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A simple and efficient test for the Pareto law


This note presents a simple and locally optimal test statistic for the Pareto law. The test is based on the Lagrange multiplier principle and can be computed easily once the maximum likelihood estimator of the scale parameter of the Pareto density has been obtained. A Monte Carlo exercise shows the good small sample properties of the test under the null of the Pareto law and also its power against some sensible and interesting alternatives. In addition, the proposed test is compared to a goodness of fit test which is powerful against more or less all alternatives. Eventually, a simple application to urban economics is performed.

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  • Aban IB, Meerschaert MM (2004) Generalized least-squares estimators for the thickness of heavy tails. J Stat Plann Inference 119: 341–352

    Article  Google Scholar 

  • Aitchison J, Silvey SD (1958) Maximum-likelihood estimation of parameters subject to restraints. Ann Math Stat 29: 813–828

    Article  Google Scholar 

  • Beirlant J, Goegebeur Y, Segers J, Teugels J (2004) Statistics of extremes: theory and applications. Wiley, Chichester

    Book  Google Scholar 

  • Bera AK, Ghosh A (2002) Neyman’s smooth test and its applications in econometrics. In: Ullah A, Wan A, Chaturvedi, A (eds) Handbook of applied econometrics. Marcel Dekker, New York, pp 177–230 Accessed Sept 2012

  • Brown JH, West GB (2000) (eds) Scaling in biology. Oxford University Press, Oxford

  • Burr IW (1942) Cumulative frequency functions. Ann Math Stat 13: 215–232

    Article  Google Scholar 

  • Champernowne D (1953) A model of income distribution. Econ J 63: 318–351

    Article  Google Scholar 

  • Clauset A, Shalizi CR, Newman MEJ (2009) Power-law distributions in empirical data. SIAM Rev 51:661–703. Accessed Sept 2012

    Google Scholar 

  • Eeckhout J (2004) Gibrat′s law for (all) cities. Am Econ Rev 94: 1429–1451

    Article  Google Scholar 

  • Embrechts P, Kluppelberg P, Mikosch C (1997) Modelling extremal events for insurance and finance. Springer, New York

    Book  Google Scholar 

  • Fan CC, Casetti E (1994) The spatial and temporal dynamics of US regional income inequality, 1950–1989. Ann Reg Sci 28: 177–196

    Article  Google Scholar 

  • Gabaix X (1999) Zipf’s law for cities: an explanation. Q J Econ 114: 739–767

    Article  Google Scholar 

  • Gabaix X (2009) Power laws in economics and finance. Annu Rev Econ 1:255–294. doi:10.1146/annurev.economics.050708.142940. Accessed Sept 2012

    Google Scholar 

  • Gabaix X, Ibragimov (2011) Rank-1/2: a simple way to improve the OLS estimation of tail exponents. J Bus Econ Stat 29:24–39. doi:10.1198/jbes.2009.06157. Accessed Sept 2012

  • Gabaix X, Ioannides YM (2004) The evolution of city size distributions. In: Henderson JV, Thisse JF (eds) Handbook of regional and urban economics, vol 4, Chap. 53. North-Holland Publishing Company, Amsterdam

  • Gibrat R (1931) Les inégalités économiques. Applications: aux inégalités des richesses, à la concentration des entreprises, aux populations des villes, aux statistiques des familles, etc., d’une loi nouvelle, la loi de l’effet proportionnel. Libraire du Recueil Sirey, Paris

  • Gini C (1912) Variabilità e mutabilità, contributo allo studio delle distribuzioni e relazioni statistiche. Studi Economico-Giuridici dell’ Universiti di Cagliari No. 3, part 2, Cagliari

  • Goerlich FJ, Mas M (2010) La distribución empírica del tamaño de las ciudades en España, 1900–2001. ¿‘Quién verifica la ley de Zipf?. Rev Econ Apl XVIII, 54, (Winter):133–159

    Google Scholar 

  • Hill BM (1975) A simple general approach to inference about the tail of a distribution. Ann Stat 3(5): 1163–1174

    Article  Google Scholar 

  • Johnson NL, Kotz S (1970) Distributions in statistics: continuous univariate distributions, vol 1. Houghton Mifflin Company, Boston

  • Kallenberg WCM, Ledwina T (1997) Data-driven smooth tests when the hypothesis is composite. J Am Stat Assoc 92:1094–1104. doi:10.1080/01621459.1997.10474065. Accessed Sept 2012

    Google Scholar 

  • Levy M, Solomon S (1996) Power laws are logarithmic Boltzmann laws. Int J Mod Phys C 7:595–601. Accessed Sept 2012

    Google Scholar 

  • Ministerio de Fomento (2000) Atlas estadístico de las áreas urbanas de España, 1st edn. Centro de Publicaciones, Ministerio de Fomento, Madrid

  • Muniruzzaman ANM (1957) On measures of location and dispersion and tests of hypothesis in a Pareto population. Bull Calcutta Stat Assoc 7: 115–123

    Google Scholar 

  • Nishiyama Y, Osada S (2004) Statistical theory of rank size rule regression under Pareto distribution. Discussion Paper No. 009 (January), 21COE, Interfaces for advanced economic analysis, Kyoto University. Accessed Sept 2012

  • Nishiyama Y, Osada S, Sato Y (2008) OLS estimation and the t test revisited in rank-size rule regression. J Reg Sci 48:691–716. [Erratum in J Reg Sci (2009) 49, 1 (February):241]

    Google Scholar 

  • Pareto V (1896) Cours d’Economie politique. Droz, Geneva

    Google Scholar 

  • Rosen KT, Resnick M (1980) The size distribution of cities: an examination of the Pareto law and primacy. J Urban Econ 8: 165–186

    Article  Google Scholar 

  • Silvey SD (1959) The lagrangian multiplier test. Ann Math Stat 30: 389–407

    Article  Google Scholar 

  • Simon HA (1955) On a class of skew distribution functions. Biometrika 42: 425–440

    Google Scholar 

  • Sornette D (2004) Critical phenomena in natural sciences. Springer, Berlin

    Google Scholar 

  • Sornette D, Cont R (1997) Convergent multiplicative processes repelled from zero: power laws and truncated power laws. J Phys I Fr 7: 431–444

    Article  Google Scholar 

  • Sutton J (1997) Gibrat’s legacy. J Econ Lit 35: 40–59

    Google Scholar 

  • Urzúa CM (2000) A simple and efficient test for Zipf’s law. Econ Lett 66: 257–260

    Article  Google Scholar 

  • Zipf G (1949) Human behavior and the principle of least effort. Addison-Wesley, Cambridge

    Google Scholar 

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Correspondence to Francisco J. Goerlich.

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Goerlich, F.J. A simple and efficient test for the Pareto law. Empir Econ 45, 1367–1381 (2013).

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