Probability Distribution Relationships
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Arxiv:2004.02679V2 [Math.OA] 17 Jul 2020
THE FREE TANGENT LAW WIKTOR EJSMONT AND FRANZ LEHNER Abstract. Nevanlinna-Herglotz functions play a fundamental role for the study of infinitely divisible distributions in free probability [11]. In the present paper we study the role of the tangent function, which is a fundamental Herglotz-Nevanlinna function [28, 23, 54], and related functions in free probability. To be specific, we show that the function tan z 1 ´ x tan z of Carlitz and Scoville [17, (1.6)] describes the limit distribution of sums of free commutators and anticommutators and thus the free cumulants are given by the Euler zigzag numbers. 1. Introduction Nevanlinna or Herglotz functions are functions analytic in the upper half plane having non- negative imaginary part. This class has been thoroughly studied during the last century and has proven very useful in many applications. One of the fundamental examples of Nevanlinna functions is the tangent function, see [6, 28, 23, 54]. On the other hand it was shown by by Bercovici and Voiculescu [11] that Nevanlinna functions characterize freely infinitely divisible distributions. Such distributions naturally appear in free limit theorems and in the present paper we show that the tangent function appears in a limit theorem for weighted sums of free commutators and anticommutators. More precisely, the family of functions tan z 1 x tan z ´ arises, which was studied by Carlitz and Scoville [17, (1.6)] in connection with the combinorics of tangent numbers; in particular we recover the tangent function for x 0. In recent years a number of papers have investigated limit theorems for“ the free convolution of probability measures defined by Voiculescu [58, 59, 56]. -
Power Comparisons of the Rician and Gaussian Random Fields Tests for Detecting Signal from Functional Magnetic Resonance Images Hasni Idayu Binti Saidi
University of Northern Colorado Scholarship & Creative Works @ Digital UNC Dissertations Student Research 8-2018 Power Comparisons of the Rician and Gaussian Random Fields Tests for Detecting Signal from Functional Magnetic Resonance Images Hasni Idayu Binti Saidi Follow this and additional works at: https://digscholarship.unco.edu/dissertations Recommended Citation Saidi, Hasni Idayu Binti, "Power Comparisons of the Rician and Gaussian Random Fields Tests for Detecting Signal from Functional Magnetic Resonance Images" (2018). Dissertations. 507. https://digscholarship.unco.edu/dissertations/507 This Text is brought to you for free and open access by the Student Research at Scholarship & Creative Works @ Digital UNC. It has been accepted for inclusion in Dissertations by an authorized administrator of Scholarship & Creative Works @ Digital UNC. For more information, please contact [email protected]. ©2018 HASNI IDAYU BINTI SAIDI ALL RIGHTS RESERVED UNIVERSITY OF NORTHERN COLORADO Greeley, Colorado The Graduate School POWER COMPARISONS OF THE RICIAN AND GAUSSIAN RANDOM FIELDS TESTS FOR DETECTING SIGNAL FROM FUNCTIONAL MAGNETIC RESONANCE IMAGES A Dissertation Submitted in Partial Fulfillment of the Requirement for the Degree of Doctor of Philosophy Hasni Idayu Binti Saidi College of Education and Behavioral Sciences Department of Applied Statistics and Research Methods August 2018 This dissertation by: Hasni Idayu Binti Saidi Entitled: Power Comparisons of the Rician and Gaussian Random Fields Tests for Detecting Signal from Functional Magnetic Resonance Images has been approved as meeting the requirement for the Degree of Doctor of Philosophy in College of Education and Behavioral Sciences in Department of Applied Statistics and Research Methods Accepted by the Doctoral Committee Khalil Shafie Holighi, Ph.D., Research Advisor Trent Lalonde, Ph.D., Committee Member Jay Schaffer, Ph.D., Committee Member Heng-Yu Ku, Ph.D., Faculty Representative Date of Dissertation Defense Accepted by the Graduate School Linda L. -
Modelling Censored Losses Using Splicing: a Global Fit Strategy With
Modelling censored losses using splicing: a global fit strategy with mixed Erlang and extreme value distributions Tom Reynkens∗a, Roel Verbelenb, Jan Beirlanta,c, and Katrien Antoniob,d aLStat and LRisk, Department of Mathematics, KU Leuven. Celestijnenlaan 200B, 3001 Leuven, Belgium. bLStat and LRisk, Faculty of Economics and Business, KU Leuven. Naamsestraat 69, 3000 Leuven, Belgium. cDepartment of Mathematical Statistics and Actuarial Science, University of the Free State. P.O. Box 339, Bloemfontein 9300, South Africa. dFaculty of Economics and Business, University of Amsterdam. Roetersstraat 11, 1018 WB Amsterdam, The Netherlands. August 14, 2017 Abstract In risk analysis, a global fit that appropriately captures the body and the tail of the distribution of losses is essential. Modelling the whole range of the losses using a standard distribution is usually very hard and often impossible due to the specific characteristics of the body and the tail of the loss distribution. A possible solution is to combine two distributions in a splicing model: a light-tailed distribution for the body which covers light and moderate losses, and a heavy-tailed distribution for the tail to capture large losses. We propose a splicing model with a mixed Erlang (ME) distribution for the body and a Pareto distribution for the tail. This combines the arXiv:1608.01566v4 [stat.ME] 11 Aug 2017 flexibility of the ME distribution with the ability of the Pareto distribution to model extreme values. We extend our splicing approach for censored and/or truncated data. Relevant examples of such data can be found in financial risk analysis. We illustrate the flexibility of this splicing model using practical examples from risk measurement. -
Moments of the Product and Ratio of Two Correlated Chi-Square Variables
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector Stat Papers (2009) 50:581–592 DOI 10.1007/s00362-007-0105-0 REGULAR ARTICLE Moments of the product and ratio of two correlated chi-square variables Anwar H. Joarder Received: 2 June 2006 / Revised: 8 October 2007 / Published online: 20 November 2007 © The Author(s) 2007 Abstract The exact probability density function of a bivariate chi-square distribu- tion with two correlated components is derived. Some moments of the product and ratio of two correlated chi-square random variables have been derived. The ratio of the two correlated chi-square variables is used to compare variability. One such applica- tion is referred to. Another application is pinpointed in connection with the distribution of correlation coefficient based on a bivariate t distribution. Keywords Bivariate chi-square distribution · Moments · Product of correlated chi-square variables · Ratio of correlated chi-square variables Mathematics Subject Classification (2000) 62E15 · 60E05 · 60E10 1 Introduction Fisher (1915) derived the distribution of mean-centered sum of squares and sum of products in order to study the distribution of correlation coefficient from a bivariate nor- mal sample. Let X1, X2,...,X N (N > 2) be two-dimensional independent random vectors where X j = (X1 j , X2 j ) , j = 1, 2,...,N is distributed as a bivariate normal distribution denoted by N2(θ, ) with θ = (θ1,θ2) and a 2 × 2 covariance matrix = (σik), i = 1, 2; k = 1, 2. The sample mean-centered sums of squares and sum of products are given by a = N (X − X¯ )2 = mS2, m = N − 1,(i = 1, 2) ii j=1 ij i i = N ( − ¯ )( − ¯ ) = and a12 j=1 X1 j X1 X2 j X2 mRS1 S2, respectively. -
ACTS 4304 FORMULA SUMMARY Lesson 1: Basic Probability Summary of Probability Concepts Probability Functions
ACTS 4304 FORMULA SUMMARY Lesson 1: Basic Probability Summary of Probability Concepts Probability Functions F (x) = P r(X ≤ x) S(x) = 1 − F (x) dF (x) f(x) = dx H(x) = − ln S(x) dH(x) f(x) h(x) = = dx S(x) Functions of random variables Z 1 Expected Value E[g(x)] = g(x)f(x)dx −∞ 0 n n-th raw moment µn = E[X ] n n-th central moment µn = E[(X − µ) ] Variance σ2 = E[(X − µ)2] = E[X2] − µ2 µ µ0 − 3µ0 µ + 2µ3 Skewness γ = 3 = 3 2 1 σ3 σ3 µ µ0 − 4µ0 µ + 6µ0 µ2 − 3µ4 Kurtosis γ = 4 = 4 3 2 2 σ4 σ4 Moment generating function M(t) = E[etX ] Probability generating function P (z) = E[zX ] More concepts • Standard deviation (σ) is positive square root of variance • Coefficient of variation is CV = σ/µ • 100p-th percentile π is any point satisfying F (π−) ≤ p and F (π) ≥ p. If F is continuous, it is the unique point satisfying F (π) = p • Median is 50-th percentile; n-th quartile is 25n-th percentile • Mode is x which maximizes f(x) (n) n (n) • MX (0) = E[X ], where M is the n-th derivative (n) PX (0) • n! = P r(X = n) (n) • PX (1) is the n-th factorial moment of X. Bayes' Theorem P r(BjA)P r(A) P r(AjB) = P r(B) fY (yjx)fX (x) fX (xjy) = fY (y) Law of total probability 2 If Bi is a set of exhaustive (in other words, P r([iBi) = 1) and mutually exclusive (in other words P r(Bi \ Bj) = 0 for i 6= j) events, then for any event A, X X P r(A) = P r(A \ Bi) = P r(Bi)P r(AjBi) i i Correspondingly, for continuous distributions, Z P r(A) = P r(Ajx)f(x)dx Conditional Expectation Formula EX [X] = EY [EX [XjY ]] 3 Lesson 2: Parametric Distributions Forms of probability -
A Multivariate Student's T-Distribution
Open Journal of Statistics, 2016, 6, 443-450 Published Online June 2016 in SciRes. http://www.scirp.org/journal/ojs http://dx.doi.org/10.4236/ojs.2016.63040 A Multivariate Student’s t-Distribution Daniel T. Cassidy Department of Engineering Physics, McMaster University, Hamilton, ON, Canada Received 29 March 2016; accepted 14 June 2016; published 17 June 2016 Copyright © 2016 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract A multivariate Student’s t-distribution is derived by analogy to the derivation of a multivariate normal (Gaussian) probability density function. This multivariate Student’s t-distribution can have different shape parameters νi for the marginal probability density functions of the multi- variate distribution. Expressions for the probability density function, for the variances, and for the covariances of the multivariate t-distribution with arbitrary shape parameters for the marginals are given. Keywords Multivariate Student’s t, Variance, Covariance, Arbitrary Shape Parameters 1. Introduction An expression for a multivariate Student’s t-distribution is presented. This expression, which is different in form than the form that is commonly used, allows the shape parameter ν for each marginal probability density function (pdf) of the multivariate pdf to be different. The form that is typically used is [1] −+ν Γ+((ν n) 2) T ( n) 2 +Σ−1 n 2 (1.[xx] [ ]) (1) ΓΣ(νν2)(π ) This “typical” form attempts to generalize the univariate Student’s t-distribution and is valid when the n marginal distributions have the same shape parameter ν . -
Statistical Distributions in General Insurance Stochastic Processes
Statistical distributions in general insurance stochastic processes by Jan Hendrik Harm Steenkamp Submitted in partial fulfilment of the requirements for the degree Magister Scientiae In the Department of Statistics In the Faculty of Natural & Agricultural Sciences University of Pretoria Pretoria January 2014 © University of Pretoria 1 I, Jan Hendrik Harm Steenkamp declare that the dissertation, which I hereby submit for the degree Magister Scientiae in Mathematical Statistics at the University of Pretoria, is my own work and has not previously been submit- ted for a degree at this or any other tertiary institution. SIGNATURE: DATE: 31 January 2014 © University of Pretoria Summary A general insurance risk model consists of in initial reserve, the premiums collected, the return on investment of these premiums, the claims frequency and the claims sizes. Except for the initial reserve, these components are all stochastic. The assumption of the distributions of the claims sizes is an integral part of the model and can greatly influence decisions on reinsurance agreements and ruin probabilities. An array of parametric distributions are available for use in describing the distribution of claims. The study is focussed on parametric distributions that have positive skewness and are defined for positive real values. The main properties and parameterizations are studied for a number of distribu- tions. Maximum likelihood estimation and method-of-moments estimation are considered as techniques for fitting these distributions. Multivariate nu- merical maximum likelihood estimation algorithms are proposed together with discussions on the efficiency of each of the estimation algorithms based on simulation exercises. These discussions are accompanied with programs developed in SAS PROC IML that can be used to simulate from the var- ious parametric distributions and to fit these parametric distributions to observed data. -
Diagonal Estimation with Probing Methods
Diagonal Estimation with Probing Methods Bryan J. Kaperick Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Matthias C. Chung, Chair Julianne M. Chung Serkan Gugercin Jayanth Jagalur-Mohan May 3, 2019 Blacksburg, Virginia Keywords: Probing Methods, Numerical Linear Algebra, Computational Inverse Problems Copyright 2019, Bryan J. Kaperick Diagonal Estimation with Probing Methods Bryan J. Kaperick (ABSTRACT) Probing methods for trace estimation of large, sparse matrices has been studied for several decades. In recent years, there has been some work to extend these techniques to instead estimate the diagonal entries of these systems directly. We extend some analysis of trace estimators to their corresponding diagonal estimators, propose a new class of deterministic diagonal estimators which are well-suited to parallel architectures along with heuristic ar- guments for the design choices in their construction, and conclude with numerical results on diagonal estimation and ordering problems, demonstrating the strengths of our newly- developed methods alongside existing methods. Diagonal Estimation with Probing Methods Bryan J. Kaperick (GENERAL AUDIENCE ABSTRACT) In the past several decades, as computational resources increase, a recurring problem is that of estimating certain properties very large linear systems (matrices containing real or complex entries). One particularly important quantity is the trace of a matrix, defined as the sum of the entries along its diagonal. In this thesis, we explore a problem that has only recently been studied, in estimating the diagonal entries of a particular matrix explicitly. For these methods to be computationally more efficient than existing methods, and with favorable convergence properties, we require the matrix in question to have a majority of its entries be zero (the matrix is sparse), with the largest-magnitude entries clustered near and on its diagonal, and very large in size. -
Intensity Distribution of Interplanetary Scintillation at 408 Mhz
Aust. J. Phys., 1975,28,621-32 Intensity Distribution of Interplanetary Scintillation at 408 MHz R. G. Milne Chatterton Astrophysics Department, School of Physics, University of Sydney, Sydney, N.S.W. 2006. Abstract It is shown that interplanetary scintillation of small-diameter radio sources at 408 MHz produces intensity fluctuations which are well fitted by a Rice-squared. distribution, better so than is usually claimed. The observed distribution can be used to estimate the proportion of flux density in the core of 'core-halo' sources without the need for calibration against known point sources. 1. Introduction The observed intensity of a radio source of angular diameter ;s 1" arc shows rapid fluctuations when the line of sight passes close to the Sun. This interplanetary scintillation (IPS) is due to diffraction by electron density variations which move outwards from the Sun at high velocities (-350 kms-1) •. In a typical IPS observation the fluctuating intensity Set) from a radio source is reCorded for a few minutes. This procedure is repeated on several occasions during the couple of months that the line of sight is within - 20° (for observatio~s at 408 MHz) of the direction of the Sun. For each observation we derive a mean intensity 8; a scintillation index m defined by m2 = «(S-8)2)/82, where ( ) denote the expectation value; intensity moments of order q QI! = «(S-8)4); and the skewness parameter Y1 = Q3 Q;3/2, hereafter referred to as y. A histogram, or probability distribution, of the normalized intensity S/8 is also constructed. The present paper is concerned with estimating the form of this distribution. -
A Guide on Probability Distributions
powered project A guide on probability distributions R-forge distributions Core Team University Year 2008-2009 LATEXpowered Mac OS' TeXShop edited Contents Introduction 4 I Discrete distributions 6 1 Classic discrete distribution 7 2 Not so-common discrete distribution 27 II Continuous distributions 34 3 Finite support distribution 35 4 The Gaussian family 47 5 Exponential distribution and its extensions 56 6 Chi-squared's ditribution and related extensions 75 7 Student and related distributions 84 8 Pareto family 88 9 Logistic ditribution and related extensions 108 10 Extrem Value Theory distributions 111 3 4 CONTENTS III Multivariate and generalized distributions 116 11 Generalization of common distributions 117 12 Multivariate distributions 132 13 Misc 134 Conclusion 135 Bibliography 135 A Mathematical tools 138 Introduction This guide is intended to provide a quite exhaustive (at least as I can) view on probability distri- butions. It is constructed in chapters of distribution family with a section for each distribution. Each section focuses on the tryptic: definition - estimation - application. Ultimate bibles for probability distributions are Wimmer & Altmann (1999) which lists 750 univariate discrete distributions and Johnson et al. (1994) which details continuous distributions. In the appendix, we recall the basics of probability distributions as well as \common" mathe- matical functions, cf. section A.2. And for all distribution, we use the following notations • X a random variable following a given distribution, • x a realization of this random variable, • f the density function (if it exists), • F the (cumulative) distribution function, • P (X = k) the mass probability function in k, • M the moment generating function (if it exists), • G the probability generating function (if it exists), • φ the characteristic function (if it exists), Finally all graphics are done the open source statistical software R and its numerous packages available on the Comprehensive R Archive Network (CRAN∗). -
An Alternative Discrete Skew Logistic Distribution
An Alternative Discrete Skew Logistic Distribution Deepesh Bhati1*, Subrata Chakraborty2, and Snober Gowhar Lateef1 1Department of Statistics, Central University of Rajasthan, 2Department of Statistics, Dibrugarh University, Assam *Corresponding Author: [email protected] April 7, 2016 Abstract In this paper, an alternative Discrete skew Logistic distribution is proposed, which is derived by using the general approach of discretizing a continuous distribution while retaining its survival function. The properties of the distribution are explored and it is compared to a discrete distribution defined on integers recently proposed in the literature. The estimation of its parameters are discussed, with particular focus on the maximum likelihood method and the method of proportion, which is particularly suitable for such a discrete model. A Monte Carlo simulation study is carried out to assess the statistical properties of these inferential techniques. Application of the proposed model to a real life data is given as well. 1 Introduction Azzalini A.(1985) and many researchers introduced different skew distributions like skew-Cauchy distribu- tion(Arnold B.C and Beaver R.J(2000)), Skew-Logistic distribution (Wahed and Ali (2001)), Skew Student's t distribution(Jones M.C. et. al(2003)). Lane(2004) fitted the existing skew distributions to insurance claims data. Azzalini A.(1985), Wahed and Ali (2001) developed Skew Logistic distribution by taking f(x) to be Logistic density function and G(x) as its CDF of standard Logistic distribution, respectively and obtained arXiv:1604.01541v1 [stat.ME] 6 Apr 2016 the probability density function(pdf) as 2e−x f(x; λ) = ; −∞ < x < 1; −∞ < λ < 1 (1 + e−x)2(1 + e−λx) They have numerically studied cdf, moments, median, mode and other properties of this distribution. -
Probability Statistics Economists
PROBABILITY AND STATISTICS FOR ECONOMISTS BRUCE E. HANSEN Contents Preface x Acknowledgements xi Mathematical Preparation xii Notation xiii 1 Basic Probability Theory 1 1.1 Introduction . 1 1.2 Outcomes and Events . 1 1.3 Probability Function . 3 1.4 Properties of the Probability Function . 4 1.5 Equally-Likely Outcomes . 5 1.6 Joint Events . 5 1.7 Conditional Probability . 6 1.8 Independence . 7 1.9 Law of Total Probability . 9 1.10 Bayes Rule . 10 1.11 Permutations and Combinations . 11 1.12 Sampling With and Without Replacement . 13 1.13 Poker Hands . 14 1.14 Sigma Fields* . 16 1.15 Technical Proofs* . 17 1.16 Exercises . 18 2 Random Variables 22 2.1 Introduction . 22 2.2 Random Variables . 22 2.3 Discrete Random Variables . 22 2.4 Transformations . 24 2.5 Expectation . 25 2.6 Finiteness of Expectations . 26 2.7 Distribution Function . 28 2.8 Continuous Random Variables . 29 2.9 Quantiles . 31 2.10 Density Functions . 31 2.11 Transformations of Continuous Random Variables . 34 2.12 Non-Monotonic Transformations . 36 ii CONTENTS iii 2.13 Expectation of Continuous Random Variables . 37 2.14 Finiteness of Expectations . 39 2.15 Unifying Notation . 39 2.16 Mean and Variance . 40 2.17 Moments . 42 2.18 Jensen’s Inequality . 42 2.19 Applications of Jensen’s Inequality* . 43 2.20 Symmetric Distributions . 45 2.21 Truncated Distributions . 46 2.22 Censored Distributions . 47 2.23 Moment Generating Function . 48 2.24 Cumulants . 50 2.25 Characteristic Function . 51 2.26 Expectation: Mathematical Details* . 52 2.27 Exercises .