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 . 74 4.2.2 Approximations to GLS . 76 4.2.2.1 B approximated by a diagonal matrix . 76 4.2.2.2 B approximated by an identity matrix . 82 4.2.2.3 BLOM’s estimator . 85 4.2.2.4 DOWNTON’s estimator . 90 4.2.3 Approximations to GLS with approximated moments of order statistics . 91 4.3 Estimators based on empirical percentiles . 97 4.3.1 Grouped data . 97 4.3.2 Randomly and multiply censored data . 101 4.4 Goodness–of–fit . 104 5 Probability plotting and linear estimation — Applications 105 5.1 What will be given for each distribution? . 105 5.2 The case of genuine location–scale distributions . 106 5.2.1 Arc–sine distribution — X ∼ AS(a, b) . 106 5.2.2 CAUCHY distribution — X ∼ CA(a, b) . 109 5.2.3 Cosine distributions . 113 5.2.3.1 Ordinary cosine distribution — X ∼ COO(a, b) . 114 5.2.3.2 Raised cosine distribution — X ∼ COR(a, b) . 116 5.2.4 Exponential distribution — X ∼ EX(a, b) . 118 5.2.5 Extreme value distributions of type I . 122 5.2.5.1 Maximum distribution (GUMBEL distribution) — X ∼ EMX1(a, b) . 124 5.2.5.2 Minimum distribution (Log–WEIBULL)— X ∼ EMN1(a, b) . 128 5.2.6 Half–distributions . 130 5.2.6.1 Half–CAUCHY distribution — X ∼ HC(a, b) . 131 Contents V 5.2.6.2 Half–logistic distribution — X ∼ HL(a, b) . 133 5.2.6.3 Half–normal distribution — X ∼ HN(a, b) . 140 5.2.7 Hyperbolic secant distribution — X ∼ HS(a, b) . 144 5.2.8 LAPLACE distribution — X ∼ LA(a, b) . 147 5.2.9 Logistic distribution — X ∼ LO(a, b) . 152 5.2.10 MAXWELL–BOLTZMANN distribution — X ∼ MB(a, b) . 157 5.2.11 Normal distribution — X ∼ NO(a, b) . 161 5.2.12 Parabolic distributions of order 2 . 167 5.2.12.1 U–shaped parabolic distribution — X ∼ PAU(a, b) . 167 5.2.12.2 Inverted U–shaped parabolic distribution — X ∼ PAI(a, b) . 169 5.2.13 RAYLEIGH distribution — X ∼ RA(a, b) . 171 5.2.14 Reflected exponential distribution — X ∼ RE(a, b) . 175 5.2.15 Semi–elliptical distribution — X ∼ SE(a, b) . 178 5.2.16 TEISSIER distribution — X ∼ TE(a, b) . 181 5.2.17 Triangular distributions . 183 5.2.17.1 Symmetric version — X ∼ TS(a, b) . 184 5.2.17.2 Right–angled and negatively skew version — X ∼ TN(a, b) . 187 5.2.17.3 Right–angled and positively skew version — X ∼ TP (a, b) . 190 5.2.18 Uniform or rectangular distribution — X ∼ UN(a, b) . 192 5.2.19 U–shaped beta distribution — X ∼ UB(a, b) . 197 5.2.20 V–shaped distribution — X ∼ VS(a, b) . 200 5.3 The case of ln–transformable distributions . 203 5.3.1 Estimation of the shift parameter . 203 5.3.2 Extreme value distributions of types II and III . 206 5.3.2.1 Type II maximum or inverse WEIBULL distribution — X ∼ EMX2(a, b, c) . 207 5.3.2.2 Type III maximum or reflected WEIBULL distribution — X ∼ EMX3(a, b, c) . 209 5.3.2.3 Type II minimum or FRECHET´ distribution — X ∼ EMN2(a, b, c) . 212 VI Contents 5.3.2.4 Type III minimum or WEIBULL distribution — X ∼ EMN3(a, b, c) . 214 5.3.3 Lognormal distributions . 219 5.3.3.1 Lognormal distribution with lower threshold — X ∼ LNL(a, ea,eb) . 219 5.3.3.2 Lognormal distribution with upper threshold — X ∼ LNU(a, ea,eb) . 223 5.3.4 PARETO distribution — X ∼ PA(a, b, c) . 225 5.3.5 Power–function distribution — X ∼ PO(a, b, c) . 229 6 The program LEPP 232 6.1 Structure of LEPP . 232 6.2 Input to LEPP . 234 6.3 Directions for using LEPP . 235 Mathematical and statistical notations 237 Bibliography 245 Author Index 255 Subject Index 257 Preface VII Preface Statistical distributions can be grouped into families or systems. Such groupings are de- scribed in JOHNSON/KOTZ/KEMP (1992, Chapter 2), JOHNSON/KOTZ/BALAKRISHNAN (1994, Chapter 12) or PATEL(KAPADIA/OWEN (1976, Chapter 4). The most popular families are those of PEARSON,JOHNSON and BURR, the exponential, the stable and the infinitely divisible distributions or those with a monotone likelihood ratio or with a mono- tone failure rate. All these categories have attracted the attention of statisticians and they are fully discussed in the statistical literature. But there is one family, the location–scale family, which hitherto has not been discussed in greater detail. To my knowledge this book is the first comprehensive monograph on one–dimensional continuous location–scale dis- tributions and it is organized as follows. Chapter 1 goes into the details of location–scale distributions and gives their properties along with a short list of those distributions which are genuinely location–scale and which — after a suitable transformation of its variable — become member of this class. We will only consider the ln–transformation. Location–scale distributions easily lend them- selves to an assessment by graphical methods. On a suitably chosen probability paper the cumulative distribution function of the universe gives a straight line and the cumulative distribution of a sample only deviates by chance from a straight line. Thus we can realize an informal goodness–of–fit test. When we fit the straight line free–hand or by eye we may read off the location and scale parameters as percentiles. Another and objective method is to find the straight line on probability paper by a least–squares technique. Then, the estimates of the location and scale parameters will be the parameters of that straight line. Because probability plotting heavily relies on ordered observations Chapter 2 gives — as a prerequisite — a detailed representation of the theory of order statistics. Probability plotting is a graphical assessment of statistical distributions. To see how this kind of graphics fits into the framework of statistical graphics we have written Chapter 3. A first core chapter is Chapter 4. It presents the theory and the methods of linear estimating the location and scale parameters. The methods to be implemented depend on the type of sample, i.e. grouped or non–grouped, censored or uncensored, the type of censoring and also whether the moments of the order statistics are easily calculable or are readily available in tabulated form or not. In the latter case we will give various approximations to the optimal method of general least–squares. Applications of the exact or approximate linear estimation procedures to a great number of location–scale distributions will be presented in Chapter 5, which is central to this book. For each of 35 distributions we give a warrant of arrest enumerating the characteristics, the underlying stochastic model and the fields of application together with the pertinent probability paper and the estimators of the location parameter and the scale parameter. Distributions which have to be transformed to location–scale type sometimes have a third parameter which has to be pre–estimated before applying probability plotting and the lin- ear estimation procedure. We will show how to estimate this third parameter. VIII Preface The calculations and graphics of Chapter 5 have been done using MATLAB,1 Version 7.4 (R2007a). The accompanying CD contains the MATLAB script M–file LEPP and all the function–files to be used by the reader when he wants to do inference on location–scale distributions. Hints how to handle the menu–driven program LEPP and how to organize the data input will be given in Chapter 6 as well as in the comments in the files on the CD. Dr. HORST RINNE, Professor Emeritus of Statistics and Econometrics Department of Economics and Management Science Justus–Liebig–University, Giessen, Germany 1 MATLAB R is a registered trade–mark of ‘The MathWorks, Inc.’ List of Figures 2/1 Structure of the variance–covariance matrix of order statistics from a distri- bution symmetric around zero . 24 3/1 Explanation of the QQ–plot and the PP–plot .