1st Edition

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

    588 Pages 191 B/W Illustrations
    by Chapman & Hall

    588 Pages 191 B/W Illustrations
    by Chapman & Hall

    588 Pages 191 B/W Illustrations
    by Chapman & Hall

    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.

    Part I: Parametric distributions and the GAMLSS family of distributions

    Types of distributions

    Properties of distributions

    The GAMLSS Family of Distributions

    Continuous distributions on (-8,8)

    Continuous distributions on (0, 8)

    Continuous distributions on (0, 1)

    Discrete count distributions

    Binomial type distributions

    Mixed distributions

    Part II: Advanced Topics

    Statistical inference

    Maximum likelihood estimation

    Robustness of parameter estimation to outlier contamination

    Methods of generating distributions

    Discussion of skewness

    Discussion of kurtosis

    Skewness and kurtosis comparisons of continuous distributions

    Heaviness of tails of distributions

    Part III: Reference Guide

    Continuous distributions on (-8,8)

    Continuous distributions on (0, 8)

    Mixed distributions on [0 to 8)

    Continuous and mixed distributions on [0, 1]

    Discrete count distributions

    Binomial type distributions and multinomial distributions



    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.

    "…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