rivista italiana di economia demografia e statistica - Sieds

Rivista Italiana di Economia Demografia e Statistica Volume LXV n. 1 – Gennaio-Marzo 2011
1. Introduction
ALTERNATIVE VARIANTS OF THE
HELIGMAN-POLLARD MODEL
Anna Maria Altavilla, Angelo Mazza, Antonio Punzo
In demographic analysis and actuarial practice, the use of crude age-specific
mortality rates is often deemed as not appropriate since the observed-data
pattern may be intended as a single realization of a stochastic process.
To cope with this issue, several graduation techniques have been proposed in
literature, based on the assumption that if the number of individuals in the group on
whose experience data are based were considerably larger, then the set of observed
indicators would display a much more regular progression with age (Copas and
Haberman, 1983).
In detail, the relation between the crude rates and the true but unknown
mortality rates may be summarized as
where denotes noise due to random variation, age misstatement, etc.. Such
random fluctuations are usually more evident when small area datasets are
investigated or at ages in which mortality is itself rarer.
In order to capture the underlying mortality pattern from the crude rates, a
graduation function
)
is used; ) aims at smoothing out irregularities in crude mortality rates .
In analogy with the usual statistical modelling, the ) function can be
specified parametrically (see, e.g. Debòn et al., 2005) or nonparametrically (see,
e.g. Debòn et al., 2006). In the following, we will address a specific class of
parametric models often employed by demographers, in which the function
described as
is
,
, (1)