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Approximate tolerance limits for Zipf–Mandelbrot distributions

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  • Young, D.S.

Abstract

Zipf–Mandelbrot distributions are commonly used to model natural phenomena where the frequency of an event’s occurrence is inversely proportional to its rank based on that frequency of occurrence. This discrete distribution typically exhibits a large number of rare events; however, it may be of interest to obtain reasonable limits that bound the majority of the number of different events. We propose the use of statistical tolerance limits as a way to quantify such a bound. The tolerance limits are constructed using Wald confidence limits for the Zipf–Mandelbrot parameters and are shown through a simulation study to have coverage probabilities near the nominal levels. We also calculate Zipf–Mandelbrot tolerance limits for two real datasets and discuss the associated computer code developed for the R programming language.

Suggested Citation

  • Young, D.S., 2013. "Approximate tolerance limits for Zipf–Mandelbrot distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(7), pages 1702-1711.
  • Handle: RePEc:eee:phsmap:v:392:y:2013:i:7:p:1702-1711
    DOI: 10.1016/j.physa.2012.11.056
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    References listed on IDEAS

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    1. Sinha, Sitabhra, 2006. "Evidence for power-law tail of the wealth distribution in India," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 359(C), pages 555-562.
    2. Montemurro, Marcelo A., 2001. "Beyond the Zipf–Mandelbrot law in quantitative linguistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 300(3), pages 567-578.
    3. Young, Derek S., 2010. "tolerance: An R Package for Estimating Tolerance Intervals," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 36(i05).
    4. Zornig, Peter & Altmann, Gabriel, 1995. "Unified representation of Zipf distributions," Computational Statistics & Data Analysis, Elsevier, vol. 19(4), pages 461-473, April.
    5. Kii, Masanobu & Akimoto, Keigo & Doi, Kenji, 2012. "Random-growth urban model with geographical fitness," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(23), pages 5960-5970.
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