Tail fit and the Zipf–Pareto law |
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Authors: | Paul Schuette Marcus C. Spruill |
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Affiliation: | (1) Department of Mathematics and Computer Sciences, Meredith College, 3800 Hillsborough Street, Raleigh, NC 27607-5298, USA;(2) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA |
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Abstract: | A limit theorem with bounds on the rate of convergence is proven. The joint distribution of a fixed number of relative decrements of the top order statistics from a random sample converges to the limit as the sample size increases if and only if the underlying distribution is in essence a Pareto. In conjunction with a chi-square test of fit, it provides an asymptotically distribution-free test of fit to the family of distributions with regularly varying tails at infinity. When the limit distribution holds, rank-size plots obey Zipf’s law. The test can be used to detect departures from this Zipf–Pareto law. |
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Keywords: | von Mises condition Regularly varying tail Asymptotic distribution Rate of convergence Chi-square fit Hill’ s estimator Exponential distribution Order statistics |
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