Alpha-Skew Generalized T Distribution
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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. -
Supplementary Materials
Int. J. Environ. Res. Public Health 2018, 15, 2042 1 of 4 Supplementary Materials S1) Re-parametrization of log-Laplace distribution The three-parameter log-Laplace distribution, LL(δ ,,ab) of relative risk, μ , is given by the probability density function b−1 μ ,0<<μ δ 1 ab δ f ()μ = . δ ab+ a+1 δ , μ ≥ δ μ 1−τ τ Let b = and a = . Hence, the probability density function of LL( μ ,,τσ) can be re- σ σ τ written as b μ ,0<<=μ δ exp(μ ) 1 ab δ f ()μ = ya+ b δ a μδ≥= μ ,exp() μ 1 ab exp(b (log(μμ )−< )), log( μμ ) = μ ab+ exp(a (μμ−≥ log( ))), log( μμ ) 1 ab exp(−−b| log(μμ ) |), log( μμ ) < = μ ab+ exp(−−a|μμ log( ) |), log( μμ ) ≥ μμ− −−τ | log( )τ | μμ < exp (1 ) , log( ) τ 1(1)ττ− σ = μσ μμ− −≥τμμ| log( )τ | exp , log( )τ . σ S2) Bayesian quantile estimation Let Yi be the continuous response variable i and Xi be the corresponding vector of covariates with the first element equals one. At a given quantile levelτ ∈(0,1) , the τ th conditional quantile T i i = β τ QY(|X ) τ of y given X is then QYτ (|iiXX ) ii ()where τ iiis the th conditional quantile and β τ i ()is a vector of quantile coefficients. Contrary to the mean regression generally aiming to minimize the squared loss function, the quantile regression links to a special class of check or loss τ βˆ τ function. The th conditional quantile can be estimated by any solution, i (), such that ββˆ τ =−ρ T τ ρ =−<τ iiii() argmini τ (Y X ()) where τ ()zz ( Iz ( 0))is the quantile loss function given in (1) and I(.) is the indicator function. -
Exploring Bimodality in Introductory Computer Science Performance Distributions
EURASIA Journal of Mathematics, Science and Technology Education, 2018, 14(10), em1591 ISSN:1305-8223 (online) OPEN ACCESS Research Paper https://doi.org/10.29333/ejmste/93190 Exploring Bimodality in Introductory Computer Science Performance Distributions Ram B Basnet 1, Lori K Payne 1, Tenzin Doleck 2*, David John Lemay 2, Paul Bazelais 2 1 Colorado Mesa University, Grand Junction, Colorado, USA 2 McGill University, Montreal, Quebec, CANADA Received 7 April 2018 ▪ Revised 19 June 2018 ▪ Accepted 21 June 2018 ABSTRACT This study examines student performance distributions evidence bimodality, or whether there are two distinct populations in three introductory computer science courses grades at a four-year southwestern university in the United States for the period 2014-2017. Results suggest that computer science course grades are not bimodal. These findings counter the double hump assertion and suggest that proper course sequencing can address the needs of students with varying levels of prior knowledge and obviate the double-hump phenomenon. Studying performance helps to improve delivery of introductory computer science courses by ensuring that courses are aligned with student needs and address preconceptions and prior knowledge and experience. Keywords: computer science performance, coding, programming, double hump, grade distribution, bimodal distribution, unimodal distribution BACKGROUND In recent years, there has been a resurgence of interest in the practice of coding (Kafai & Burke, 2013), with many pushing for making it a core competency for students (Lye & Koh, 2014). There are inherent challenges in learning to code evidenced by high failure and dropout rates in programming courses (Ma, Ferguson, Roper, & Wood, 2011; (Qian & Lehman, 2017; Robins, 2010). -
A Study of Non-Central Skew T Distributions and Their Applications in Data Analysis and Change Point Detection
A STUDY OF NON-CENTRAL SKEW T DISTRIBUTIONS AND THEIR APPLICATIONS IN DATA ANALYSIS AND CHANGE POINT DETECTION Abeer M. Hasan A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 2013 Committee: Arjun K. Gupta, Co-advisor Wei Ning, Advisor Mark Earley, Graduate Faculty Representative Junfeng Shang. Copyright c August 2013 Abeer M. Hasan All rights reserved iii ABSTRACT Arjun K. Gupta, Co-advisor Wei Ning, Advisor Over the past three decades there has been a growing interest in searching for distribution families that are suitable to analyze skewed data with excess kurtosis. The search started by numerous papers on the skew normal distribution. Multivariate t distributions started to catch attention shortly after the development of the multivariate skew normal distribution. Many researchers proposed alternative methods to generalize the univariate t distribution to the multivariate case. Recently, skew t distribution started to become popular in research. Skew t distributions provide more flexibility and better ability to accommodate long-tailed data than skew normal distributions. In this dissertation, a new non-central skew t distribution is studied and its theoretical properties are explored. Applications of the proposed non-central skew t distribution in data analysis and model comparisons are studied. An extension of our distribution to the multivariate case is presented and properties of the multivariate non-central skew t distri- bution are discussed. We also discuss the distribution of quadratic forms of the non-central skew t distribution. In the last chapter, the change point problem of the non-central skew t distribution is discussed under different settings. -
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. -
Some Properties of the Log-Laplace Distribution* V
SOME PROPERTIES OF THE LOG-LAPLACE DISTRIBUTION* V, R. R. Uppuluri Mathematics and Statistics Research Department Computer Sciences Division Union Carbide Corporation, Nuclear Division Oak Ridge, Tennessee 37830 B? accSpBficS of iMS artlclS* W8 tfcHteHef of Recipient acKnowta*g«9 th« U.S. Government's right to retain a non - excluslva, royalty - frw Jlcqns* In JMWl JS SOX 50j.yrt«ltt BfiXSflng IM -DISCLAIMER . This book w« prtMtM n in nonim nl mrk moniond by m vncy of At UnllM Sum WM»> ih« UWIrt SUM Oow.n™»i nor an, qprcy IMrw, w «v ol Ifmf mptovMi. mkn «iy w»ti«y, nptoi Of ImolM. « »u~i •ny U»l liability or tBpomlbllliy lof Ihe nary, ramoloweB. nr unhtlm at mi Womwlon, appmlui, product, or pram dliclowl. or rwromti that lu u« acuU r»i MMngt prhiltly ownad rljM<- flihmot Mi to any malic lomnwclal product. HOCOI. 0. »tvlc by ttada naw. tradtimr», mareittciunr. or otktmte, taa not rucaaainy coralltun or Imply In anoonamnt, nKommendailon. o» Iworlnj by a, Unilid SUM Oowrnment or m atncy limml. Th, Amnd ooMora ol «iihon ncrtwd fmtin dn not tauaaruv iu» or nlfcn thoaot tt» Unltad Sum GiMmrmt or any agmv trmot. •Research sponsored by the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy, under contract W-74Q5-eng-26 with the Union Carbide Corporation. DISTRIBii.il i yf T!it3 OGCUHEiU IS UHLIKITEOr SOME PROPERTIES OF THE LOS-LAPLACE DISTRIBUTION V. R. R. Uppuluri ABSTRACT A random variable Y is said to have the Laplace distribution or the double exponential distribution whenever its probability density function is given by X exp(-A|y|), where -» < y < ~ and X > 0. -
Recombining Partitions Via Unimodality Tests
Working Paper 13-07 Departamento de Estadística Statistics and Econometrics Series (06) Universidad Carlos III de Madrid March 2013 Calle Madrid, 126 28903 Getafe (Spain) Fax (34) 91 624-98-48 RECOMBINING PARTITIONS VIA UNIMODALITY TESTS Adolfo Álvarez (1) , Daniel Peña(2) Abstract In this article we propose a recombination procedure for previously split data. It is based on the study of modes in the density of the data, since departing from unimodality can be a sign of the presence of clusters. We develop an algorithm that integrates a splitting process inherited from the SAR algorithm (Peña et al., 2004) with unimodality tests such as the dip test proposed by Hartigan and Hartigan (1985), and finally, we use a network configuration to visualize the results. We show that this can be a useful tool to detect heterogeneity in the data, but limited to univariate data because of the nature of the dip test. In a second stage we discuss the use of multivariate mode detection tests to avoid dimensionality reduction techniques such as projecting multivariate data into one dimension. The results of the application of multivariate unimodality tests show that is possible to detect the cluster structure of the data, although more research can be oriented to estimate the proper fine-tuning of some parameters of the test for a given dataset or distribution. Keywords: Cluster analysis, unimodality, dip test. (1) Álvarez, Adolfo, Departamento de Estadística, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid), Spain, e-mail: [email protected] . (2) Peña, Daniel, Departamento de Estadística, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid), Spain, e-mail: [email protected] . -
Approximating the Distribution of the Product of Two Normally Distributed Random Variables
S S symmetry Article Approximating the Distribution of the Product of Two Normally Distributed Random Variables Antonio Seijas-Macías 1,2 , Amílcar Oliveira 2,3 , Teresa A. Oliveira 2,3 and Víctor Leiva 4,* 1 Departamento de Economía, Universidade da Coruña, 15071 A Coruña, Spain; [email protected] 2 CEAUL, Faculdade de Ciências, Universidade de Lisboa, 1649-014 Lisboa, Portugal; [email protected] (A.O.); [email protected] (T.A.O.) 3 Departamento de Ciências e Tecnologia, Universidade Aberta, 1269-001 Lisboa, Portugal 4 Escuela de Ingeniería Industrial, Pontificia Universidad Católica de Valparaíso, Valparaíso 2362807, Chile * Correspondence: [email protected] or [email protected] Received: 21 June 2020; Accepted: 18 July 2020; Published: 22 July 2020 Abstract: The distribution of the product of two normally distributed random variables has been an open problem from the early years in the XXth century. First approaches tried to determinate the mathematical and statistical properties of the distribution of such a product using different types of functions. Recently, an improvement in computational techniques has performed new approaches for calculating related integrals by using numerical integration. Another approach is to adopt any other distribution to approximate the probability density function of this product. The skew-normal distribution is a generalization of the normal distribution which considers skewness making it flexible. In this work, we approximate the distribution of the product of two normally distributed random variables using a type of skew-normal distribution. The influence of the parameters of the two normal distributions on the approximation is explored. When one of the normally distributed variables has an inverse coefficient of variation greater than one, our approximation performs better than when both normally distributed variables have inverse coefficients of variation less than one. -
UNIMODALITY and the DIP STATISTIC; HANDOUT I 1. Unimodality Definition. a Probability Density F Defined on the Real Line R Is C
UNIMODALITY AND THE DIP STATISTIC; HANDOUT I 1. Unimodality Definition. A probability density f defined on the real line R is called unimodal if the following hold: (i) f is nondecreasing on the half-line ( ,b) for some real b; (ii) f is nonincreasing on the half-line (−∞a, + ) for some real a; (iii) For the largest possible b in (i), which∞ exists, and the smallest possible a in (ii), which also exists, we have a b. ≤ Here a largest possible b exists since if f is nondecreasing on ( ,bk) −∞ for all k and bk b then f is also nondecreasing on ( ,b). Also b < + since f is↑ a probability density. Likewise a smallest−∞ possible a exists∞ and is finite. In the definition of unimodal density, if a = b it is called the mode of f. It is possible that f is undefined or + there, or that f(x) + as x a and/or as x a. ∞ ↑ ∞ If a<b↑ then f is constant↓ on the interval (a,b) and we can take it to be constant on [a,b], which will be called the interval of modes of f. Any x [a,b] will be called a mode of f. When a = b the interval of modes∈ reduces to the singleton a . { } Examples. Each normal distribution N(µ,σ2) has a unimodal density with a unique mode at µ. Each uniform U[a,b] distribution has a unimodal density with an interval of modes equal to [a,b]. Consider a−1 −x the gamma density γa(x) = x e /Γ(a) for x > 0 and 0 for x 0, where the shape parameter a > 0. -
Copulaesque Versions of the Skew-Normal Andskew-Student
S S symmetry Article Copulaesque Versions of the Skew-Normal and Skew-Student Distributions Christopher Adcock Sheffield University Management School, University College Dublin, Sheffield S10 1FL, UK; c.j.adcock@sheffield.ac.uk Abstract: A recent paper presents an extension of the skew-normal distribution which is a copula. Under this model, the standardized marginal distributions are standard normal. The copula itself depends on the familiar skewing construction based on the normal distribution function. This paper is concerned with two topics. First, the paper presents a number of extensions of the skew-normal copula. Notably these include a case in which the standardized marginal distributions are Student’s t, with different degrees of freedom allowed for each margin. In this case the skewing function need not be the distribution function for Student’s t, but can depend on certain of the special functions. Secondly, several multivariate versions of the skew-normal copula model are presented. The paper contains several illustrative examples. Keywords: copula; multivariate distributions; skew-normal distributions; skew-Student distributions JEL Classification: C18; G01; G10; G12 Citation: Adcock, C.J. Copulaesque 1. Introduction Versions of the Skew-Normal and The recent paper by [1] presents an extension of the skew-normal distribution which Skew-Student Distributions. has subsequently been referred to as a copula. Under this model, the standardized marginal Symmetry 2021, 13, 815. https:// distributions are standard normal and some of the conditional distributions are skew- doi.org/10.3390/sym13050815 normal.The skew-normal distribution itself was introduced in two landmark papers [2] and [3]. These papers have led to a very substantial research effort by numerous authors Academic Editor: Nicola Maria over the last thirty five years. -
Machine Learning - HT 2016 3
Machine learning - HT 2016 3. Maximum Likelihood Varun Kanade University of Oxford January 27, 2016 Outline Probabilistic Framework � Formulate linear regression in the language of probability � Introduce the maximum likelihood estimate � Relation to least squares estimate Basics of Probability � Univariate and multivariate normal distribution � Laplace distribution � Likelihood, Entropy and its relation to learning 1 Univariate Gaussian (Normal) Distribution The univariate normal distribution is defined by the following density function (x µ)2 1 − 2 p(x) = e− 2σ2 X (µ,σ ) √2πσ ∼N Hereµ is the mean andσ 2 is the variance. 2 Sampling from a Gaussian distribution Sampling fromX (µ,σ 2) ∼N X µ By settingY= − , sample fromY (0, 1) σ ∼N Cumulative distribution function x 1 t2 Φ(x) = e− 2 dt √2π �−∞ 3 Covariance and Correlation For random variableX andY the covariance measures how the random variable change jointly. cov(X, Y)=E[(X E[X])(Y E[Y ])] − − Covariance depends on the scale of the random variable. The (Pearson) correlation coefficient normalizes the covariance to give a value between 1 and +1. − cov(X, Y) corr(X, Y)= , σX σY 2 2 2 2 whereσ X =E[(X E[X]) ] andσ Y =E[(Y E[Y]) ]. − − 4 Multivariate Gaussian Distribution Supposex is an-dimensional random vector. The covariance matrix consists of all pariwise covariances. var(X ) cov(X ,X ) cov(X ,X ) 1 1 2 ··· 1 n cov(X2,X1) var(X2) cov(X 2,Xn) T ··· cov(x) =E (x E[x])(x E[x]) = . . − − . � � . cov(Xn,X1) cov(Xn,X2) var(X n,Xn) ··· Ifµ=E[x] andΣ = cov[x], the multivariate normal is defined by the density 1 1 T 1 (µ,Σ) = exp (x µ) Σ− (x µ) N (2π)n/2 Σ 1/2 − 2 − − | | � � 5 Bivariate Gaussian Distribution 2 2 SupposeX 1 (µ 1,σ ) andX 2 (µ 2,σ ) ∼N 1 ∼N 2 What is the joint probability distributionp(x 1, x2)? 6 Suppose you are given three independent samples: x1 = 1, x2 = 2.7, x3 = 3. -
Field Guide to Continuous Probability Distributions
Field Guide to Continuous Probability Distributions Gavin E. Crooks v 1.0.0 2019 G. E. Crooks – Field Guide to Probability Distributions v 1.0.0 Copyright © 2010-2019 Gavin E. Crooks ISBN: 978-1-7339381-0-5 http://threeplusone.com/fieldguide Berkeley Institute for Theoretical Sciences (BITS) typeset on 2019-04-10 with XeTeX version 0.99999 fonts: Trump Mediaeval (text), Euler (math) 271828182845904 2 G. E. Crooks – Field Guide to Probability Distributions Preface: The search for GUD A common problem is that of describing the probability distribution of a single, continuous variable. A few distributions, such as the normal and exponential, were discovered in the 1800’s or earlier. But about a century ago the great statistician, Karl Pearson, realized that the known probabil- ity distributions were not sufficient to handle all of the phenomena then under investigation, and set out to create new distributions with useful properties. During the 20th century this process continued with abandon and a vast menagerie of distinct mathematical forms were discovered and invented, investigated, analyzed, rediscovered and renamed, all for the purpose of de- scribing the probability of some interesting variable. There are hundreds of named distributions and synonyms in current usage. The apparent diver- sity is unending and disorienting. Fortunately, the situation is less confused than it might at first appear. Most common, continuous, univariate, unimodal distributions can be orga- nized into a small number of distinct families, which are all special cases of a single Grand Unified Distribution. This compendium details these hun- dred or so simple distributions, their properties and their interrelations.