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On the parametric description of the French, German, Italian and Spanish city size distributions

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Abstract

We study the parametric description of the city size distribution of four European countries: France, Germany, Italy and Spain. The parametric models used are the lognormal, the double Pareto lognormal, the normal-Box–Cox and the threshold double Pareto Singh–Maddala (last two of these are defined in this paper). The results are quite regular. The preferred model is always the threshold double Pareto Singh–Maddala in the four countries. However, the dPln is not rejected always for the case of France, and in the case of Italy, the dPln is the runner-up distribution.

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Notes

  1. We have the data for Italian comuni on the whole period 1901–2011 in, generally decennial, intervals. Likewise we have the data for Spanish municipios for the whole period 1900–2010. The results for the years not shown in this paper, for the sake of brevity, are similar to those reported, and they are available from the authors upon request.

  2. The \(f_{\mathrm{SM}}\) is directly related to the Burr Type XII distribution (Burr 1942). See also Kleiber and Kotz (2003).

  3. We have performed the estimations with MATLAB as in González-Val et al. (2013c) and Ramos et al. (2014). The standard errors are computed as indicated in Efron and Hinkley (1978) and McCullough and Vinod (2003).

  4. The estimations are close to those obtained by Giesen and Suedekum (2012) although not equal as they take the data for the year 2008 which are not at our disposal.

  5. We have checked that these \(\mathrm{msd}\) and \(R^2\) metrics provide essentially the same results for our distributions and the whole range of populations and cities even for a very low number of generated samples, for example, 10 or 20. This is because we generate samples of the same size of the empirical samples covering all of the cities, not only the upper tail. We have chosen a number of generated samples reasonably high enough while maintaining computational feasibility.

  6. The results for the AIC and BIC of the lgn and the dPln for France are qualitatively similar to those of Giesen and Suedekum (2012).

  7. We have computed the standard errors following the procedure of Efron and Hinkley (1978) and McCullough and Vinod (2003).

  8. In Giesen and Suedekum (2012), it has already been pointed out that Paris is a remarkable outlier for the Pareto distribution at the upper tail.

  9. The details are available from the authors upon request.

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Acknowledgments

We thank two anonymous referees for constructive comments that have helped us to improve the paper notably. We thank Fernando Sanz-Gracia for sharing data sets with us and for useful comments, and Rafael González-Val, Christian Schluter and Mark Trede for sharing data sets with us. However, all remaining errors are ours. This work is supported by the Spanish Ministry of Economy and Competitiveness (ECO2013- 45969-P) and Aragon Government (ADETRE Consolidated Group).

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  1. Department of Economic Analysis, Universidad de Zaragoza, Zaragoza, Spain

    Miguel Puente-Ajovín & Arturo Ramos

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  1. Miguel Puente-Ajovín
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  2. Arturo Ramos
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Correspondence to Arturo Ramos.

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Puente-Ajovín, M., Ramos, A. On the parametric description of the French, German, Italian and Spanish city size distributions. Ann Reg Sci 54, 489–509 (2015). https://doi.org/10.1007/s00168-015-0663-3

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