Hierarchical and multidimensional smoothing with applications to longitudinal and mortality data

Abstract

This thesis is concerned with two themes: (a) smooth mixed models in hierarchical settings with applications to grouped longitudinal data and (b) multi-dimensional smoothing with reference to the modelling and forecasting of mortality data. In part (a), we examine a popular method to smooth models for longitudinal data, which consists of expressing the model as a mixed model. This approach is particularly appealing when truncated polynomials are used as a basis for the smoothing, as the mixed model representation is almost immediate. We show that this approach can lead to a severely biased estimate of the group and subject effects, and to confidence intervals with undesirable properties. We use penalization to investigate an alternative approach with either B-spline or truncated polynomial bases and show that this new approach does not suffer from the same defects. Our models are defined in terms of B-splines or truncated polynomials with appropriate penalties, but we re-parametrize them as mixed models and this gives access to fitting with standard procedures. In part (b), we first demonstrate the adverse impact of over-dispersion (and heterogeneity) in the modelling of mortality data, and describe the resolution of this problem through a two-stage smoothing of mean and dispersion effects via penalized quasi-likelihoods. Next, we propose a method for the joint modelling of several mortality tables (e.g. male and female mortality in Demography, mortality by lives and by amounts in Life Insurance, etc) and describe how this joint approach leads to the classification and simple comparison of these tables. Finally, we deal with the smooth modelling of mortality improvement factors, which are two-dimensional correlated data; here we first form a basic flexible model incorporating the correlation structure, and then extend this model to cope with cohort and period shock effects