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Tight bounds for the median of a gamma distribution

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  • Richard F Lyon

Abstract

The median of a standard gamma distribution, as a function of its shape parameter k, has no known representation in terms of elementary functions. In this work we prove the tightest upper and lower bounds of the form 2−1/k(A + k): an upper bound with A = e−γ (with γ being the Euler–Mascheroni constant) and a lower bound with A = log ( 2 ) - 1 3. These bounds are valid over the entire domain of k > 0, staying between 48 and 55 percentile. We derive and prove several other new tight bounds in support of the proofs.

Suggested Citation

  • Richard F Lyon, 2023. "Tight bounds for the median of a gamma distribution," PLOS ONE, Public Library of Science, vol. 18(9), pages 1-18, September.
    • Handle: RePEc:plo:pone00:0288601
    DOI: 10.1371/journal.pone.0288601
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    References listed on IDEAS

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    1. Frédéric Ouimet, 2023. "A refined continuity correction for the negative binomial distribution and asymptotics of the median," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(7), pages 827-849, October.
    2. Richard F Lyon, 2021. "On closed-form tight bounds and approximations for the median of a gamma distribution," PLOS ONE, Public Library of Science, vol. 16(5), pages 1-18, May.
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