A Mathematical Analysis of Benford's Law and its Generalization

We explain Kossovsky's generalization of Benford's law which is a formula that approximates the distribution of leftmost digits in finite sequences of natural data and apply it to six sequences of data including populations of US cities and towns and times between earthquakes. We model the natural logarithms of these two data sequences as samples of random variables having normal and reflected Gumbel densities respectively. We show that compliance with the general law depends on how nearly constant the periodized density functions are and that the models are generally more compliant than the natural data. This surprising result suggests that the generalized law might be used to improve density estimation which is the basis of statistical pattern recognition, machine learning and data science.Comment: 15 pages, 8 figure

Overweight and obesity in a Swiss city: 10-year trends

Abstract Background Increased rates of overweight/obesity have been reported in recent years in developed countries. This population study of healthy subjects evaluated the changes in overweight/obesity prevalence in 2003, compared with 1993, and determined the association of age, sex and leisure-time activity with body mass index (BMI), fat-free mass index (FFMI) and fat mass index (FMI). Design Two transversal samples of convenience. Participants Healthy volunteers (1993, n=802; 2003, n=1631). Methods Fat-free mass was determined using the bioelectrical impedance multiple regression equation. Multivariable linear regression, including confounding variables (age, sex, leisure-time activity), was used to model the body composition evolution between the 1993 and the 2003 subjects. Results BMI and FMI were higher in 2003 than in 1993, P<0.001. FFMI was not higher in 2003 than in 1993, P=0.38. More subjects were overweight/obese in 2003 than in 1993 (27.5 versus 17.2%, chi-square P<0.001), and had a high FFMI (30.2 versus 21.8%, chi-square P<0.001) and high FMI (28.0 versus 20.3%, chi-square P<0.001). Multivariate linear regressions showed that leisure-time activity was negatively, and sex, age and inclusion year were positively associated with BMI, FFMI and FMI (the exception was a negative association with sex) (P<0.001). Conclusion Overweight prevalence increased between 1993 and 2003 in a Swiss city, and was associated with a higher fat mass. This observation remained statistically significant after adjustment for age, sex and leisure-time activit