Lognormal Distributions

Theory and Applications

Crow, Edwin L. Crow, Kunio Shimizu

• Publisher: CRC Press
• Year: 2018
• eBook ISBN: 9781351434683
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Presenting the first comprehensive review of the subject's theory and applications inmore than 15 years, this outstanding reference encompasses the most-up-to-date advancesin lognormal distributions in thorough, detailed contributions by specialists in statistics,business and economics , industry, biology , ecology, geology, and meteorology.Lognormal Distributions describes the theory and methods of point and intervalestimation as well as the testing of hypotheses clearly and precisely from a modemviewpoint-not only for the basic two-parameter lognormal distribution but also for itsgeneralizations, including three parameters, truncated distributions, delta-lognormaldistributions, and two or more dimensions.Featuring over 600 references plus author and subject indexes, this volume rev iews thesubject's history .. . gives explicit formulas for minimum variance unbiased estimates ofparameters and their variances ... provides optimal tests of hypotheses and confidenceinterval procedures for various functions of the parameters in the two-parameter model. .. and discusses practical methods of analysis for truncated, censored, or groupedsamples.

1

History, Genesis, and Properties

KUNIO SHIMIZU Department of Information Sciences, Faculty of Science and Technology, Science University of Tokyo, Noda City, Chiba, Japan

L. CROW EDWIN Institute for Telecommunication Sciences, National Telecommunications and Information Administration, U.S. Department of Commerce, Boulder, Colorado

1. INTRODUCTION

The lognormal distribution (with two parameters) may be defined as the distribution of a random variable whose logarithm is normally distributed. Such a variable is necessarily positive. Since many variables in real life, from the sizes of organisms and the numbers of species in biology to rainfalls in meteorology and sizes of incomes in economics, are inherently positive, the lognormal distribution has been widely applied in an empirical way for fitting data. In addition, it has been derived theoretically from qualitative assumptions; Gibrat (1930, 1931) did this in 1930, calling it the law of proportionate effect, but Kapteyn (1903) had described a machine that was the mechanical equivalent. Kolmogoroff (1941) derived the distribution as the asymptotic result of an iterative process of successive breakage of a particle into two randomly sized particles.

Thus there is a theoretical basis as well as empirical application of lognormal distributions, but why is there much to say about them if the data analysis can be referred to the intensively studied normal distribution by taking the logarithm? There are several reasons:

(1) The parameter estimates resulting from the inverse transformation are biased.

(2) The two-parameter distribution is often not a sufficient description; a third parameter, the threshold or location parameter, is needed, for example, for the distribution of ages at first marriage.

(3) The distribution may be censored or tr...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Contributors
• 1. History, Genesis, and Properties
• 2. Point Estimation
• 3. Hypothesis Tests and Interval Estimates
• 4. Three-Parameter Estimation
• 5. Censored, Truncated, and Grouped Estimation
• 6. Bayesian Estimation
• 7. Poisson-Lognormal Distributions
• 8. The Lognormal as Event-Time Distribution
• 9. Applications in Economics and Business
• 10. Applications in Industry
• 11. Applications in Biology: Simple Growth Models
• 12. Applications in Ecology
• 13. Applications in Atmospheric Sciences
• 14. Applications in Geology
• Author Index
• Subject Index