Cover, T. King, R.
Abstract
In his original paper on the subject, Shannon found upper and lower bounds for the entropy of printed English based on the number of trials required for a subject to guess subsequent symbols in a given text. The guessing approach precludes asymptotic consistency of either the upper or lower bounds except for degenerate ergodic processes. Shannon's technique of guessing the next symbol is altered by having the subject place sequential bets on the next symbol of text.....
Maximum likelihood estimation for constrained parameters of multinomial distributions—Application to Zipf–Mandelbrot models
F. Izsáka, b, Corresponding Author Contact Information, E-mail The Corresponding Author
aELTE, Institute of Mathematics, P.O. Box 120, 1518 Budapest, Hungary
bUniversity of Twente, EWI, P.O. Box 217, 7500 AE Enschede, Netherlands
Received 3 June 2005; revised 10 May 2006; accepted 11 May 2006. Available online 12 June 2006.
Abstract
A numerical maximum likelihood (ML) estimation procedure is developed for the constrained parameters of multinomial distributions. The main difficulty involved in computing the likelihood function is the precise and fast determination of the multinomial coefficients. For this the coefficients are rewritten into a telescopic product. The presented method is applied to the ML estimation of the Zipf–Mandelbrot (ZM) distribution, which provides a true model in many real-life cases. The examples discussed arise from ecological and medical observations. Based on the estimates, the hypothesis that the data is ZM distributed is tested using a chi-square test. The computer code of the presented procedure is available on request by the author.
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