M.: Using the Box-Cox t distribution in GAMLSS to model skewness and kurtosis

Abstract: 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 Yν 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. Key words: centile estimation; cubic smoothing splines; generalized additive models; LMS method; non-linear model; penalized likelihood; reference curves; regression quantile