Comparisons of log-normal mixture and Pareto tails, GB2 or log-normal body of Romania’s all cities size distribution

Abstract

Modeling demographic data has been on the agenda of statisticians for many years. Some of the distributions used are Pareto, reverse Pareto, q-exponential and log-normal models. An approach to this problem is to consider three statistical models: one for the upper tail, one for the middle range, and another for the lower tail. This paper deals with the size distribution of urban and rural agglomerations in Romania for the 1992–2017 period, by comparing the recently introduced three log-normal mixture (3LN), Pareto tails log-normal (PTLN), and threshold double Pareto Generalized Beta of second kind (tdPGB2) models. The tdPGB2 statistical model has the PTLN distribution as a limiting case. The maximum likelihood estimates of the distributions are computed, and goodness-of-fit tests are performed using the Kolmogorov–Smirnov (KS), Cramér–von Mises (CM) and Anderson–Darling (AD) statistics. Also, we use the Vuong and Bayes factor log-likelihood tests. Using both graphical and formal statistical tests, our results rigorously confirm that the 3LN model is statistically equivalent to PTLN and tdPGB2 distributions, the preferred model being the PTLN probability law. Both the PTLN and tdPGB2 distributions have Pareto tails but the 3LN model does not. All the three models prove to be very well suited parameterizations of Romania’s city size data.