Pareto or Log‐normal? Best Fit and Truncation in the Distribution of All Cities*

Abstract

ABSTRACT. In the literature, the distribution of city size is a controversial issue with two common contenders: the Pareto and the log-normal. While the first is most accredited when the distribution is truncated above a certain threshold, the latter is usually considered a better representation for the untruncated distribution of all cities. In this paper, we reassess the empirical evidence on the best-fitting distribution in relation to the truncation point issue. Specifically, we provide a comparison among four recently proposed approaches and alternative definitions of U.S. cities. Our results highlight the importance to look at issue of the best-fitting distribution together with the truncation issue and provide guidance with respect to the existing tests of the truncation point.

Reconsidering Zipf’s law for regional development: The case of settlements and cities in Croatia

“…Although Eeckhout (2004) argues that the entire distribution of city sizes is lognormal, studies by authors that affirm that the distribution of city sizes have a Pareto tail and a lognormal body (Bee, Riccaboni & Schiavo 2011Clauset, Shalizi & Newman 2009;Ioannides and Skouras, 2013;Luckstead and Devadoss, 2017;Malavergne, Pisarenko & Sornette 2011) are consistent with our results. Our study follows the study of Fazio and Modica (2015), which examined alternative city size distributions depending on differing truncation points and proved the validity of Eeckhout's (2009) study, stating that the choice of Pareto or lognormal distribution depends on the truncation point, wherein the upper tail is longer than assumed.…”

City size distribution across the OECD: Does the definition of cities matter?

“…In the latter case, it was found that the city size distribution follows a lognormal distribution rather than a Pareto one (Eeckhout, 2004;Parr & Suzuki, 1973). This can be at the basis of the high sensitivity of ζ to the minimum city size threshold in the data (Fazio & Modica, 2015). In an analysis of the United States, González-Val (2010) found that Zipf's law holds only if the sample is sufficiently restricted at the top.…”

Historical urban growth in Europe (1300–1800)

“…Furthermore, the literature contains other ways to estimate the population threshold that switches between the body of the distribution and the Pareto upper‐tail. Fazio and Modica () compare four different methodologies (including the CSN method), concluding that the Ioannides and Skouras () approach tends to underestimate the truncation point moderately, providing the closest prediction. However, Fazio and Modica () show that the differences between thresholds estimated with different methodologies increase with the sample size by running simulations with sample sizes from 1,000 to 25,000 observations.…”

“…Fazio and Modica () compare four different methodologies (including the CSN method), concluding that the Ioannides and Skouras () approach tends to underestimate the truncation point moderately, providing the closest prediction. However, Fazio and Modica () show that the differences between thresholds estimated with different methodologies increase with the sample size by running simulations with sample sizes from 1,000 to 25,000 observations. In this paper our sample sizes are considerably smaller (only in the last two periods are there more than 1,000 cities), so the differences between the results obtained with the different methodologies should be small.…”