Semiparametric additive models under symmetric distributions

In this paper we discuss estimation and diagnostic procedures in semiparametric additive models with symmetric errors in order to permit distributions with heavier and lighter tails than the normal ones, such as Student-t, Pearson VII, power exponential, logistics I and II, and contaminated normal, among others. Such models belong to the general class of statistical models GAMLSS proposed by Rigby and Stasinopoulos (Appl. Stat. 54:507–554, 2005). A back-fitting algorithm to attain the maximum penalized likelihood estimates (MPLEs) by using natural cubic smoothing splines is presented. In particular, the score functions and Fisher information matrices for the parameters of interest are expressed in a similar notation of that used in parametric symmetric models. Sufficient conditions on the existence of the MPLEs are presented as well as some inferential results and discussions on degrees of freedom and smoothing parameter estimation. Diagnostic quantities such as leverage, standardized residual and normal curvatures of local influence under two perturbation schemes are derived. A real data set previously analyzed under normal linear models is reanalyzed under semiparametric additive models with symmetric errors.