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A fixed-point algorithm to estimate the Yule-Simon distribution parameter. (English) Zbl 1217.62027

Summary: The Yule-Simon distribution is a discrete probability distribution related to preferential attachment processes such as the growth in the number of species per genus in some higher taxons of biotic organisms, the distributions of the sizes of cities, the wealth distribution among individuals, the number of links to pages in the World Wide Web, among others. We present an algorithm, given a set of observations stemmed from a Yule process, to obtain the parameter of the Yule-Simon distribution [see H. A. Simon, Biometrika 42, 425–440 (1955; Zbl 0066.11201)] with maximum likelihood. In order to test our algorithm, we use modified Polya urn process simulation to generate some data that was used as input to our algorithm. We make a comparison of our algorithm with other methods and also we show an application to some empirical data.

MSC:

62F10 Point estimation
65C60 Computational problems in statistics (MSC2010)
62-04 Software, source code, etc. for problems pertaining to statistics

Citations:

Zbl 0066.11201

Software:

plfit
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Full Text: DOI

References:

[1] G.U. Yule, A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J.C. Willis, F.R.S., Philosophical Transactions of the Royal Society of London. Series B, Containing Papers of a Biological Character, vol. 213, 1925, pp. 21-87.; G.U. Yule, A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J.C. Willis, F.R.S., Philosophical Transactions of the Royal Society of London. Series B, Containing Papers of a Biological Character, vol. 213, 1925, pp. 21-87.
[2] Simon, H. A., On A Class of Skew Distribution Functions, Biometrika, 42, 425-440 (1955) · Zbl 0066.11201
[3] Estoup, J. B., Gammes Stenographique (1916), Institute Stenographique de France: Institute Stenographique de France Paris
[4] Condon, E. U., Statistics of vocabulary, Science, 67, 300-+ (1928)
[5] Zipf, G. K., The Psycho-Biology of Language (1935), MIT Press: MIT Press Cambridge, Massachusetts, USA
[6] Barabasi, A. L.; Albert, R., Emergence of scaling in random networks, Science (New York, N.Y.), 286, 509-512 (1999) · Zbl 1226.05223
[7] S. Bornholdt, H. Ebel, World wide web scaling exponent from simon’s 1955 model, Phys. Rev. E 64 (2001) 035104-1-035104-4.; S. Bornholdt, H. Ebel, World wide web scaling exponent from simon’s 1955 model, Phys. Rev. E 64 (2001) 035104-1-035104-4.
[8] Simkin, M. V.; Roychowdhury, V. P., Re-inventing willis, ArXiv Physics e-prints (2006)
[9] Andrews, G. E.; Askey, R. A.; Roy, R., Special Functions; Encyclopaedia of Mathematics and its Applications (2001), Cambridge Univ.Press: Cambridge Univ.Press Cambridge
[10] Albert, R.; Jeong, H.; Barabasi, A. L., The diameter of the world wide web, Nature, 401, 130-131 (1999)
[11] Mahmoud, H., Polya Urn Models (2008), Chapman & Hall/CRC
[12] Chung, F.; Handjani, S.; Jungreis, D., Generalizations of polya’s urn problem, Annals of Combinatorics, 7, 141-153 (2003), 10.1007/s00026-003-0178-y · Zbl 1022.60005
[13] Clauset, A.; Shalizi, C. R.; Newman, M. E.J., Power-law distributions in empirical data, SIAM Review, 51, 661-703 (2009) · Zbl 1176.62001
[14] Arnold, B. C., Pareto Distribution, Encyclopedia of Statistical Sciences (2004), John Wiley & Sons, Inc.
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