Splitting Models for Multivariate Count Data
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Relationships Among Some Univariate Distributions
IIE Transactions (2005) 37, 651–656 Copyright C “IIE” ISSN: 0740-817X print / 1545-8830 online DOI: 10.1080/07408170590948512 Relationships among some univariate distributions WHEYMING TINA SONG Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan, Republic of China E-mail: [email protected], wheyming [email protected] Received March 2004 and accepted August 2004 The purpose of this paper is to graphically illustrate the parametric relationships between pairs of 35 univariate distribution families. The families are organized into a seven-by-five matrix and the relationships are illustrated by connecting related families with arrows. A simplified matrix, showing only 25 families, is designed for student use. These relationships provide rapid access to information that must otherwise be found from a time-consuming search of a large number of sources. Students, teachers, and practitioners who model random processes will find the relationships in this article useful and insightful. 1. Introduction 2. A seven-by-five matrix An understanding of probability concepts is necessary Figure 1 illustrates 35 univariate distributions in 35 if one is to gain insights into systems that can be rectangle-like entries. The row and column numbers are modeled as random processes. From an applications labeled on the left and top of Fig. 1, respectively. There are point of view, univariate probability distributions pro- 10 discrete distributions, shown in the first two rows, and vide an important foundation in probability theory since 25 continuous distributions. Five commonly used sampling they are the underpinnings of the most-used models in distributions are listed in the third row. -
On Two-Echelon Inventory Systems with Poisson Demand and Lost Sales
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Universiteit Twente Repository On two-echelon inventory systems with Poisson demand and lost sales Elisa Alvarez, Matthieu van der Heijden Beta Working Paper series 366 BETA publicatie WP 366 (working paper) ISBN ISSN NUR 804 Eindhoven December 2011 On two-echelon inventory systems with Poisson demand and lost sales Elisa Alvarez Matthieu van der Heijden University of Twente University of Twente The Netherlands1 The Netherlands 2 Abstract We derive approximations for the service levels of two-echelon inventory systems with lost sales and Poisson demand. Our method is simple and accurate for a very broad range of problem instances, including cases with both high and low service levels. In contrast, existing methods only perform well for limited problem settings, or under restrictive assumptions. Key words: inventory, spare parts, lost sales. 1 Introduction We consider a two-echelon inventory system for spare part supply chains, consisting of a single central depot and multiple local warehouses. Demand arrives at each local warehouse according to a Poisson process. Each location controls its stocks using a one-for-one replenishment policy. Demand that cannot be satisfied from stock is served using an emergency shipment from an external source with infinite supply and thus lost to the system. It is well-known that the analysis of such lost sales inventory systems is more complex than the equivalent with full backordering (Bijvank and Vis [3]). In particular, the analysis of the central depot is complex, since (i) the order process is not Poisson, and (ii) the order arrival rate depends on the inventory states of the local warehouses: Local warehouses only generate replenishment orders if they have stock on hand. -
Distance Between Multinomial and Multivariate Normal Models
Chapter 9 Distance between multinomial and multivariate normal models SECTION 1 introduces Andrew Carter’s recursive procedure for bounding the Le Cam distance between a multinomialmodelandits approximating multivariatenormal model. SECTION 2 develops notation to describe the recursive construction of randomizations via conditioning arguments, then proves a simple Lemma that serves to combine the conditional bounds into a single recursive inequality. SECTION 3 applies the results from Section 2 to establish the bound involving randomization of the multinomial distribution in Carter’s inequality. SECTION 4 sketches the argument for the bound involving randomization of the multivariate normalin Carter’s inequality. SECTION 5 outlines the calculation for bounding the Hellinger distance between a smoothed Binomialand its approximating normaldistribution. 1. Introduction The multinomial distribution M(n,θ), where θ := (θ1,...,θm ), is the probability m measure on Z+ defined by the joint distribution of the vector of counts in m cells obtained by distributing n balls independently amongst the cells, with each ball assigned to a cell chosen from the distribution θ. The variance matrix nVθ corresponding to M(n,θ) has (i, j)th element nθi {i = j}−nθi θj . The central limit theorem ensures that M(n,θ) is close to the N(nθ,nVθ ), in the sense of weak convergence, for fixed m when n is large. In his doctoral dissertation, Andrew Carter (2000a) considered the deeper problem of bounding the Le Cam distance (M, N) between models M :={M(n,θ) : θ ∈ } and N :={N(nθ,nVθ ) : θ ∈ }, under mild regularity assumptions on . For example, he proved that m log m maxi θi <1>(M, N) ≤ C √ provided sup ≤ C < ∞, n θ∈ mini θi for a constant C that depends only on C. -
The Exciting Guide to Probability Distributions – Part 2
The Exciting Guide To Probability Distributions – Part 2 Jamie Frost – v1.1 Contents Part 2 A revisit of the multinomial distribution The Dirichlet Distribution The Beta Distribution Conjugate Priors The Gamma Distribution We saw in the last part that the multinomial distribution was over counts of outcomes, given the probability of each outcome and the total number of outcomes. xi f(x1, ... , xk | n, p1, ... , pk)= [n! / ∏xi!] ∏pi The count of The probability of each outcome. each outcome. That’s all smashing, but suppose we wanted to know the reverse, i.e. the probability that the distribution underlying our random variable has outcome probabilities of p1, ... , pk, given that we observed each outcome x1, ... , xk times. In other words, we are considering all the possible probability distributions (p1, ... , pk) that could have generated these counts, rather than all the possible counts given a fixed distribution. Initial attempt at a probability mass function: Just swap the domain and the parameters: The RHS is exactly the same. xi f(p1, ... , pk | n, x1, ... , xk )= [n! / ∏xi!] ∏pi Notational convention is that we define the support as a vector x, so let’s relabel p as x, and the counts x as α... αi f(x1, ... , xk | n, α1, ... , αk )= [n! / ∏ αi!] ∏xi We can define n just as the sum of the counts: αi f(x1, ... , xk | α1, ... , αk )= [(∑αi)! / ∏ αi!] ∏xi But wait, we’re not quite there yet. We know that probabilities have to sum to 1, so we need to restrict the domain we can draw from: s.t. -
556: MATHEMATICAL STATISTICS I 1 the Multinomial Distribution the Multinomial Distribution Is a Multivariate Generalization of T
556: MATHEMATICAL STATISTICS I MULTIVARIATE PROBABILITY DISTRIBUTIONS 1 The Multinomial Distribution The multinomial distribution is a multivariate generalization of the binomial distribution. The binomial distribution can be derived from an “infinite urn” model with two types of objects being sampled without replacement. Suppose that the proportion of “Type 1” objects in the urn is θ (so 0 · θ · 1) and hence the proportion of “Type 2” objects in the urn is 1 ¡ θ. Suppose that n objects are sampled, and X is the random variable corresponding to the number of “Type 1” objects in the sample. Then X » Bin(n; θ). Now consider a generalization; suppose that the urn contains k + 1 types of objects (k = 1; 2; :::), with θi being the proportion of Type i objects, for i = 1; :::; k + 1. Let Xi be the random variable corresponding to the number of type i objects in a sample of size n, for i = 1; :::; k. Then the joint T pmf of vector X = (X1; :::; Xk) is given by e n! n! kY+1 f (x ; :::; x ) = θx1 ....θxk θxk+1 = θxi X1;:::;Xk 1 k x !:::x !x ! 1 k k+1 x !:::x !x ! i 1 k k+1 1 k k+1 i=1 where 0 · θi · 1 for all i, and θ1 + ::: + θk + θk+1 = 1, and where xk+1 is defined by xk+1 = n ¡ (x1 + ::: + xk): This is the mass function for the multinomial distribution which reduces to the binomial if k = 1. 2 The Dirichlet Distribution The Dirichlet distribution is a multivariate generalization of the Beta distribution. -
Probability Distributions and Related Mathematical Constructs
Appendix B More probability distributions and related mathematical constructs This chapter covers probability distributions and related mathematical constructs that are referenced elsewhere in the book but aren’t covered in detail. One of the best places for more detailed information about these and many other important probability distributions is Wikipedia. B.1 The gamma and beta functions The gamma function Γ(x), defined for x> 0, can be thought of as a generalization of the factorial x!. It is defined as ∞ x 1 u Γ(x)= u − e− du Z0 and is available as a function in most statistical software packages such as R. The behavior of the gamma function is simpler than its form may suggest: Γ(1) = 1, and if x > 1, Γ(x)=(x 1)Γ(x 1). This means that if x is a positive integer, then Γ(x)=(x 1)!. − − − The beta function B(α1,α2) is defined as a combination of gamma functions: Γ(α1)Γ(α2) B(α ,α ) def= 1 2 Γ(α1+ α2) The beta function comes up as a normalizing constant for beta distributions (Section 4.4.2). It’s often useful to recognize the following identity: 1 α1 1 α2 1 B(α ,α )= x − (1 x) − dx 1 2 − Z0 239 B.2 The Poisson distribution The Poisson distribution is a generalization of the binomial distribution in which the number of trials n grows arbitrarily large while the mean number of successes πn is held constant. It is traditional to write the mean number of successes as λ; the Poisson probability density function is λy P (y; λ)= eλ (y =0, 1,.. -
Binomial and Multinomial Distributions
Binomial and multinomial distributions Kevin P. Murphy Last updated October 24, 2006 * Denotes more advanced sections 1 Introduction In this chapter, we study probability distributions that are suitable for modelling discrete data, like letters and words. This will be useful later when we consider such tasks as classifying and clustering documents, recognizing and segmenting languages and DNA sequences, data compression, etc. Formally, we will be concerned with density models for X ∈{1,...,K}, where K is the number of possible values for X; we can think of this as the number of symbols/ letters in our alphabet/ language. We assume that K is known, and that the values of X are unordered: this is called categorical data, as opposed to ordinal data, in which the discrete states can be ranked (e.g., low, medium and high). If K = 2 (e.g., X represents heads or tails), we will use a binomial distribution. If K > 2, we will use a multinomial distribution. 2 Bernoullis and Binomials Let X ∈{0, 1} be a binary random variable (e.g., a coin toss). Suppose p(X =1)= θ. Then − p(X|θ) = Be(X|θ)= θX (1 − θ)1 X (1) is called a Bernoulli distribution. It is easy to show that E[X] = p(X =1)= θ (2) Var [X] = θ(1 − θ) (3) The likelihood for a sequence D = (x1,...,xN ) of coin tosses is N − p(D|θ)= θxn (1 − θ)1 xn = θN1 (1 − θ)N0 (4) n=1 Y N N where N1 = n=1 xn is the number of heads (X = 1) and N0 = n=1(1 − xn)= N − N1 is the number of tails (X = 0). -
Discrete Probability Distributions
Machine Learning Srihari Discrete Probability Distributions Sargur N. Srihari 1 Machine Learning Srihari Binary Variables Bernoulli, Binomial and Beta 2 Machine Learning Srihari Bernoulli Distribution • Expresses distribution of Single binary-valued random variable x ε {0,1} • Probability of x=1 is denoted by parameter µ, i.e., p(x=1|µ)=µ – Therefore p(x=0|µ)=1-µ • Probability distribution has the form Bern(x|µ)=µ x (1-µ) 1-x • Mean is shown to be E[x]=µ Jacob Bernoulli • Variance is var[x]=µ (1-µ) 1654-1705 • Likelihood of n observations independently drawn from p(x|µ) is N N x 1−x p(D | ) = p(x | ) = n (1− ) n µ ∏ n µ ∏µ µ n=1 n=1 – Log-likelihood is N N ln p(D | ) = ln p(x | ) = {x ln +(1−x )ln(1− )} µ ∑ n µ ∑ n µ n µ n=1 n=1 • Maximum likelihood estimator 1 N – obtained by setting derivative of ln p(D|µ) wrt m equal to zero is = x µML ∑ n N n=1 • If no of observations of x=1 is m then µML=m/N 3 Machine Learning Srihari Binomial Distribution • Related to Bernoulli distribution • Expresses Distribution of m – No of observations for which x=1 • It is proportional to Bern(x|µ) Histogram of Binomial for • Add up all ways of obtaining heads N=10 and ⎛ ⎞ m=0.25 ⎜ N ⎟ m N−m Bin(m | N,µ) = ⎜ ⎟µ (1− µ) ⎝⎜ m ⎠⎟ Binomial Coefficients: ⎛ ⎞ N ! ⎜ N ⎟ • Mean and Variance are ⎜ ⎟ = ⎝⎜ m ⎠⎟ m!(N −m)! N E[m] = ∑mBin(m | N,µ) = Nµ m=0 Var[m] = N (1− ) 4 µ µ Machine Learning Srihari Beta Distribution • Beta distribution a=0.1, b=0.1 a=1, b=1 Γ(a + b) Beta(µ |a,b) = µa−1(1 − µ)b−1 Γ(a)Γ(b) • Where the Gamma function is defined as ∞ Γ(x) = ∫u x−1e−u du -
Statistical Distributions
CHAPTER 10 Statistical Distributions In this chapter, we shall present some probability distributions that play a central role in econometric theory. First, we shall present the distributions of some discrete random variables that have either a finite set of values or that take values that can be indexed by the entire set of positive integers. We shall also present the multivariate generalisations of one of these distributions. Next, we shall present the distributions of some continuous random variables that take values in intervals of the real line or over the entirety of the real line. Amongst these is the normal distribution, which is of prime importance and for which we shall consider, in detail, the multivariate extensions. Associated with the multivariate normal distribution are the so-called sam- pling distributions that are important in the theory of statistical inference. We shall consider these distributions in the final section of the chapter, where it will transpire that they are special cases of the univariate distributions described in the preceding section. Discrete Distributions Suppose that there is a population of N elements, Np of which belong to class and N(1 p) to class c. When we select n elements at random from the populationA in n −successive trials,A we wish to know the probability of the event that x of them will be in and that n x of them will be in c. The probability will Abe affected b−y the way in which theA n elements are selected; and there are two ways of doing this. Either they can be put aside after they have been sampled, or else they can be restored to the population. -
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. -
Probability Distributions Used in Reliability Engineering
Probability Distributions Used in Reliability Engineering Probability Distributions Used in Reliability Engineering Andrew N. O’Connor Mohammad Modarres Ali Mosleh Center for Risk and Reliability 0151 Glenn L Martin Hall University of Maryland College Park, Maryland Published by the Center for Risk and Reliability International Standard Book Number (ISBN): 978-0-9966468-1-9 Copyright © 2016 by the Center for Reliability Engineering University of Maryland, College Park, Maryland, USA All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from The Center for Reliability Engineering, Reliability Engineering Program. The Center for Risk and Reliability University of Maryland College Park, Maryland 20742-7531 In memory of Willie Mae Webb This book is dedicated to the memory of Miss Willie Webb who passed away on April 10 2007 while working at the Center for Risk and Reliability at the University of Maryland (UMD). She initiated the concept of this book, as an aid for students conducting studies in Reliability Engineering at the University of Maryland. Upon passing, Willie bequeathed her belongings to fund a scholarship providing financial support to Reliability Engineering students at UMD. Preface Reliability Engineers are required to combine a practical understanding of science and engineering with statistics. The reliability engineer’s understanding of statistics is focused on the practical application of a wide variety of accepted statistical methods. Most reliability texts provide only a basic introduction to probability distributions or only provide a detailed reference to few distributions. -
5. the Multinomial Distribution
The Multinomial Distribution http://www.math.uah.edu/stat/bernoulli/Multinomial.xhtml Virtual Laboratories > 11. Bernoulli Trials > 1 2 3 4 5 6 5. The Multinomial Distribution Basic Theory Multinomial trials A multinomial trials process is a sequence of independent, identically distributed random variables X=(X 1 ,X 2 , ...) each taking k possible values. Thus, the multinomial trials process is a simple generalization of the Bernoulli trials process (which corresponds to k=2). For simplicity, we will denote the set of outcomes by {1, 2, ...,k}, and we will denote the c ommon probability density function of the trial variables by pi = ℙ(X j =i ), i ∈ {1, 2, ...,k} k Of course pi > 0 for each i and ∑i=1 pi = 1. In statistical terms, the sequence X is formed by sampling from the distribution. As with our discussion of the binomial distribution, we are interested in the random variables that count the number of times each outcome occurred. Thus, let Yn,i = #({ j ∈ {1, 2, ...,n}:X j =i }), i ∈ {1, 2, ...,k} k Note that ∑i=1 Yn,i =n so if we know the values of k−1 of the counting variables , we can find the value of the remaining counting variable. As with any counting variable, we can express Yn,i as a sum of indicator variables: n 1. Show that Yn,i = ∑j=1 1(X j =i ). Basic arguments using independence and combinatorics can be used to derive the joint, marginal, and conditional densities of the counting variables. In particular, recall the definition of the multinomial coefficient: k for positive integers (j 1 ,j 2 , ...,j k) with ∑i=1 ji =n, n n! ( ) = j1 ,j 2 , ...,j k j1 ! j2 ! ··· jk ! Joint Distribution k 2.