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