Estimating Functions Inference to Semi‐Parametric Additive Models for Skewed And/Or Heavy‐Tailed Longitudinal Data

ABSTRACT

In many fields of research, it is common to conduct experiments where several observations of the response variable are made on the same subject over non‐random conditions, generating longitudinal studies. A wide variety of techniques deal with this type of data when analysing population variations, many based on estimating functions. It is also common to find longitudinal data where the assumption of asymmetry and/or heavy tails is necessary for the distribution associated with the outcome(s) and also where we do not know the relationship between these outcome(s) and the covariates. In these situations we can use the scale mixture of skew‐normal distributions and the partially linear models. Said that, in this article, we developed estimating functions for the additive partially linear models based on the scale mixture of skew‐normal distributions using a centred parameterisation. This parameterisation does not present inferential problems, as the non‐quadratic form of the log‐likelihood, that the classical skew‐normal distribution can present. Additionally, it provides a better interpretation of the related parameters, since they represent the mean, variance and skewness coefficient. Our methodology focuses on the flexibility of the functional part when using semi‐parametric models and also on the distributional flexibility and robustness when using the aforementioned class of distributions. Additionally, we presented residuals and influence diagnostic tools. A Monte Carlo experiment is conducted to evaluate the performances of the estimators in finite samples. The methodology is illustrated with the analysis of the Framingham cholesterol study.