DOCSLIB.ORG

Importance of Generalized Logistic Distribution in Extreme Value Modeling

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Importance of Generalized Logistic Distribution in Extreme Value Modeling Applied Mathematics, 2013, 4, 560-573 http://dx.doi.org/10.4236/am.2013.43080 Published Online March 2013 (http://www.scirp.org/journal/am) Importance of Generalized Logistic Distribution in Extreme Value Modeling K. Nidhin1, C. Chandran2 1Department of Mathematics, Indian Institute of Science, Bangalore, India 2Department of Statistics, University of Calicut, Calicut, India Email: [email protected], [email protected] Received September 10, 2012; revised February 1, 2013; accepted February 8, 2013 ABSTRACT We consider a problem from stock market modeling, precisely, choice of adequate distribution of modeling extremal behavior of stock market data. Generalized extreme value (GEV) distribution and generalized Pareto (GP) distribution are the classical distributions for this problem. However, from 2004, [1] and many other researchers have been empiri- cally showing that generalized logistic (GL) distribution is a better model than GEV and GP distributions in modeling extreme movement of stock market data. In this paper, we show that these results are not accidental. We prove the theoretical importance of GL distribution in extreme value modeling. For proving this, we introduce a general multi- variate limit theorem and deduce some important multivariate theorems in probability as special cases. By using the theorem, we derive a limit theorem in extreme value theory, where GL distribution plays central role instead of GEV distribution. The proof of this result is parallel to the proof of classical extremal types theorem, in the sense that, it pos- sess important characteristic in classical extreme value theory, for e.g. distributional property, stability, convergence and multivariate extension etc. Keywords: Financial Risk Modeling; Stock Market Analysis; Generalized Logistic Distribution; Generalized Extreme Value Distribution; Tail Equivalence; Maximum Stability; Random Sample Size; Limit Distribution 1. Introduction different from normal and lognormal to model large val- ues of finance data. For example t-distribution, alpha sta- An important problem from the field of stock market modeling is the determination of adequate model for ex- ble distribution, etc. In 1990’s the interest in modeling treme stock movement. In financial literature, the choice large values of finance data have been diverted to ex- of using an appropriate probability model for financial treme value theory where modeling maximum of the data, returns are clearly exemplified rather than selecting a generalized extreme value distribution for maxima (GEV conventional model (see for ex. [2]). The fitted tale dis- (max)) or generalized Pareto (GP) distribution are used tribution is crucially important in financial studies. For and for minimum of the data, generalized extreme value Value at Risk (VaR) estimation, one requires appropriate distribution for minima (GEV(min)) is used. These dis- probability distribution of extremes as input. A vast num- tributions enjoy strong theoretical support for analyzing ber of literature show the importance of best fitted dis- extreme movement of a data compared to other models. tribution in VaR analysis (see [3]). Another important Below we give an outline of these theoretical properties area of application of probability models of extremes is of the distributions. in hedging procedure. Hedging procedure is clearly based The theoretical representation for GEV(max) and on the probability of fitted distribution (see [4]). Also GEV(min) have been established by Jenkinson and Von measuring the risk attached to a share or portfolio will Mises (see von Mises (1954) and Jenkinson (1955)). We critically depends on the tale distribution. [5] largely il- define GEV(max) below. lustrates the importance of appropriate probability mod- Definition 1.1 A random variable X is said to follow els for extremes of financial returns data. the generalized extreme value distribution for maximum Initially, normal and lognormal distributions were used (GEV(max)) if its distribution function is given by, to model data which arise from financial sector. In the 1 k last five decades, different authors have been showing exp 1kx if k 0 Gx that the distribution of extreme daily returns is far from k exp expxk if 0 normal (see [5-11]. They used a number of distributions Copyright © 2013 SciRes. AM K. NIDHIN, C. CHANDRAN 561 and the supports are Theorem 1.1 (see [12]): Let Xnn , 1 be a se- quence of i.i.d.r.v.’s and M =maxXX , ,,X . If xk1 for k 0 nn12 there exist sequences of norming constants xk1 for k 0 abn 0, n , and a non-degenerate d.f. G such that, xRfor k 0. Mb w PxGnn x, (1.1) The related location-scale family can be introduced by an x replacing the argument x above by for R then G , in Equation (1.1), is the GEV(max) distribution defined in Definition 1.4. and 0 . The parameter k is known as the shape Since parameter of the distribution and different values of k min X12 ,XX , ,nn = max XXX1 , 2 , , , the leads to different well known distributions. That is, for asymptotic distribution of minima of i.i.d.r.v.’s can be k 0 , it is the Gumbel (Type I) distribution which is derived as 1 Gx , where G is the GEV(max). interpreted as limk0 Gk, For k 0 , it is the reverse- Hence the asymptotic distribution for minima is the Weibull (Type III) distribution, for k 0 , it is Frechet GEV(min) distribution defined in Definition 1.2. These (Type II) distribution. Also GEV(min) can be defined as theories have been extended to multivariate case, where follows. multivariate generalized extreme value distributions play Definition 1.2 A random variable X is said to follow central role (see [13,14]). the generalized extreme value distribution for minimum Another alternative model for maxima is the gene- (GEV(min)), if its distribution function is given by, ralized Pareto distribution, has been suggested by two in- 1 k dependent works [15] and [16]. Below we define GP dis- 1exp1kx if k 0 tribution. Hx k Definition 1.4 A random variable X is said to follow 1expexpxk if 0 the generalized Pareto (GP) distribution if its distribu- and the supports are tion function is given by, xk1 for k 0 1 k 1 11kx if k 0 xkfor k 0 Qx xRfor k 0. 1exp xk if 0 The parameter k is the shape parameter and k . and the supports are One can introduce the related location-scale family by xk1 for k 0 x replacing the argument x above by for R xk 1 for k 0 xk0for0. and 0 . For k 0 (which is interpreted as limk0 Fk), the distribution is called reverse-Gumbel, for k 0 , it is called reverse-Frechet distribution and for The parameter k is the shape parameter and k . k 0 , the distribution is the Weibull distribution. One can introduce the related location-scale family by x The above two distributions possess the characterizing replacing the argument x above by for R properties called max-stability and min-stability respec- tively. In the following we define max-stability for non- and 0 . With respect to k 0 , k 0 and k 0 random sample size. we get the exponential distribution, beta distribution and Definition 1.3 A non-degenerate random variable X Pareto distribution. is said to be max-stable if, for each n 2,3, there are GP distribution possess characterizing property called Peak over Threshold (POT) stability w.r.t non-random constants an 0 and bn such that d sample size. In the following we define POT stability for maxX12 ,XXaXb , , nn n. non-random sample size. Definition 1.5 A non-degenerate random variable X where X ,,,XX are same copies of X . 12 n is said to be POT-stable if, for each u 0 there are Similarly, one can define min-stability. By the extre- constants a 0 and b such that mal types theorem, asymptotic distribution of maxima of u u d independent and identically distributed random variables X uX u aXb. (i.i.d.r.v.’s) under proper normalization can be approxi- uu mated by the GEV(max) distribution. Below we give an By Peak over threshold theorem, under proper norma- outline of the theorem. lization, the asymptotic distribution of the observations Copyright © 2013 SciRes. AM 562 K. NIDHIN, C. CHANDRAN above exceedances of a level u has a generalized pareto variate extremal types theorem and multivariate random distribution iff asymptotic distribution of maximum of sum convergence theorem are special cases of this theo- the observations, under proper normalization, is the rem. In Section 5, we use the above general limit theo- GEV(max) distribution. We give an outline of the theo- rem to prove the convergence of random maxima and rem below. random minima to GL distribution. For an additional Theorem 1.2 (see [15,16]) let Xnn ,1 be a se- support to our claim, we present a data analysis in Sec- quence of i.i.d.r.v.’s with common continuous distri- tion 6. We introduce a multivariate generalized logistic bution function F x . Suppose there exists a pair of se- distribution in Section 7, and also we prove a charac- quences an and bn with an 0 for all n and a teristic property of this multivariate distribution by using non-degenerate distribution function Gx such that the general multivariate theorem in Section 4. To prove the results we require some basic concepts lim PMnnn b a x n (1.2) which we discuss below. n Definition 1.6 The right end point and left end point of lim F axnn b G x n a d.f. F , denoted by xF and yF respectively, are for all x at which Gx is continuous.
Load more

Recommended publications
  • A Tail Quantile Approximation Formula for the Student T and the Symmetric Generalized Hyperbolic Distribution
    A Service of Leibniz-Informationszentrum econstor Wirtschaft Leibniz Information Centre Make Your Publications Visible. zbw for Economics Schlüter, Stephan; Fischer, Matthias J. Working Paper A tail quantile approximation formula for the student t and the symmetric generalized hyperbolic distribution IWQW Discussion Papers, No. 05/2009 Provided in Cooperation with: Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics Suggested Citation: Schlüter, Stephan; Fischer, Matthias J. (2009) : A tail quantile approximation formula for the student t and the symmetric generalized hyperbolic distribution, IWQW Discussion Papers, No. 05/2009, Friedrich-Alexander-Universität Erlangen-Nürnberg, Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung (IWQW), Nürnberg This Version is available at: http://hdl.handle.net/10419/29554 Standard-Nutzungsbedingungen: Terms of use: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Documents in EconStor may be saved and copied for your Zwecken und zum Privatgebrauch gespeichert und kopiert werden. personal and scholarly purposes. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle You are not to copy documents for public or commercial Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich purposes, to exhibit the documents publicly, to make them machen, vertreiben oder anderweitig nutzen. publicly available on the internet, or to distribute or otherwise use the documents in public. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, If the documents have been made available under an Open gelten abweichend von diesen Nutzungsbedingungen die in der dort Content Licence (especially Creative Commons Licences), you genannten Lizenz gewährten Nutzungsrechte. may exercise further usage rights as specified in the indicated licence. www.econstor.eu IWQW Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung Diskussionspapier Discussion Papers No.
    [Show full text]
  • Asymmetric Generalizations of Symmetric Univariate Probability Distributions Obtained Through Quantile Splicing
    Asymmetric generalizations of symmetric univariate probability distributions obtained through quantile splicing a* a b Brenda V. Mac’Oduol , Paul J. van Staden and Robert A. R. King aDepartment of Statistics, University of Pretoria, Pretoria, South Africa; bSchool of Mathematical and Physical Sciences, University of Newcastle, Callaghan, Australia *Correspondence to: Brenda V. Mac’Oduol; Department of Statistics, University of Pretoria, Pretoria, South Africa. email: [email protected] Abstract Balakrishnan et al. proposed a two-piece skew logistic distribution by making use of the cumulative distribution function (CDF) of half distributions as the building block, to give rise to an asymmetric family of two-piece distributions, through the inclusion of a single shape parameter. This paper proposes the construction of asymmetric families of two-piece distributions by making use of quantile functions of symmetric distributions as building blocks. This proposition will enable the derivation of a general formula for the L-moments of two-piece distributions. Examples will be presented, where the logistic, normal, Student’s t(2) and hyperbolic secant distributions are considered. Keywords Two-piece; Half distributions; Quantile functions; L-moments 1. Introduction Balakrishnan, Dai, and Liu (2017) introduced a skew logistic distribution as an alterna- tive to the model proposed by van Staden and King (2015). It proposed taking the half distribution to the right of the location parameter and joining it to the half distribution to the left of the location parameter which has an inclusion of a single shape parameter, a > 0. The methodology made use of the cumulative distribution functions (CDFs) of the parent distributions to obtain the two-piece distribution.
    [Show full text]
  • A Comparison Between Skew-Logistic and Skew-Normal Distributions 1
    MATEMATIKA, 2015, Volume 31, Number 1, 15–24 c UTM Centre for Industrial and Applied Mathematics A Comparison Between Skew-logistic and Skew-normal Distributions 1Ramin Kazemi and 2Monireh Noorizadeh 1,2Department of Statistics, Imam Khomeini International University Qazvin, Iran e-mail: [email protected] Abstract Skew distributions are reasonable models for describing claims in property- liability insurance. We consider two well-known datasets from actuarial science and fit skew-normal and skew-logistic distributions to these dataset. We find that the skew-logistic distribution is reasonably competitive compared to skew-normal in the literature when describing insurance data. The value at risk and tail value at risk are estimated for the dataset under consideration. Also, we compare the skew distributions via Kolmogorov-Smirnov goodness-of-fit test, log-likelihood criteria and AIC. Keywords Skew-logistic distribution; skew-normal distribution; value at risk; tail value at risk; log-likelihood criteria; AIC; Kolmogorov-Smirnov goodness-of-fit test. 2010 Mathematics Subject Classification 62P05, 91B30 1 Introduction Fitting an adequate distribution to real data sets is a relevant problem and not an easy task in actuarial literature, mainly due to the nature of the data, which shows several fea- tures to be accounted for. Eling [1] showed that the skew-normal and the skew-Student t distributions are reasonably competitive compared to some models when describing in- surance data. Bolance et al. [2] provided strong empirical evidence in favor of the use of the skew-normal, and log-skew-normal distributions to model bivariate claims data from the Spanish motor insurance industry.
    [Show full text]
  • Logistic Distribution Becomes Skew to the Right and When P>1 It Becomes Skew to the Left
    FITTING THE VERSATILE LINEARIZED, COMPOSITE, AND GENERALIZED LOGISTIC PROBABILITY DISTRIBUTION TO A DATA SET R.J. Oosterbaan, 06-08-2019. On www.waterlog.info public domain. Abstract The logistic probability distribution can be linearized for easy fitting to a data set using a linear regression to determine the parameters. The logistic probability distribution can also be generalized by raising the data values to the power P that is to be optimized by an iterative method based on the minimization of the squares of the deviations of calculated and observed data values (the least squares or LS method). This enhances its applicability. When P<1 the logistic distribution becomes skew to the right and when P>1 it becomes skew to the left. When P=1, the distribution is symmetrical and quite like the normal probability distribution. In addition, the logistic distribution can be made composite, that is: split into two parts with different parameters. Composite distributions can be used favorably when the data set is obtained under different external conditions influencing its probability characteristics. Contents 1. The standard logistic cumulative distribution function (CDF) and its linearization 2. The logistic cumulative distribution function (CDF) generalized 3. The composite generalized logistic distribution 4. Fitting the standard, generalized, and composite logistic distribution to a data set, available software 5. Construction of confidence belts 6. Constructing histograms and the probability density function (PDF) 7. Ranking according to goodness of fit 8. Conclusion 9. References 1. The standard cumulative distribution function (CDF) and its linearization The cumulative logistic distribution function (CDF) can be written as: Fc = 1 / {1 + e (A*X+B)} where Fc = cumulative logistic distribution function or cumulative frequency e = base of the natural logarithm (Ln), e = 2.71 .
    [Show full text]
  • The Extended Log-Logistic Distribution: Properties and Application
    Anais da Academia Brasileira de Ciências (2017) 89(1): 3-17 (Annals of the Brazilian Academy of Sciences) Printed version ISSN 0001-3765 / Online version ISSN 1678-2690 http://dx.doi.org/10.1590/0001-3765201720150579 www.scielo.br/aabc The Extended Log-Logistic Distribution: Properties and Application STÊNIO R. LIMA and GAUSS M. CORDEIRO Universidade Federal de Pernambuco, Departamento de Estatística, Av. Prof. Moraes Rego, 1235, Cidade Universitária, 50740-540 Recife, PE, Brazil Manuscript received on September 3, 2015; accepted for publication on October 7, 2016 ABSTRACT We propose a new four-parameter lifetime model, called the extended log-logistic distribution, to generalize the two-parameter log-logistic model. The new model is quite flexible to analyze positive data. We provide some mathematical properties including explicit expressions for the ordinary and incomplete moments, probability weighted moments, mean deviations, quantile function and entropy measure. The estimation of the model parameters is performed by maximum likelihood using the BFGS algorithm. The flexibility of the new model is illustrated by means of an application to a real data set. We hope that the new distribution will serve as an alternative model to other useful distributions for modeling positive real data in many areas. Key words: exponentiated generalize, generating function, log-logistic distribution, maximum likelihood estimation, moment. INTRODUCTION Numerous classical distributions have been extensively used over the past decades for modeling data in several areas. In fact, the statistics literature is filled with hundreds of continuous univariate distributions (see, e.g, Johnson et al. 1994, 1995). However, in many applied areas, such as lifetime analysis, finance and insurance, there is a clear need for extended forms of these distributions.
    [Show full text]
  • Logistic Distribution
    Logistic Distribution Raminta Stockute, Andrea Veaux, Paul Johnson June 10, 2013 1 Introduction The Logistic distribution is a continuous probability density function that is symmetric and uni-modal. It is similar in appearance to the Normal distribution and in practical applications, the two distributions cannot be distinguished from one another. 2 Mathematical Definition This distribution is characterized by two main parameters: location µ and scale σ. The probability density function is: (Raminta: I believe these signs are wrong. Do you agree?) 1 e−(x−µ)/σ f(x) = ∗ . σ 1 + e−(x−µ)/σ)2 The cumulative distribution of the Logistic is the famous S-shaped curve on which Logistic regression is based: 1 F (x) = . 1 + e−(x−µ)/σ 3 Moments The Compendium of the distributions (citation?) points out that the expected value is equal to the location parameter, µ: E(x) = µ Because the distribution is symmetric, the median and the mode are also equal to α. The variance of this distribution is: 1 V ar(x) = (πσ)2. 3 Compare that value against the Normal distribution, N(µ,σ2). The variance of the Logistic is different from the variance of the normal only by the scaling value of π2/3 1 4 Illustrations In Figure 1, some examples of Logistic distributions are offered. In Figures 2 and 3, the density functions that correspond to various location and scale parameters are presented. If we look at Figure 2, we see that scale controls the width and the height of the distribution. As you see in Figure 3, if we vary the location of the density, the density curve shifts to the left and right, while the “spread” and “height” remains the same.
    [Show full text]
  • Type Ii Half Logistic Exponentiated Exponential Distribution: Properties and Applications
    Pak. J. Statist. 2020 Vol. 36(1), 29-56 TYPE II HALF LOGISTIC EXPONENTIATED EXPONENTIAL DISTRIBUTION: PROPERTIES AND APPLICATIONS Muritala Abdulkabir1 and Ipinyomi, R.A.2 Department of Statistics, University of Ilorin, Ilorin, Nigeria Email: [email protected] [email protected] ABSTRACT This paper introduce a new distribution called Type II half logistic exponentiated exponential (TIIHLEE) distribution established from Type II half logistic G family of distribution . Some mathematical properties; moments, probability weighted moments, mean deviation, quantile function, Renyi entropy of TIIHLEE distribution are investigated. The expressions of order statistics are derived. Parameters of the derived distribution are obtained using maximum likelihood method. We compare the fits of the TIIHLEE model with some models which shows that the new model is strong and better fit than other models. KEYWORD Type II half logistic-G family, Moments, Order statistic, Estimation, Maximum likelihood, quantile function, Renyi Entropy. 1. INTRODUCTION Since real world data are usually complex and can take a variety of shapes, existing distributions do not always provide an adequate fit. Hence, generalizing distributions and studying their flexibility are of interest of researchers for last decades, the generated family of continuous distributions is a new improvement for producing and extending the usual classical distributions. These families have been broadly studied in several areas as well as yield more flexibility in many applications. Some of the generators are: Beta-generated (B-G) (Eugene et al. (2002)), Gamma-G (Zografos and Balakrishnan (2009)) and Ristic and Balakrishnan (2011)), Kumaraswamy-G (Cordeiro and de Castro (2011)), exponentiated generalized class (Cordeiro et al. (2013)), Weibull-G (Bourguignon et al.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Inequalities and Comparisons of the Cauchy, Gauss, and Logistic Distributions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Available at Applied www.ElsevierMathematics.com Mathematics DIRECT- Letters Applied Mathematics Letters 17 (2004) 77-82 www.elsevier.com/locate/aml Inequalities and Comparisons of the Cauchy, Gauss, and Logistic Distributions B. 0. OLUYEDE Department of Mathematics and Computer Science Georgia Southern University Statesboro, GA 30460-8093, U.S.A. boluyedeQgasou.edu (Received July 2001; revised and accepted November 2002) Abstract-In this note, some inequalities and basic results including characterizations and com- parisons of the Cauchy, logistic, and normal distributions are given. These results lead to necessary and sufficient conditions for the stochastic and dispersive ordering of the corresponding absolute random variables. @ 2004 Elsevier Ltd. All rights reserved. Keywords-Characteristic function, Inequalities, Stochastic order, Dispersive order 1. INTRODUCTION In this note, some inequalities and comparisons of the Cauchy, logistic, and normal distributions are presented. These results shed further insight into these well-known distributions. Also, necessary and sufficient conditions for dispersive and stochastic ordering of the corresponding absolute random variables are given. A general Cauchy law C(q p) is a distribution with the form Y = PZ + cr, where 2 has the density function of the form f(z) = (l/rr)(l + .z2), --co < z < co, denoted by C(0, 1). The characteristic function is given by @y(t) = exp{icut - p(ti}. A random variable X has a logistic distribution with parameters cr and p if the distribution function can be written as F(z)=P(X<z)= [l+exp{-y}]-‘, -oo < a < oo, /3 > 0.
    [Show full text]
  • On Families of Generalized Pareto Distributions: Properties and Applications
    Journal of Data Science 377-396 , DOI: 10.6339/JDS.201804_16(2).0008 On Families of Generalized Pareto Distributions: Properties and Applications D. Hamed1 and F. Famoye2 and C. Lee2 1 Department of Mathematics, Winthrop University, Rock Hill, SC, USA; 2Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan, USA In this paper, we introduce some new families of generalized Pareto distributions using the T-R{Y} framework. These families of distributions are named T-Pareto{Y} families, and they arise from the quantile functions of exponential, log-logistic, logistic, extreme value, Cauchy and Weibull distributions. The shapes of these T-Pareto families can be unimodal or bimodal, skewed to the left or skewed to the right with heavy tail. Some general properties of the T-Pareto{Y} family are investigated and these include the moments, modes, mean deviations from the mean and from the median, and Shannon entropy. Several new generalized Pareto distributions are also discussed. Four real data sets from engineering, biomedical and social science are analyzed to demonstrate the flexibility and usefulness of the T-Pareto{Y} families of distributions. Key words: Shannon entropy; quantile function; moment; T-X family. 1. Introduction The Pareto distribution is named after the well-known Italian-born Swiss sociologist and economist Vilfredo Pareto (1848-1923). Pareto [1] defined Pareto’s Law, which can be stated as N Axa , where N represents the number of persons having income x in a population. Pareto distribution is commonly used in modelling heavy tailed distributions, including but not limited to income, insurance and city size populations.
    [Show full text]
  • Exponentiated–Exponential Logistic Distribution: Some Properties And
    Exponentiated–Exponential Logistic Distribution: Some Properties and Application Laxmi Prasad Sapkota Lecturer, Tribhuvan University, Tribhuvan Multiple Campus, Palpa Email: [email protected] Article History Received 29 March 2020 Revised 24 October 2020 Accepted 16 November 2020 Abstract This study proposes new distribution which is generated from exponentiated-exponential-X family of distribution. It is explored various shape and behavior of the observed distribution through probability density plot, hazard rate function and quantile function. Further we have investigated some mathematical properties, estimation of the parameters and associated confidence interval using maximum likelihood estimation (MLE) method of the exponentiated- exponential-logistic distribution (EELD). Keywords: Exponentiated-exponential-X family, Hazard function, Logistic distribution, MLE. Introduction In the last few years, new generated families of continuous distributions have attracted several statisticians to develop new models. To describe the real world phenomena, we generally use statistical distribution. Since real data are usually complex and they have a variety of shapes, existing distribution do not always provide an adequate fit. Hence investigating new distribution and studying their behavior and flexibility are of interest of researchers for last decades. To improve the flexibility of the statistical model, the families are obtained by introducing one or more additional shape parameter(s) to the baseline distribution. In probability and
    [Show full text]