Statistical Modelling 7 (2007), 29–48
Smoothing the Lee-Carter and Poisson log-bilinear models for
mortality forecasting
Antoine Delwarde
Institut des Sciences Actuarielles,
Université Catholique de Louvain
Belgium.
Michel Denuit
Institut de Statistique,
Université Catholique de Louvain,
Voie du Roman Pays, 20
B–1348 Louvain-la-Neuve
Belgium.
E-mail: denuit@stat.ucl.ac.be
eMail:
denuit@stat.ucl.ac.be
Paul Eilers
Medical Statistics,
Leiden University Medical Centre,
The Netherlands
Abstract:
Mortality improvements pose a challenge for the planning of
public retirement systems as well as for the private life
annuities business. For public policy, as well as for the
management of financial institutions, it is important to
forecast future mortality rates. Standard models for mortality
forecasting assume that the force of mortality at age x in
calendar year t is of the form exp({alpha}x + ßx{kappa}t ).
The log of the time series of age-specific death rates is thus
expressed as the sum of an age-specific component {alpha}x that
is independent of time and another component that is the product
of time-varying parameter {kappa}t reflecting the general level
of mortality, and an age-specific component ßx that represents
how rapidly or slowly mortality at each age varies when the
general level of mortality changes. This model is fitted to
historical data. The resulting estimated {kappa}t 's are then
modeled and projected as stochastic time series using standard
Box-Jenkins methods. However, the estimated ßx's exhibit an irregular
pattern in most cases, and this produces irregular projected life
tables. This article demonstrates that it is possible to smooth
the estimated ßx's in the Lee-Carter and Poisson
log-bilinear models for mortality projection. To this end,
penalized least-squares/maximum likelihood analysis is performed.
The optimal value of the smoothing parameter is selected with
the help of cross validation.
Keywords:
cross validation; life table; mortality projection;
roughness penalty; smoothing
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