IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v388y2009i19p4179-4191.html
   My bibliography  Save this article

The Pareto-positive stable distribution: A new descriptive model for city size data

Author

Listed:
  • Sarabia, José María
  • Prieto, Faustino

Abstract

The Pareto-positive stable (PPS) distribution is introduced as a new model for describing city size data in a country. The PPS distribution provides a flexible model for fitting the entire range of a set of city size data, where zero and unimodality are possible, and the classical Pareto and Zipf distributions are included as a particular case. Expressions for the shape, moments and other descriptive probabilistic measures are given. Estimation methods are discussed and a simple graphical method for studying the adequacy of the data to model is given. Finally, we consider city size data for Spain for several different years. The new distribution is compared with three classical models: Pareto, lognormal and Tsallis distributions. In all the data sets considered, the PPS distribution outperforms the fits of these three previous distributions.

Suggested Citation

  • Sarabia, José María & Prieto, Faustino, 2009. "The Pareto-positive stable distribution: A new descriptive model for city size data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(19), pages 4179-4191.
    • Handle: RePEc:eee:phsmap:v:388:y:2009:i:19:p:4179-4191
    DOI: 10.1016/j.physa.2009.06.047
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437109005020
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2009.06.047?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Bosker, Maarten & Brakman, Steven & Garretsen, Harry & Schramm, Marc, 2008. "A century of shocks: The evolution of the German city size distribution 1925-1999," Regional Science and Urban Economics, Elsevier, vol. 38(4), pages 330-347, July.
    2. Rosen, Kenneth T. & Resnick, Mitchel, 1980. "The size distribution of cities: An examination of the Pareto law and primacy," Journal of Urban Economics, Elsevier, vol. 8(2), pages 165-186, September.
    3. Bercher, J.-F. & Vignat, C., 2008. "A new look at q-exponential distributions via excess statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(22), pages 5422-5432.
    4. Fujiwara, Yoshi & Di Guilmi, Corrado & Aoyama, Hideaki & Gallegati, Mauro & Souma, Wataru, 2004. "Do Pareto–Zipf and Gibrat laws hold true? An analysis with European firms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 335(1), pages 197-216.
    5. Stanley, Michael H. R. & Buldyrev, Sergey V. & Havlin, Shlomo & Mantegna, Rosario N. & Salinger, Michael A. & Eugene Stanley, H., 1995. "Zipf plots and the size distribution of firms," Economics Letters, Elsevier, vol. 49(4), pages 453-457, October.
    6. Alberto F. Ades & Edward L. Glaeser, 1995. "Trade and Circuses: Explaining Urban Giants," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 110(1), pages 195-227.
    7. Xavier Gabaix, 1999. "Zipf's Law for Cities: An Explanation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 114(3), pages 739-767.
    8. Donald R. Davis & David E. Weinstein, 2002. "Bones, Bombs, and Break Points: The Geography of Economic Activity," American Economic Review, American Economic Association, vol. 92(5), pages 1269-1289, December.
    9. Xavier Gabaix, 1999. "Zipf's Law and the Growth of Cities," American Economic Review, American Economic Association, vol. 89(2), pages 129-132, May.
    10. Nitsch, Volker, 2005. "Zipf zipped," Journal of Urban Economics, Elsevier, vol. 57(1), pages 86-100, January.
    11. Benguigui, Lucien & Blumenfeld-Lieberthal, Efrat, 2009. "The temporal evolution of the city size distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(7), pages 1187-1195.
    12. Anderson, Gordon & Ge, Ying, 2005. "The size distribution of Chinese cities," Regional Science and Urban Economics, Elsevier, vol. 35(6), pages 756-776, November.
    13. Gabaix, Xavier & Ioannides, Yannis M., 2004. "The evolution of city size distributions," Handbook of Regional and Urban Economics, in: J. V. Henderson & J. F. Thisse (ed.), Handbook of Regional and Urban Economics, edition 1, volume 4, chapter 53, pages 2341-2378, Elsevier.
    14. Soo, Kwok Tong, 2005. "Zipf's Law for cities: a cross-country investigation," Regional Science and Urban Economics, Elsevier, vol. 35(3), pages 239-263, May.
    15. Moura, Newton J. & Ribeiro, Marcelo B., 2006. "Zipf law for Brazilian cities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 441-448.
    16. Urzua, Carlos M., 2000. "A simple and efficient test for Zipf's law," Economics Letters, Elsevier, vol. 66(3), pages 257-260, March.
    17. Tsallis, Constantino & Mendes, RenioS. & Plastino, A.R., 1998. "The role of constraints within generalized nonextensive statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 261(3), pages 534-554.
    18. Gangopadhyay, Kausik & Basu, B., 2009. "City size distributions for India and China," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(13), pages 2682-2688.
    19. S. Narayan, 2009. "India," Chapters, in: Peter Draper & Philip Alves & Razeen Sally (ed.), The Political Economy of Trade Reform in Emerging Markets, chapter 7, Edward Elgar Publishing.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Deepesh Bhati & Enrique Calderín-Ojeda & Mareeswaran Meenakshi, 2019. "A New Heavy Tailed Class of Distributions Which Includes the Pareto," Risks, MDPI, vol. 7(4), pages 1-17, September.
    2. Kwong, Hok Shing & Nadarajah, Saralees, 2019. "A note on “Pareto tails and lognormal body of US cities size distribution”," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 55-62.
    3. Arshad, Sidra & Hu, Shougeng & Ashraf, Badar Nadeem, 2019. "Zipf’s law, the coherence of the urban system and city size distribution: Evidence from Pakistan," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 87-103.
    4. Asif, Muhammad & Hussain, Zawar & Asghar, Zahid & Hussain, Muhammad Irfan & Raftab, Mariya & Shah, Said Farooq & Khan, Akbar Ali, 2021. "A statistical evidence of power law distribution in the upper tail of world billionaires’ data 2010–20," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 581(C).
    5. Sen, Hu & Chunxia, Yang & Xueshuai, Zhu & Zhilai, Zheng & Ya, Cao, 2015. "Distributions of region size and GDP and their relation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 430(C), pages 46-56.
    6. Kii, Masanobu & Akimoto, Keigo & Doi, Kenji, 2012. "Random-growth urban model with geographical fitness," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(23), pages 5960-5970.
    7. Sáez, Antonio José & Prieto, Faustino & Sarabia, José María, 2012. "A two-tail version of the PPS distribution with application to current account balance data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(21), pages 5160-5171.
    8. Calderín-Ojeda, Enrique, 2016. "The distribution of all French communes: A composite parametric approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 450(C), pages 385-394.
    9. Faustino Prieto & José María Sarabia & Enrique Calderín-Ojeda, 2021. "The nonlinear distribution of employment across municipalities," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 16(2), pages 287-307, April.
    10. Guillen, Montserrat & Prieto, Faustino & Sarabia, José María, 2011. "Modelling losses and locating the tail with the Pareto Positive Stable distribution," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 454-461.
    11. Catana, Luigi-Ionut, 2022. "Stochastic orders of multivariate Jones–Larsen distribution family with empirical applications in physics, economy and social sciences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 603(C).
    12. Gómez-Déniz, Emilio & Calderín-Ojeda, Enrique, 2015. "On the use of the Pareto ArcTan distribution for describing city size in Australia and New Zealand," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 821-832.
    13. Sarabia, José María & Prieto, Faustino & Trueba, Carmen & Jordá, Vanesa, 2013. "About the modified Gaussian family of income distributions with applications to individual incomes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(6), pages 1398-1408.
    14. Joseph L. Gastwirth & Richard Luo & Qing Pan, 2024. "A statistical examination of wealth inequality within the Forbes 400 richest families in the United States from 2000 to 2023," METRON, Springer;Sapienza Università di Roma, vol. 82(3), pages 329-344, December.
    15. Francisco Louzada & Daniele C. T. Granzotto, 2016. "The transmuted log-logistic regression model: a new model for time up to first calving of cows," Statistical Papers, Springer, vol. 57(3), pages 623-640, September.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kii, Masanobu & Akimoto, Keigo & Doi, Kenji, 2012. "Random-growth urban model with geographical fitness," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(23), pages 5960-5970.
    2. Valente J. Matlaba & Mark J. Holmes & Philip McCann & Jacques Poot, 2013. "A Century Of The Evolution Of The Urban System In Brazil," Review of Urban & Regional Development Studies, Wiley Blackwell, vol. 25(3), pages 129-151, November.
    3. Arshad, Sidra & Hu, Shougeng & Ashraf, Badar Nadeem, 2019. "Zipf’s law, the coherence of the urban system and city size distribution: Evidence from Pakistan," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 513(C), pages 87-103.
    4. Chen, Zhihong & Fu, Shihe & Zhang, Dayong, 2010. "Searching for the parallel growth of cities," MPRA Paper 21528, University Library of Munich, Germany.
    5. repec:wyi:journl:002175 is not listed on IDEAS
    6. Sebastien TERRA, 2009. "Zipf's Law for Cities: On a New Testing Procedure," Working Papers 200920, CERDI.
    7. Zhihong Chen & Shihe Fu & Dayong Zhang, 2013. "Searching for the Parallel Growth of Cities in China," Urban Studies, Urban Studies Journal Limited, vol. 50(10), pages 2118-2135, August.
    8. Luckstead, Jeff & Devadoss, Stephen, 2014. "A comparison of city size distributions for China and India from 1950 to 2010," Economics Letters, Elsevier, vol. 124(2), pages 290-295.
    9. Ramos, Arturo & Sanz-Gracia, Fernando & González-Val, Rafael, 2013. "A new framework for the US city size distribution: Empirical evidence and theory," MPRA Paper 52190, University Library of Munich, Germany.
    10. González-Val, Rafael & Lanaspa, Luis & Sanz, Fernando, 2008. "New Evidence on Gibrat’s Law for Cities," MPRA Paper 10411, University Library of Munich, Germany.
    11. Xavier Gabaix, 2009. "Power Laws in Economics and Finance," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 255-294, May.
    12. Bosker, Maarten & Brakman, Steven & Garretsen, Harry & Schramm, Marc, 2008. "A century of shocks: The evolution of the German city size distribution 1925-1999," Regional Science and Urban Economics, Elsevier, vol. 38(4), pages 330-347, July.
    13. Cerqueti, Roy & Ausloos, Marcel, 2015. "Evidence of economic regularities and disparities of Italian regions from aggregated tax income size data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 421(C), pages 187-207.
    14. Kristian Giesen & Jens Südekum, 2011. "Zipf's law for cities in the regions and the country," Journal of Economic Geography, Oxford University Press, vol. 11(4), pages 667-686, July.
    15. Kristian GIESEN & Jens SÜDEKUM, 2012. "The French Overall City Size Distribution," Region et Developpement, Region et Developpement, LEAD, Universite du Sud - Toulon Var, vol. 36, pages 107-126.
    16. Ronan Lyons & Elisa Maria Tirindelli, 2022. "The Rise & Fall of Urban Concentration in Britain: Zipf, Gibrat and Gini across two centuries," Trinity Economics Papers tep0522, Trinity College Dublin, Department of Economics.
    17. Chen, Yanguang, 2012. "The rank-size scaling law and entropy-maximizing principle," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(3), pages 767-778.
    18. Chen, Yanguang & Wang, Jiejing, 2014. "Recursive subdivision of urban space and Zipf’s law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 395(C), pages 392-404.
    19. Luckstead, Jeff & Devadoss, Stephen, 2014. "A nonparametric analysis of the growth process of Indian cities," Economics Letters, Elsevier, vol. 124(3), pages 516-519.
    20. Daniel ARRIBAS-BEL & Fernando SANZ GRACIA & Domingo P. XIMENEZ-DE-EMBUN, 2012. "Kangaroos, Cities And Space: A First Approach To The Australian Urban System," Region et Developpement, Region et Developpement, LEAD, Universite du Sud - Toulon Var, vol. 36, pages 165-187.
    21. Cuberes David, 2009. "A Model of Sequential City Growth," The B.E. Journal of Macroeconomics, De Gruyter, vol. 9(1), pages 1-41, May.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:388:y:2009:i:19:p:4179-4191. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.