It is commonly found that distributions that seem to be lognormal over a broad range change to a power-law (Pareto) distribution for the last few percentiles. The distributions of many physical, natural, and social events (earthquake size, species abundance, income and wealth, as well as file, city, and firm sizes) display this structure. We present a test for the occurrence of power-law tails in statistical distributions based on maximum entropy. This methodology allows one to identify the true data-generating processes even in the case when it is neither lognormal nor Pareto. The maximum entropy approach is then compared with other widely used methods and applied to different levels of aggregation of complex systems. Our results provide support for the theory that distributions with lognormal body and Pareto tail can be generated as mixtures of lognormally distributed units.

Forecasting the creditworthiness of customers is a central issue of banking activity. This task requires the analysis of large datasets with many variables, for which machine learning algorithms and feature selection techniques are a crucial tool. Moreover, the percentages of “good” and “bad” customers are typically imbalanced such that over- and undersampling techniques should be employed. In the literature, most investigations tackle these three issues individually. Since there is little evidence about their joint performance, in this paper, we try to fill this gap. We use five machine learning classifiers, and each of them is combined with different feature selection techniques and various data-balancing approaches. According to the empirical analysis of a retail credit bank dataset, we find that the best combination is given by random forests, random forest recursive feature elimination and random oversampling.