Normalized multiple variability indices for statistical rates: studying the global demographic convergence

We introduce some statistical methods for studying in a multidimensional viewpoint the global demographic convergence of the populations towards a common pattern across time. Many demographers have empirically tested the convergence using statistical indices of variability (the so-called “σ-convergence”) but they focused on a unidimensional viewpoint, that is testing separately the convergence of each variable at a time. Let consider a set of k variables each observed on n populations across time. Since the demographic transition theory refers to the changes of births, mortality and age structure over time, we consider the crude birth rate, the crude death rate, the infant mortality rate and the aging index. As these variables are statistical rates, we define suitably the variance and covariance matrix S and the correlation matrix R. Then, we take the determinant of S and the determinant of R as absolute multiple variability indices. In aim to evaluate the magnitude of the convergence, we apply a linear normalization procedure to each absolute index, obtaining the corresponding normalized one, with values comprised between 0 and 1. Here, we use the normalized indices for testing the demographic convergence of the European populations.