Tight bounds for the median of a gamma distribution

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.sup.-1/k (A + k): an upper bound with A = e.sup.-[gamma] (with [gamma] being the...

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Bibliographic Details
Published in:PloS one 2023-09, Vol.18 (9), p.e0288601
Main Author: Lyon, Richard F
Format: Article
Language:English
Subjects:
Algorithms
Biology and Life Sciences
Evaluation
Functions, Gamma
Kao Tsu, Emperor of China
Lower bounds
Physical Sciences
Probability
Probability distribution
Probability distribution functions
Research and Analysis Methods
Social Sciences
Theorems
Upper bounds
X chromosome inactivation
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Summary: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.sup.-1/k (A + k): an upper bound with A = e.sup.-[gamma] (with [gamma] 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.
ISSN:1932-6203
1932-6203
DOI:10.1371/journal.pone.0288601