Generalized Additive Models for Location, Scale and Shape (GAMLSS) were introduced by Rigby and Stasinopoulos (2005). GAMLSS is a general framework for univariate regression type statistical problems. In GAMLSS the exponential family distribution assumption used in Generalized Linear Model (GLM) and Generalized Additive Model (GAM), (see Nelder andWedderburn, 1972 andHastie andTibshirani, 1990, respectively) is relaxed and replaced by a very general distribution family including highly skew and kurtotic discrete and continuous distributions. The systematic part of the model is expanded to allow modelling not only the mean (or location) but other parameters of the distribution of y as linear parametric, non-linear parametric or additive non-parametric functions of explanatory variables and/or random effects terms. Maximum (penalized) likelihood estimation is used to fit the models. The algorithms used to fit the model are described in detail in Rigby and Stasinopoulos (2005). For medium to large size data, GAMLSS allow flexibility in statistical modelling far beyond other currently available methods. The most important application of GAMLSS up to now is its use by the Department of Nutrition for Health and Development of the World Health Organization to construct the worldwide standard growth curves. The range of possible applications for GAMLSS though is a lot more general and examples will be given of its usefulness in modelling medical and insurance data. In the talk we will describe the GAMLSS model, the variety of different (two, three and four) distributions that are implemented within the GAMLSS package and the variety of different additive terms that can be used in the current implementation. New distributions and new additive terms can be easily added to the package. We shall also discuss the difference of GAMLSS with other available packages in R such as gam and mgcv. More recent work, for example the inclusion of non linear parameter components as additive terms and the inclusion of truncated distributions and censored data within the GAMLSS family, will be also discussed.

GAMLSS is a general framework for fitting regression type models where the distribution of the response variable does not have to belong to the exponential family and includes highly skew and kurtotic continuous and discrete distribution. GAMLSS allows all the parameters of the distribution of the response variable to be modelled as linear/non-linear or smooth functions of the explanatory variables. This paper starts by defining the statistical framework of GAMLSS, then describes the current implementation of GAMLSS in R and finally gives four different data examples to demonstrate how GAMLSS can be used for statistical modelling.

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The Box-Cox power exponential (BCPE) distribution, developed in this paper, provides a model for a dependent variable Y exhibiting both skewness and kurtosis (leptokurtosis or platykurtosis). The distribution is defined by a power transformation Y(nu) having a shifted and scaled (truncated) standard power exponential distribution with parameter tau. The distribution has four parameters and is denoted BCPE (mu,sigma,nu,tau). The parameters, mu, sigma, nu and tau, may be interpreted as relating to location (median), scale (approximate coefficient of variation), skewness (transformation to symmetry) and kurtosis (power exponential parameter), respectively. Smooth centile curves are obtained by modelling each of the four parameters of the distribution as a smooth non-parametric function of an explanatory variable. A Fisher scoring algorithm is used to fit the non-parametric model by maximizing a penalized likelihood. The first and expected second and cross derivatives of the likelihood, with respect to mu, sigma, nu and tau, required for the algorithm, are provided. The centiles of the BCPE distribution are easy to calculate, so it is highly suited to centile estimation. This application of the BCPE distribution to smooth centile estimation provides a generalization of the LMS method of the centile estimation to data exhibiting kurtosis (as well as skewness) different from that of a normal distribution and is named here the LMSP method of centile estimation. The LMSP method of centile estimation is applied to modelling the body mass index of Dutch males against age.

The Box-Cox t (BCT) distribution is presented as a model for a dependent variable Y exhibiting both skewness and leptokurtosis. The distribution is defined by a power transformation Y v having a shifted and scaled (truncated) t distribution with degrees of freedom parameter τ. The distribution has four parameters and is denoted by BCT( μ, σ,ν, τ). The parameters μ, σ,ν and τ may be interpreted as relating to location (median), scale (centile-based coefficient of variation), skewness (power transformation to symmetry) and kurtosis (degrees of freedom), respectively. The generalized additive model for location, scale and shape (GAMLSS) is extended to allow each of the parameters of the distribution to be modelled as linear and /or non-linear parametric and/or smooth non-parametric functions of explanatory variables. A Fisher scoring algorithm is used to fit the model by maximizing a (penalized) likelihood. The first and expected second and cross derivatives of the likelihood with respect to μ, σ,ν and τ, required for the algorithm, are provided. The use of the BCT distribution is illustrated by two data applications.