Testing the Pareto against the lognormal distributions with the uniformly most powerful unbiased test applied to the distribution of cities

Abstract

Fat-tail distributions of sizes abound in natural, physical, economic, and social systems. The lognormal and the power laws have historically competed for recognition with sometimes closely related generating processes and hard-to-distinguish tail properties. This state-of-affair is illustrated with the debate between Eeckhout [Amer. Econ. Rev. 94, 1429 (2004)] and Levy [Amer. Econ. Rev. 99, 1672 (2009)] on the validity of Zipf's law for US city sizes. By using a uniformly most powerful unbiased (UMPU) test between the lognormal and the power-laws, we show that conclusive results can be achieved to end this debate. We advocate the UMPU test as a systematic tool to address similar controversies in the literature of many disciplines involving power laws, scaling, "fat" or "heavy" tails. In order to demonstrate that our procedure works for data sets other than the US city size distribution, we also briefly present the results obtained for the power-law tail of the distribution of personal identity (ID) losses, which constitute one of the major emergent risks at the interface between cyberspace and reality.