Smooth estimation of mean and dispersion function in extended generalized additive models with application to Italian induced abortion data

Abstract

We analyse data on abortion rate (AR) in Italy with a particular focus on different behaviours in different regions in Italy. The aim is to try to reveal the relationship between the AR and several covariates that describe in some way the modernity of the region and the condition of the women there. The data are mostly underdispersed and the degree of underdispersion also varies with the covariates. To analyse these data, recent techniques for flexible modelling of a mean and dispersion function in a double exponential family framework are further developed now in a generalized additive model context for dealing with the multivariate set-up. The appealing unified framework and approach even allow to semi-parametric modelling of the covariates without any additional efforts. The methodology is illustrated on ozone-level data and leads to interesting findings in the Italian abortion data.

Keywords:

• dispersion function
• heteroscedasticity
• mean function
• nonparametric estimation
• overdispersion
• P-splines
• underdispersion

Acknowledgements

The authors are grateful to the reviewers for the valuable comments which led to an improvement of the manuscript. Support from the GOA/07/04-project of the Research Fund KULeuven is gratefully acknowledged, as well as support from the IAP research network nr. P6/03 of the Federal Science Policy, Belgium. The authors thank Professor Ester Lucia Rizzi of the Université catholique de Louvain for helpful discussions on the subject.