Distributions for Modeling Location, Scale, and Shape: Using GAMLSS in R

by Robert A. Rigby, Mikis D. Stasinopoulos, Gillian Z. Heller, Fernanda De Bastiani

Overview

This is a book about statistical distributions, their properties, and their application to modelling the dependence of the location, scale, and shape of the distribution of a response variable on explanatory variables. It will be especially useful to applied statisticians and data scientists in a wide range of application areas, and also to those interested in the theoretical properties of distributions. This book follows the earlier book ‘Flexible Regression and Smoothing: Using GAMLSS in R’, [Stasinopoulos et al., 2017], which focused on the GAMLSS model and software. GAMLSS (the Generalized Additive Model for Location, Scale, and Shape, [Rigby and Stasinopoulos, 2005]), is a regression framework in which the response variable can have any parametric distribution and all the distribution parameters can be modelled as linear or smooth functions of explanatory variables. The current book focuses on distributions and their application.

Key features:

• Describes over 100 distributions, (implemented in the GAMLSS packages in R), including continuous, discrete and mixed distributions.

• Comprehensive summary tables of the properties of the distributions.

• Discusses properties of distributions, including skewness, kurtosis, robustness and an important classification of tail heaviness.

• Includes mixed distributions which are continuous distributions with additional specific values with point probabilities.

• Includes many real data examples, with R code integrated in the text for ease of understanding and replication.

• Supplemented by the gamlss website.

This book will be useful for applied statisticians and data scientists in selecting a distribution for a univariate response variable and modelling its dependence on explanatory variables, and to those interested in the properties of distributions.

Product Details

ISBN-13:	9781032089423
Publisher:	CRC Press
Publication date:	06/30/2021
Series:	Chapman & Hall/CRC The R Series
Pages:	588
Product dimensions:	7.00(w) x 10.00(h) x (d)

About the Author

Robert Rigby was researching in Statistics at London Metropolitan University for over 30 years specializing in distributions and advanced regression and smoothing models (for supervised learning). He is one of the two original developers of GAMLSS models. He is currently a freelance consultant.

Mikis Stasinopoulos is a statistician. He has a considerable experience in applied statistics and he is one of the two creators of GAMLSS. He worked as the director of STORM, the statistics and mathematics research centre of London Metropolitan University and now he is working as an independent statistical consultant.

Gillian Heller is Professor of Statistics at Macquarie University, Sydney. Her research interests are mainly in flexible regression models for heavy-tailed count data, with applications in biostatistics and insurance.

Fernanda De Bastiani is a permanent lecturer in the Statistics Department at Universidade Federal de Pernambuco, Brazil. Her research interests are mainly in flexible regression models, spatial data analysis and influential diagnostics in regression models.

Table of Contents

Part I: Parametric distributions and the GAMLSS family of distributions Chapter 1 Types of distributions Chapter 2 Properties of distributions Chapter 3 The GAMLSS Family of Distributions Chapter 4 Continuous distributions on (−1,1) Chapter 5 Continuous distributions on (0, ∞) Chapter 6 Continuous distributions on (0, 1) Chapter 7 Discrete distributions for count data Chapter 8 Binomial type distributions Chapter 9 Mixed distributions Part II: Advanced Topics Chapter 10 Maximum likelihood Chapter 11 Robustness of parameter estimation to outlier Chapter 12 Methods of generating Chapter 13 Discussion of skewness Chapter 14 Discussion of Kurtosis Chapter 15 Skewness and kurtosis comparisons of continuous distributions Chapter 16 Heaviness of tails of continuous Part III: Reference Guide Chapter 17 Continuous distributions on (−∞,∞) Chapter 18 Continuous distributions on (0, ∞) Chapter 19 Mixed distributions on 0 to ∞, including 0 Chapter 20 Continuous and mixed distributions on [0, 1] Chapter 21 Count data Chapter 22 Count data distributions 23 Binomial type distributions and multinomial distributions

Editorial Reviews

"…focuses on all probability distributions that can be used in GAMLSS modelling…a distributional regression framework making inroads in different fields due to its flexibility…GAMLSS’s power rests on its capability to apply smoothers to numeric and categorical covariates and model numeric response variables via probability distributions other than the usual exponential family…including continuous distributions, ...discrete distributions and mixtures of continuous and discrete (mixed) distributions...This last type of distribution, although commonplace in practice, is rather ignored by applied researchers…The book has three parts. Part II ("Advanced topics") contains eight Chapters, and is perhaps the most exciting section. It deals with topics that link the GAMLSS framework and probability distributions to ‘hot’ topics in statistical learning."
~ Fernando Marmolejo-Ramos, Raydonal Ospina, and Freddy Hernández-Barajas, respectively University of South Australia, Universidade Federal de Pernambuco, and Universidad Nacional de Colombia sede Medellín, appeared in Australian and New Zealand Journal of Statistics, September 2022

From the Publisher