The significant digit law: a paradigm of statistical scale symmetries

The significant digit law
Interdisciplinary Physics

Abstract.

In many different topics, the most significant digits of data series display a non-uniform distribution which points to an equiprobability of logarithms. This surprising ubiquitous property, known as the significant digit law, is shown here to follow from two similar, albeit different, scale symmetries: the scale-invariance and the scale-ratio invariance. After having legitimized these symmetries in the present context, the corresponding symmetric distributions are determined by implementing a covariance criterion. The logarithmic distribution is identified as the only distribution satisfying both symmetries. Attraction of other distributions to this most symmetric distribution by dilation, stretching and merging is investigated and clarified. The natures of both the scale-invariance and the scale-ratio invariance are further analyzed by determining the structure of the sets composed by the corresponding symmetric distributions. Altogether, these results provide new insights into the meaning and the role of scale symmetries in statistics.

PACS.

02.50.-r Probability theory, stochastic processes, and statistics  07.05.Kf Data analysis: algorithms and implementation; data management 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion  

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References

  1. S. Newcomb, Am. J. Math. 4, 39 (1881) MATHMathSciNetGoogle Scholar
  2. F. Benford, Proc. Amer. Philos. Soc. 78, 551 (1938) MATHGoogle Scholar
  3. T. Hill, Stat. Sci. 10, 354 (1995); T. Hill, Proc. Amer. Math. Soc. 123, 887 (1995); T. Hill, Amer. Math. Monthly 102, 322 (1976) MATHGoogle Scholar
  4. Eur. Phys. J. B 22, number 1, 2, 3 and 4, (2001) Google Scholar
  5. P. Schatte, J. Inform. Process. Cybernet. 24, 433 (1988) MathSciNetGoogle Scholar
  6. Ph.D. Drake, M.J. Nigrini, J. of Acc. Ed. 18, 127 (2000) CrossRefGoogle Scholar
  7. M.J. Nigrini, J. Amer. Taxation Assoc. 18, 72 (1996) Google Scholar
  8. C.L. Geiger, P.P. Williamson, Communication in Statistics, Simulation and Computation 33, 229 (2004) CrossRefGoogle Scholar
  9. E. Ley, Amer. Stat. 50, 311 (1996) CrossRefADSGoogle Scholar
  10. R. Raimi, Sci. Amer. 221, 109 (1969); R. Raimi, Amer. Math. Monthly 83, 521 (1976) CrossRefGoogle Scholar
  11. R. Pinckham, Ann. Math. Statist. 32, 1223 (1961) MathSciNetGoogle Scholar
  12. P. Diaconis, Ann. Probab. 5, 72 (1977) MATHMathSciNetGoogle Scholar
  13. A. Adhikari, B. Sarkar, Sankhya Ser. B 30, 47 (1968) MathSciNetGoogle Scholar
  14. L. Pietronero, E. Tosatti, V. Tosatti, A. Vespigliani, Physica A 293, 297 (2001) CrossRefADSMATHGoogle Scholar
  15. M.A. Snyder, J.H. Curry, A.M. Dougherty, Phys. Rev. E 64, 026222 (2001) CrossRefADSGoogle Scholar
  16. C.R. Tolle, J.L. Budzien, R.A. LaViolette, Chaos 10, 331 (2000) CrossRefADSGoogle Scholar
  17. D. Knuth, The Art of Computer Programming (Addisson-Wesley, Reading, MA, 1969), Vol. 2, p. 219 Google Scholar
  18. B. Buck, A. Merchant, S. Perez, Eur. J. Phys. 14, 59 (1993) CrossRefGoogle Scholar
  19. W. Feller, An introduction to probability Theory and its Applications (Wiley and Sons, 1966) Google Scholar
  20. A. Pocheau, in Mixing, Chaos and Turbulence, edited by H. Chaté, J.-M. Chomaz, E. Villermaux Nato ASI Series B (Physics), Vol. 373 (Plenum Press, 2000) pp. 187–204; A. Pocheau, Phys. Rev. E 49, 1109 (1994); A. Pocheau, D. Queiros-Condé, Phys. Rev. Lett. 76, 3352 (1996) Google Scholar
  21. A. Pocheau, Europhys. Lett. 43, 410 (1998) CrossRefADSGoogle Scholar
  22. M.J.K. de Ceuster, G. Dhaene, T. Shatteman, J. Empirical Finance 5, 263 (1998) CrossRefGoogle Scholar

Copyright information

© EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2006

Authors and Affiliations

  1. 1.IRPHE, CNRS & Universités Aix-Marseille I & IIMarseille Cedex 13France

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