the GARCH formulations. Chapters 1 and 2 provide an overview of my dissertation and recent developments in the volatil- ity literature. In Chapter 3 we provide an empirical illustration of the PR approach for modeling univariate volatility. Estimation results suggest that the Student’s t AR model is a parsimonious and statistically adequate representation of exchange rate returns and Dow Jones returns data.
    [Show full text]
  • Determination of the Weibull Parameters from the Mean Value and the Coefficient of Variation of the Measured Strength for Brittle Ceramics
    Journal of Advanced Ceramics 2017, 6(2): 149–156 ISSN 2226-4108 DOI: 10.1007/s40145-017-0227-3 CN 10-1154/TQ Research Article Determination of the Weibull parameters from the mean value and the coefficient of variation of the measured strength for brittle ceramics Bin DENGa, Danyu JIANGb,* aDepartment of the Prosthodontics, Chinese PLA General Hospital, Beijing 100853, China bAnalysis and Testing Center for Inorganic Materials, State Key Laboratory of High Performance Ceramics and Superfine Microstructure, Shanghai Institute of Ceramics, Shanghai 200050, China Received: January 11, 2017; Accepted: April 7, 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract: Accurate estimation of Weibull parameters is an important issue for the characterization of the strength variability of brittle ceramics with Weibull statistics. In this paper, a simple method was established for the determination of the Weibull parameters, Weibull modulus m and scale parameter 0 , based on Monte Carlo simulation. It was shown that an unbiased estimation for Weibull modulus can be yielded directly from the coefficient of variation of the considered strength sample. Furthermore, using the yielded Weibull estimator and the mean value of the strength in the considered sample, the scale parameter 0 can also be estimated accurately. Keywords: Weibull distribution; Weibull parameters; strength variability; unbiased estimation; coefficient of variation 1 Introduction likelihood (ML) method. Many studies [9–18] based on Monte Carlo simulation have shown that each of these Weibull statistics [1] has been widely employed to methods has its benefits and drawbacks. model the variability in the fracture strength of brittle According to the theory of statistics, the expected ceramics [2–8].
    [Show full text]