DOCSLIB.ORG

A Generalized Rayleigh Distribution and Its Application

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Generalized Rayleigh Distribution and Its Application Biometrics & Biostatistics International Journal Research Article Open Access A generalized Rayleigh distribution and its application Abstract Volume 8 Issue 4 - 2019 This paper studies properties and applications of the generalized Rayleigh-truncated 1 2 1 Lishamol Tomy, Jiju Gillariose negative binomial distribution established by Jiju & Lishamol. For the concerned objective, 1Department of Statistics, Deva Matha College, India we applied the model to a real life data set and its performance is compared with that of 2Department of Statistics, St.Thomas College, India other four-parameter generalized Rayleigh distributions which are derived using different generators. Correspondence: Lishamol Tomy, Department of Statistics, Deva Matha College, Kuravilangad, Kerala-686633, India, Keywords: hazard rate function, marshall-olkin family, maximum likelihood estimation, Email rayleigh-truncated negative binomial distribution Received: May 31, 2019 | Published: July 08, 2019 Introduction distributed random variables with survival function F(x) and N be a truncated negative binomial random variable, independent of X ’s, 2 i The Rayleigh distribution was introduced by Rayleigh and with parameters 0<α <1 and β >0, such that originally proposed in the fields of acoustics and optics. It has emerged β β +− as a special case of the Weibull distribution. The Rayleigh distribution α n1 n P(N = n) = β (1−α ) ;n = 1,2,... (3) has widely used in communication theory to describe hourly median 1−α β −1 and instantaneous peak power of received radio signals. Moreover, It If U = min(X ,X ,...X ), then the survival function of U is has received a considerable attention from engineers and physicists N 12 N N for modeling wave propagation, radiation, synthetic aperture radar α β αβ +α −β −∈αβ images, and other related phenomena. There have been many forms G(x, , ) =β [F(x) F(x)] 1 x R, , > 0 (4) 1−α { } for the Rayleigh distribution to provide flexibility for modeling data. Vod 3,4 proposed a powerful extension of the Rayleigh distribution and Similarly, if α >1 and N is a truncated negative binomial 1 studied its properties. Its probability density function (pdf) is given by random variable with parameters and β >0, then α −−2λλ 2 2 +− 1 (x/ θ )2 2θ xe WN = max(X 12 ,X ,...X N) also has the same survival function f (x,θλ , ) = , x,θ > 0, λ >− 1. (1) Γ+(λ 1) given in (4). If α →1 in (4), then G(x,αβ , )→ F(x) . If β = 1 , where θ is the scale parameter, λ is the shape parameter and then this family reduces to the Marshall-Olkin family of distributions. ∞ Γ(a) = ta1−− e t dt is the complete gamma function. Its survival The pdf of survival function given in equation (4) is ∫0 (1−α ) βα β f (x) function is αβ 2 g(x, , ) = β +1 (5) Γ+(λθ 1,(x / ) ) β F(x,θλ , ) = 1−+−=γλ ( 1,(x / θ )2 ), x, θ > 0, λ > 1. (1−+αα ){ F(x) F(x)} Γ+(λ 1) Recently, several authors have used this approach to introduce new (2) distributions. Jayakumar & Sankaran10 defined a generalized uniform x 8 11 where Γ(a,x) = ta1−− e t dt is the incomplete gamma function, distribution using the approach of Nadarajah et al., Babu introduced ∫0 Weibull truncated negative binomial distribution. Further, Jayakumar hence statistical software’s can be used for various values of θ and & Sankaran12 introduced generalized exponential truncated negative λ . In literature, there are many studies based on extensions of this binomial distribution and studied its properties. Also, Jose & Sivadas13 generalized Rayleigh (GR) distribution using different generators. used the family given by equation (4) to introduce the negative Cordeiro et al.,5 derived four-parameter beta-GR distribution, Gomes binomial Marshall-Olkin Rayleigh distribution. et al.,6 proposed the four-parameter Kumaraswamy-GR distribution, and MirMostafaee et al.,7 introduced Marshall-Olkin extended GR The contents of this paper are organized as follows. Section distribution respectively. 2 deals with a generalized Rayleigh-truncated negative binomial distribution and its properties. Section 3 gives a real-life application. In modern era, the literature has suggested several ways of The concluding remarks are given in Section 4. extending well-known distributions to generate a more flexible of distributions. Recently new generator approach introduced by Generalized Rayleigh-truncated negative Nadarajah et al.,8 as a generalization of the family of Marshall-Olkin extended (MOE) distributions by Marshall & Olkin.9 This approach binomial distribution deals with the shape parameter induction in parent (or baseline) In this section, we are focused on generalized Rayleigh truncated distribution to explore tail properties and to improve goodness-of- negative binomial (GR-TNB) distribution introduced by Jiju & 1 fits. Let X12 ,X ,... be a sequence of independent and identically Lishamol. The distribution is derived by using generator approach of Submit Manuscript | http://medcraveonline.com Biom Biostat Int J. 2019;8(4):139‒143. 139 © 2019 Lishamol et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Copyright: A generalized Rayleigh distribution and its application ©2019 Lishamol et al. 140 Nadarajah et al.,8 They have examined various statistical properties function of X follows GR-TNB distribution, it can be expressed as of this distribution including estimation of parameters and have 1 showed that this distribution is more flexible comparing to other − 1 (1−− u)(1α β ) β generalizations of the rayleigh distribution. The survival function of Q(u) =θα QG β +−1 the GR-TNB distribution is given by 1−α α β α 22−β G(x,αβλθ , , , ) =β [ γλ (+ 1,(x / θ ) ) ++ αγλ ( 1,(x / θ ) )] − 1 , x,αβθ , , > 0, λ >− 1 ⋅ −α { } where u is generated from the uniform (0, 1) distribution and QG () 1 is the (standardized) gamma quantile function with shape parameter (6) λ +1 and unit scale parameter (Figure 1). corresponding pdf is Simulation study 2 (1−α ) βαβ 2 θ −2( λλ + 1) x 2 +− 1 e (x/ θ ) In this section, we carry out Monte Carlo simulation study to αβλθ αβθ λ − g(x, , , , ) = β +1 , x, , , > 0, > 1 β 22 assess the performance of the maximum likelihood estimates (MLE). Γ+(λ 1)(1 − α ){ γλ ( + 1,(x / θ ) ) + αγλ ( + 1,(x / θ ) )} The results are obtained from generating 1000 samples from the GR- TNB distribution. For each replication, a random sample of size n = (7) 50, 100 and 200 is drawn from the GR-TNB distribution. The GR- TNB random number generation was performed using the quantile and the hazard rate function (hrf) of the GR-TNB distribution function of GR-TNB distribution and the parameters are estimated by becomes using the method of MLE by using package nlm in R, we get MLEs ˆ ˆ ˆ 2 xx22β , θ and λ for fixed α = 0.2 or α = 0.02 . The evaluation of the (1−αβθ ) 2−−2λλ 2 x 2 +− 1 e (x/ θ ) [γλ (+++ 1, ) αγλ ( 1, )]−1 22performance is based on the bias and the mean squared errors (MSE) h(x,α , β , λθ , ) = θθ Γ+(λ 1)(1 − [ γλ ( + 1,(x / θ )22 ) + αγλ ( + 1,(x / θ ) )]β ) defined as follows: Average bias of the simulated N estimates of R : (8) N 1 ˆ The GR-TNB distribution enfolds some sub models such as MOE ∑(Ri − R) GR distribution, MOE half normal distribution and MOE Reyleigh N i=1 distribution. Figure 1 shows the plots for pdf and hrf for GR-TNB Average Mean square error of the simulated N estimates of R : distribution with various parameter values. As seen in Figure 1, the pdf N and the hrf of the GR-TNB distribution have several different shapes 1 ˆ 2 ∑(Ri − R) according to the values of the parameters. This shows that GR-TNB N i=1 distribution is more flexible than Rayleigh distribution. The quantile Figure 1 Graphs of pdf and hrf the GR-TNB distribution for different values of α , β , λ and θ . where R is the true value of parameters β ,θ and λ and also Data analysis N is the number of replications. The initial values of parameter are In this section, we consider a real data set on breaking stress β = 1.5 ,θ = 0.1 and λ = 0.2 . The results of our simulation study are of carbon fibres of 50 mm length (GPa) to assess the flexibility of summarized. From this table, we can see that the bias and MSE of the the GR-TNB distribution over some well-known generalizations of MLEs converge to zero when the sample size is increased (Table 1). Rayleigh distribution. The data have been previously used by Nichols & Padgettcit.14 The data are as follows: Citation: Lishamol T, Jiju G. A generalized Rayleigh distribution and its application. Biom Biostat Int J. 2019;8(4):139‒143. DOI: 10.15406/bbij.2019.08.00282 Copyright: A generalized Rayleigh distribution and its application ©2019 Lishamol et al. 141 0.39, 0.85, 1.08, 1.25, 1.47, 1.57, 1.61, 1.61, 1.69, 1.80, 1.84 ,1.87, We compare the results of GR-TNB distribution with following 1.89, 2.03, 2.03, 2.05, 2.12, 2.35, 2.41, 2.43, 2.48, 2.50, 2.53, 2.55, four-parameter generalizations of Rayleigh distribution which are 2.55, 2.56, 2.59, 2.67, 2.73, 2.74, 2.79, 2.81, 2.82, 2.85, 2.87, 2.88, generalized by using different generators: 2.93, 2.95, 2.96, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19, 3.22, 3.22, 1.
Load more

Recommended publications
  • The Poisson Burr X Inverse Rayleigh Distribution and Its Applications
    Journal of Data Science,18(1). P. 56 – 77,2020 DOI:10.6339/JDS.202001_18(1).0003 The Poisson Burr X Inverse Rayleigh Distribution And Its Applications Rania H. M. Abdelkhalek* Department of Statistics, Mathematics and Insurance, Benha University, Egypt. ABSTRACT A new flexible extension of the inverse Rayleigh model is proposed and studied. Some of its fundamental statistical properties are derived. We assessed the performance of the maximum likelihood method via a simulation study. The importance of the new model is shown via three applications to real data sets. The new model is much better than other important competitive models. Keywords: Inverse Rayleigh; Poisson; Simulation; Modeling. * [email protected] Rania H. M. Abdelkhalek 57 1. Introduction and physical motivation The well-known inverse Rayleigh (IR) model is considered as a distribution for a life time random variable (r.v.). The IR distribution has many applications in the area of reliability studies. Voda (1972) proved that the distribution of lifetimes of several types of experimental (Exp) units can be approximated by the IR distribution and studied some properties of the maximum likelihood estimation (MLE) of the its parameter. Mukerjee and Saran (1984) studied the failure rate of an IR distribution. Aslam and Jun (2009) introduced a group acceptance sampling plans for truncated life tests based on the IR. Soliman et al. (2010) studied the Bayesian and non- Bayesian estimation of the parameter of the IR model Dey (2012) mentioned that the IR model has also been used as a failure time distribution although the variance and higher order moments does not exist for it.
    [Show full text]
  • New Proposed Length-Biased Weighted Exponential and Rayleigh Distribution with Application
    CORE Metadata, citation and similar papers at core.ac.uk Provided by International Institute for Science, Technology and Education (IISTE): E-Journals Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.7, 2014 New Proposed Length-Biased Weighted Exponential and Rayleigh Distribution with Application Kareema Abed Al-kadim1 Nabeel Ali Hussein 2 1,2.College of Education of Pure Sciences, University of Babylon/Hilla 1.E-mail of the Corresponding author: [email protected] 2.E-mail: [email protected] Abstract The concept of length-biased distribution can be employed in development of proper models for lifetime data. Length-biased distribution is a special case of the more general form known as weighted distribution. In this paper we introduce a new class of length-biased of weighted exponential and Rayleigh distributions(LBW1E1D), (LBWRD).This paper surveys some of the possible uses of Length - biased distribution We study the some of its statistical properties with application of these new distribution. Keywords: length- biased weighted Rayleigh distribution, length- biased weighted exponential distribution, maximum likelihood estimation. 1. Introduction Fisher (1934) [1] introduced the concept of weighted distributions Rao (1965) [2]developed this concept in general terms by in connection with modeling statistical data where the usual practi ce of using standard distributions were not found to be appropriate, in which various situations that can be modeled by weighted distributions, where non-observe ability of some events or damage caused to the original observation resulting in a reduced value, or in a sampling procedure which gives unequal chances to the units in the original That means the recorded observations cannot be considered as a original random sample.
    [Show full text]
  • Study of Generalized Lomax Distribution and Change Point Problem
    STUDY OF GENERALIZED LOMAX DISTRIBUTION AND CHANGE POINT PROBLEM Amani Alghamdi 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 2018 Committee: Arjun K. Gupta, Committee Co-Chair Wei Ning, Committee Co-Chair Jane Chang, Graduate Faculty Representative John Chen Copyright c 2018 Amani Alghamdi All rights reserved iii ABSTRACT Arjun K. Gupta and Wei Ning, Committee Co-Chair Generalizations of univariate distributions are often of interest to serve for real life phenomena. These generalized distributions are very useful in many fields such as medicine, physics, engineer- ing and biology. Lomax distribution (Pareto-II) is one of the well known univariate distributions that is considered as an alternative to the exponential, gamma, and Weibull distributions for heavy tailed data. However, this distribution does not grant great flexibility in modeling data. In this dissertation, we introduce a generalization of the Lomax distribution called Rayleigh Lo- max (RL) distribution using the form obtained by El-Bassiouny et al. (2015). This distribution provides great fit in modeling wide range of real data sets. It is a very flexible distribution that is related to some of the useful univariate distributions such as exponential, Weibull and Rayleigh dis- tributions. Moreover, this new distribution can also be transformed to a lifetime distribution which is applicable in many situations. For example, we obtain the inverse estimation and confidence intervals in the case of progressively Type-II right censored situation. We also apply Schwartz information approach (SIC) and modified information approach (MIC) to detect the changes in parameters of the RL distribution.
    [Show full text]
  • A New Generalization of the Generalized Inverse Rayleigh Distribution with Applications
    S S symmetry Article A New Generalization of the Generalized Inverse Rayleigh Distribution with Applications Rana Ali Bakoban * and Ashwaq Mohammad Al-Shehri Department of Statistics, College of Science, University of Jeddah, Jeddah 22254, Saudi Arabia; [email protected] * Correspondence: [email protected] Abstract: In this article, a new four-parameter lifetime model called the beta generalized inverse Rayleigh distribution (BGIRD) is defined and studied. Mixture representation of this model is derived. Curve’s behavior of probability density function, reliability function, and hazard function are studied. Next, we derived the quantile function, median, mode, moments, harmonic mean, skewness, and kurtosis. In addition, the order statistics and the mean deviations about the mean and median are found. Other important properties including entropy (Rényi and Shannon), which is a measure of the uncertainty for this distribution, are also investigated. Maximum likelihood estimation is adopted to the model. A simulation study is conducted to estimate the parameters. Four real-life data sets from difference fields were applied on this model. In addition, a comparison between the new model and some competitive models is done via information criteria. Our model shows the best fitting for the real data. Keywords: beta generalized inverse Rayleigh distribution; statistical properties; mean deviations; quantile; Rényi entropy; Shannon entropy; incomplete beta function; Montecarlo Simulation Citation: Bakoban, R.A.; Al-Shehri, MSC: Primary 62E10; Secondary 60E05 A.M. A New Generalization of the Generalized Inverse Rayleigh Distribution with Applications. Symmetry 2021, 13, 711. https:// 1. Introduction doi.org/10.3390/sym13040711 Modeling and analysis of lifetime phenomena are important aspects of statistical Academic Editor: Sergei D.
    [Show full text]
  • Location-Scale Distributions
    Location–Scale Distributions Linear Estimation and Probability Plotting Using MATLAB Horst Rinne Copyright: Prof. em. Dr. Horst Rinne Department of Economics and Management Science Justus–Liebig–University, Giessen, Germany Contents Preface VII List of Figures IX List of Tables XII 1 The family of location–scale distributions 1 1.1 Properties of location–scale distributions . 1 1.2 Genuine location–scale distributions — A short listing . 5 1.3 Distributions transformable to location–scale type . 11 2 Order statistics 18 2.1 Distributional concepts . 18 2.2 Moments of order statistics . 21 2.2.1 Definitions and basic formulas . 21 2.2.2 Identities, recurrence relations and approximations . 26 2.3 Functions of order statistics . 32 3 Statistical graphics 36 3.1 Some historical remarks . 36 3.2 The role of graphical methods in statistics . 38 3.2.1 Graphical versus numerical techniques . 38 3.2.2 Manipulation with graphs and graphical perception . 39 3.2.3 Graphical displays in statistics . 41 3.3 Distribution assessment by graphs . 43 3.3.1 PP–plots and QQ–plots . 43 3.3.2 Probability paper and plotting positions . 47 3.3.3 Hazard plot . 54 3.3.4 TTT–plot . 56 4 Linear estimation — Theory and methods 59 4.1 Types of sampling data . 59 IV Contents 4.2 Estimators based on moments of order statistics . 63 4.2.1 GLS estimators . 64 4.2.1.1 GLS for a general location–scale distribution . 65 4.2.1.2 GLS for a symmetric location–scale distribution . 71 4.2.1.3 GLS and censored samples .
    [Show full text]
  • Slashed Rayleigh Distribution
    Revista Colombiana de Estadística January 2015, Volume 38, Issue 1, pp. 31 to 44 DOI: http://dx.doi.org/10.15446/rce.v38n1.48800 Slashed Rayleigh Distribution Distribución Slashed Rayleigh Yuri A. Iriarte1;a, Héctor W. Gómez2;b, Héctor Varela2;c, Heleno Bolfarine3;d 1Instituto Tecnológico, Universidad de Atacama, Copiapó, Chile 2Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Antofagasta, Chile 3Departamento de Estatística, Instituto de Matemática y Estatística, Universidad de Sao Paulo, Sao Paulo, Brasil Abstract In this article we study a subfamily of the slashed-Weibull family. This subfamily can be seen as an extension of the Rayleigh distribution with more flexibility in terms of the kurtosis of distribution. This special feature makes the extension suitable for fitting atypical observations. It arises as the ratio of two independent random variables, the one in the numerator being a Rayleigh distribution and a power of the uniform distribution in the denominator. We study some probability properties, discuss maximum likelihood estimation and present real data applications indicating that the slashed-Rayleigh distribution can improve the ordinary Rayleigh distribution in fitting real data. Key words: Kurtosis, Rayleigh Distribution, Slashed-elliptical Distribu- tions, Slashed-Rayleigh Distribution, Slashed-Weibull Distribution, Weibull Distribution. Resumen En este artículo estudiamos una subfamilia de la familia slashed-Weibull. Esta subfamilia puede ser vista como una extensión de la distribución Ray- leigh con más flexibilidad en cuanto a la kurtosis de la distribución. Esta particularidad hace que la extensión sea adecuada para ajustar observa- ciones atípicas. Esto surge como la razón de dos variables aleatorias in- dependientes, una en el numerador siendo una distribución Rayleigh y una aLecturer.
    [Show full text]
  • Hand-Book on STATISTICAL DISTRIBUTIONS for Experimentalists
    Internal Report SUF–PFY/96–01 Stockholm, 11 December 1996 1st revision, 31 October 1998 last modification 10 September 2007 Hand-book on STATISTICAL DISTRIBUTIONS for experimentalists by Christian Walck Particle Physics Group Fysikum University of Stockholm (e-mail: [email protected]) Contents 1 Introduction 1 1.1 Random Number Generation .............................. 1 2 Probability Density Functions 3 2.1 Introduction ........................................ 3 2.2 Moments ......................................... 3 2.2.1 Errors of Moments ................................ 4 2.3 Characteristic Function ................................. 4 2.4 Probability Generating Function ............................ 5 2.5 Cumulants ......................................... 6 2.6 Random Number Generation .............................. 7 2.6.1 Cumulative Technique .............................. 7 2.6.2 Accept-Reject technique ............................. 7 2.6.3 Composition Techniques ............................. 8 2.7 Multivariate Distributions ................................ 9 2.7.1 Multivariate Moments .............................. 9 2.7.2 Errors of Bivariate Moments .......................... 9 2.7.3 Joint Characteristic Function .......................... 10 2.7.4 Random Number Generation .......................... 11 3 Bernoulli Distribution 12 3.1 Introduction ........................................ 12 3.2 Relation to Other Distributions ............................. 12 4 Beta distribution 13 4.1 Introduction .......................................
    [Show full text]
  • Journal of Science Log-Gamma
    Research Article GU J Sci 31(3): 967-983 (2018) Gazi University Journal of Science http://dergipark.gov.tr/gujs Log-Gamma - Rayleigh Distribution: Properties and Applications Samir ASHOUR 1, Darwish DARWISH 1, Ahmad SAAD 1,* 1 Institute of Statistical Studies & Research, Cairo University, 5 Ahmed Zowil St., Orman, Giza, Egypt Article Info Abstract In this paper, for the first time we defined and studied a new two parameter lifetime model by Received: 14/08/2017 using the T-X method, called the Log-Gamma Rayleigh distribution. This distribution can be Accepted: 20/04/2018 considered as a new generalization of the Gamma distribution and the Rayleigh distribution. We obtain some of its mathematical properties. Some structural properties of the new distribution are studied. Maximum likelihood estimation method is used for estimating the model Keywords parameters. An application to real data set is given to show the flexibility and potentiality of the T-X method new model. Gamma distribution Rayleigh distribution Quantile function Hazard function Moments Maximum likelihood estimation 1. INTRODUCTION Eugene et al. [6] for the first time introduced the beta-generated family of distributions. They noted that the probability density function pdf of the beta random variable and the cumulative distribution function CDF of any distribution are between 0 and 1. The beta-generated random variable X is defined with the following CDF and pdf Fx() G()() x b t dt (1) 0 and 1 g() x f () x F11 ()[1 x F ()] x B(,) where b(t) is the pdf of the beta random variable with parameters and , F(x) and f(x) are the CDF and the pdf of any random variable.
    [Show full text]
  • Slashed Exponentiated Rayleigh Distribution
    Revista Colombiana de Estadística July 2015, Volume 38, Issue 2, pp. 453 to 466 DOI: http://dx.doi.org/10.15446/rce.v38n2.51673 Slashed Exponentiated Rayleigh Distribution Distribución recortada Rayleigh exponenciada Hugo S. Salinas1;a, Yuri A. Iriarte2;b, Heleno Bolfarine3;c 1Departamento de Matemática, Facultad de Ingeniería, Universidad de Atacama, Copiapó, Chile 2Instituto Tecnológico, Universidad de Atacama, Copiapó, Chile 3Departamento de Estatística, IME, Universidad de Sao Paulo, Sao Paulo, Brasil Abstract In this paper we introduce a new distribution for modeling positive data with high kurtosis. This distribution can be seen as an extension of the exponentiated Rayleigh distribution. This extension builds on the quotient of two independent random variables, one exponentiated Rayleigh in the numerator and Beta(q; 1) in the denominator with q > 0. It is called the slashed exponentiated Rayleigh random variable. There is evidence that the distribution of this new variable can be more flexible in terms of modeling the kurtosis regarding the exponentiated Rayleigh distribution. The properties of this distribution are studied and the parameter estimates are calculated using the maximum likelihood method. An application with real data re- veals good performance of this new distribution. Key words: Exponentiated Rayleigh Distribution, Kurtosis, Maximum Likelihood, Rayleigh Distribution, Slash Distribution. Resumen En este trabajo presentamos una nueva distribución para modelizar datos positivos con alta curtosis. Esta distribución puede ser vista como una exten- sión de la distribución Rayleigh exponenciada. Esta extensión se construye en base al cuociente de dos variables aleatorias independientes, una Raileigh exponenciada en el numerador y una Beta(q; 1) en el denominador con q > 0.
    [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]
  • On the Arcsecant Hyperbolic Normal Distribution. Properties, Quantile Regression Modeling and Applications
    S S symmetry Article On the Arcsecant Hyperbolic Normal Distribution. Properties, Quantile Regression Modeling and Applications Mustafa Ç. Korkmaz 1 , Christophe Chesneau 2,* and Zehra Sedef Korkmaz 3 1 Department of Measurement and Evaluation, Artvin Çoruh University, City Campus, Artvin 08000, Turkey; [email protected] 2 Laboratoire de Mathématiques Nicolas Oresme, University of Caen-Normandie, 14032 Caen, France 3 Department of Curriculum and Instruction Program, Artvin Çoruh University, City Campus, Artvin 08000, Turkey; [email protected] * Correspondence: [email protected] Abstract: This work proposes a new distribution defined on the unit interval. It is obtained by a novel transformation of a normal random variable involving the hyperbolic secant function and its inverse. The use of such a function in distribution theory has not received much attention in the literature, and may be of interest for theoretical and practical purposes. Basic statistical properties of the newly defined distribution are derived, including moments, skewness, kurtosis and order statistics. For the related model, the parametric estimation is examined through different methods. We assess the performance of the obtained estimates by two complementary simulation studies. Also, the quantile regression model based on the proposed distribution is introduced. Applications to three real datasets show that the proposed models are quite competitive in comparison to well-established models. Keywords: bounded distribution; unit hyperbolic normal distribution; hyperbolic secant function; normal distribution; point estimates; quantile regression; better life index; dyslexia; IQ; reading accu- racy modeling Citation: Korkmaz, M.Ç.; Chesneau, C.; Korkmaz, Z.S. On the Arcsecant Hyperbolic Normal Distribution. 1. Introduction Properties, Quantile Regression Over the past twenty years, many statisticians and researchers have focused on propos- Modeling and Applications.
    [Show full text]
  • Weighted Inverse Rayleigh Distribution
    International Journal of Statistics and Systems ISSN 0973-2675 Volume 12, Number 1 (2017), pp. 119-137 © Research India Publications http://www.ripublication.com Weighted Inverse Rayleigh Distribution Kawsar Fatima and S.P Ahmad Department of Statistics, University of Kashmir, Srinagar, India. Corresponding author Abstract In this paper, we have introduced weighted inverse Rayleigh (WIR) distribution and investigated its different statistical properties. Expressions for the Mode and entropy have also been derived. In addition, it also contains some special cases that are well known. Moreover, we apply the maximum likelihood method to estimate the parameter , and applications to two real data sets show the superiority of this new distribution by comparing the fitness with its special cases. Keywords: Inverse Rayleigh distribution, weighted distribution, Reliability Analysis, Entropy, Maximum likelihood estimation, Real life data sets. 1. INTRODUCTION The IR distribution was proposed by Voda (1972). He studies some properties of the MLE of the scale parameter of inverse Rayleigh distribution which is also being used in lifetime experiments. If X has IR distribution, its probability density function (pdf) takes the following form: 2 2 g() x ex ; x 0, 0 )1.1( x3 The corresponding cumulative distribution function is 2 G( x ) ex ; x 0, 0 )2.1( Where x 0 the scale parameter 0 The kth moment of the IR distribution is given as the following: 120 Kawsar Fatima and S.P Ahmad 2 k k 2 x Eg () x x e dx. 0 x3 1 1 2 Making the substitution y , dx dy , so that x 2/1 ,we obtain x 2 x3 y y E() xk y(1k / 2) 1 e dy.
    [Show full text]