Hypothesis testing in rank-size rule regression

Yoko Konishi, Yoshihiko Nishiyama

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


This note examines testing methods for Paretoness in the framework of rank-size rule regression. Rank-size rule regression describes a relationship found in the analysis of various topics such as city population, words in texts, scale of companies and so on. In terms of city population, it is basically an empirical rule that log (S(i)) is approximately a linear function of log (i) where S(i) is the number of population of ith largest city in a country. This is closely related to the so-called Zipf's law. It is known that this kind of empirical observation is found when the city population is a random variable following a Pareto distribution. Thus one may be willing to test if city size has a Pareto distribution or not. Rosen and Resnick [K.T. Rosen, M. Resnick, The size distribution of cities: an explanation of the Pareto law and primacy, Journal of Urban Economics 8 (1980), 165-186] and Soo [K.T. Soo, Zipf's law for cities: a cross country investigation, Regional Science and Urban Economics (35) 2005, 239-263] regress log (S(i)) on log (i) and log 2 (i) and test the null of Paretoness by standard t-test for the latter regressor. It is found that t-statistics take large values and the Paretoness is rejected in many countries. We study the statistical properties of the t-statistic and show that it explodes asymptotically, in fact, by simulation and thus the t-test does not provide a reasonable testing procedure. We propose an alternative test statistic which seems to be asymptotically normally distributed. We also propose a test with the null hypothesis that the city size distribution is Pareto with exponent unity, which is a modification of the F-test.

Original languageEnglish
Pages (from-to)2869-2878
Number of pages10
JournalMathematics and Computers in Simulation
Issue number9
Publication statusPublished - 2009 May
Externally publishedYes


  • Paretoness
  • Rank-size rule regression

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)
  • Numerical Analysis
  • Modelling and Simulation
  • Applied Mathematics


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