rivista italiana di economia demografia e statistica - Sieds

6
Volume LXV n. 1 – Gennaio-Marzo 2011
where measures infant mortality, represents senescent mortality and
reflects accidental mortality, which is additional mortality not described by the
other two terms, such as accident mortality and maternal mortality.
Following this approach, we will propose three variations of the nineparameters
model
proposed in Heligman and Pollard (1980, p.60); in the same paper, the
mathematical aspects and demographical interpretation of parameters are also
discussed.
The models proposed will be fitted to the male and female Australian national
mortality datasets for the periods 1946-48, 1960-62 and 1970-72, as in Heligman
and Pollard (1980), and to the merged-gender mortality table of the Valencia
Region for the three-year period 1999–2001 (see Debòn et al., 2005, 2006, and
Mazza and Punzo, 2011, for details on these data).
According to the leading literature, parameters will be estimated by least
squares, using the Levenberg-Marquardt algorithm (Marquardt, 1963) to minimize
the quantity
where, naturally, the summation runs across all the observed ages. The library
minpack within the R environment has been used to perform these estimates.
2. Alternative graduation models
Our first proposal consists in using a logistic function to modelize . Thus,
model (2) becomes
(2)
. (1)
The number of parameters in (3) remains unchanged with respect to model (2) and
so does the meaning of the first two components. As regards the third component,
H represents the age (not necessarily belonging to the adopted age range) at which