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On closed-form tight bounds and approximations for the median of a gamma distribution

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  • Richard F Lyon

Abstract

The median of a gamma distribution, as a function of its shape parameter k, has no known representation in terms of elementary functions. In this work we use numerical simulations and asymptotic analyses to bound the median, finding bounds of the form 2−1/k(A + Bk), including an upper bound that is tight for low k and a lower bound that is tight for high k. These bounds have closed-form expressions for the constant parameters A and B, and are valid over the entire range of k > 0, staying between 48 and 55 percentile. Furthermore, an interpolation between these bounds yields closed-form expressions that more tightly bound the median, with absolute and relative margins to both upper and lower bounds approaching zero at both low and high values of k. These bound results are not supported with analytical proofs, and hence should be regarded as conjectures. Simple approximation expressions between the bounds are also found, including one in closed form that is exact at k = 1 and stays between 49.97 and 50.03 percentile.

Suggested Citation

  • 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.
    • Handle: RePEc:plo:pone00:0251626
    DOI: 10.1371/journal.pone.0251626
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    References listed on IDEAS

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    1. Chen, Jeesen & Rubin, Herman, 1986. "Bounds for the difference between median and mean of gamma and poisson distributions," Statistics & Probability Letters, Elsevier, vol. 4(6), pages 281-283, October.
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    Cited by:

    1. 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.
    2. 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.

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