Asymptotically Normal Estimators for Zipf’s Law
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We study an infinite urn scheme with probabilities corresponding to a power function. Urns here represent words from an infinitely large vocabulary. We propose asymptotically normal estimators of the exponent of the power function. The estimators use the number of different elements and a few similar statistics. If we use only one of the statistics we need to know asymptotics of a normalizing constant (a function of a parameter). All the estimators are implicit in this case. If we use two statistics then the estimators are explicit, but their rates of convergence are lower than those for estimators with the known normalizing constant.
Keywords and phrases.Infinite urn scheme Zipf’s law Asymptotic normality.
AMS (2000) subject classification.Primary 62F10; Secondary 62F12
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Our research was partially supported by RFBR grant 17-01-00683 and by the program of fundamental scientific researches of the SB RAS No. I.1.3., project No. 0314-2016-0008.
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