Measuring the demographic convergence of the European populations by a normalized variability index

The research has been originated from an empirical problem regarding the demographic transition of the national populations belonging to the European Union (EU). The aim is to verify if they converge across time towards one European population with a common demographic pattern, measuring the variability of some demographic indicators by means of a normalized index that takes values comprised between 0 and 1. If the variability is low, it means that the populations converge towards a common pattern and therefore the demographic transition theory is complied. Since the demographic transition theory refers to the changes of births and mortality over time, for every population we observe the crude birth rate, the crude death rate and the infant mortality rate from 1960 to 2007. We consider six different frameworks of the EU (namely, with 6, 10, 12, 15, 25 and 27 members). In the literature there are many variability indices for distributions of a statistical variable X; here we define a new absolute variability index that seems to be more suitable for series of statistical rates. In general, after defining an absolute index, in order to apply a normalization procedure, we must determinate both the minimum and the maximum of the index. This problem has been discussed widely in the literature for statistical distributions of a non-negative variable. The corresponding solutions can not be applied without modifications to series of rates, above all for rates that take only values comprised between 0 and 1, as is the case of the rates considered here. We investigate the minimum and the maximum of the absolute index determining the analytical solutions and implement an algorithm in R.