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|1 pages

Parametric distributions and the GAMLSS family of distributions

chapter 1|18 pages

Types of distributions

chapter 2|17 pages

Properties of distributions

chapter 3|21 pages

The GAMLSS family of distributions

chapter 4|25 pages

Continuous distributions on (–∞, ∞)

chapter 5|34 pages

Continuous distributions on (0, ∞)

chapter 6|12 pages

Continuous distributions on (0, 1)

chapter 7|40 pages

Discrete count distributions

chapter 8|10 pages

Binomial type distributions

chapter 9|30 pages

Mixed distributions

part II|2 pages

Advanced topics

chapter 10|14 pages

Statistical inference

chapter 11|26 pages

Maximum likelihood estimation

chapter 13|22 pages

Methods of generating distributions

chapter 14|12 pages

Discussion of skewness

chapter 15|12 pages

Discussion of kurtosis

chapter 17|36 pages

Heaviness of tails of distributions

part III|1 pages

Reference guide

chapter 18|56 pages

Continuous distributions on (−∞, ∞)

chapter 19|34 pages

Continuous distributions on (0, ∞)

chapter 20|4 pages

Mixed distributions on [0, ∞)

chapter 21|12 pages

Continuous and mixed distributions on [0,1]

chapter 22|50 pages

Discrete count distributions