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

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...

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Autor principal:	Richard F Lyon
Formato:	article
Lenguaje:	EN
Publicado:	Public Library of Science (PLoS) 2021
Materias:	Medicine R Science Q
Acceso en línea:	https://doaj.org/article/d78c3f0a19a5406eb704150616b6f42c
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spelling	oai:doaj.org-article:d78c3f0a19a5406eb704150616b6f42c2021-12-02T20:04:03ZOn closed-form tight bounds and approximations for the median of a gamma distribution.1932-620310.1371/journal.pone.0251626https://doaj.org/article/d78c3f0a19a5406eb704150616b6f42c2021-01-01T00:00:00Zhttps://doi.org/10.1371/journal.pone.0251626https://doaj.org/toc/1932-6203The 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.Richard F LyonPublic Library of Science (PLoS)articleMedicineRScienceQENPLoS ONE, Vol 16, Iss 5, p e0251626 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Medicine R Science Q
spellingShingle	Medicine R Science Q Richard F Lyon On closed-form tight bounds and approximations for the median of a gamma distribution.
description	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.
format	article
author	Richard F Lyon
author_facet	Richard F Lyon
author_sort	Richard F Lyon
title	On closed-form tight bounds and approximations for the median of a gamma distribution.
title_short	On closed-form tight bounds and approximations for the median of a gamma distribution.
title_full	On closed-form tight bounds and approximations for the median of a gamma distribution.
title_fullStr	On closed-form tight bounds and approximations for the median of a gamma distribution.
title_full_unstemmed	On closed-form tight bounds and approximations for the median of a gamma distribution.
title_sort	on closed-form tight bounds and approximations for the median of a gamma distribution.
publisher	Public Library of Science (PLoS)
publishDate	2021
url	https://doaj.org/article/d78c3f0a19a5406eb704150616b6f42c
work_keys_str_mv	AT richardflyon onclosedformtightboundsandapproximationsforthemedianofagammadistribution
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On closed-form tight bounds and approximations for the median of a gamma distribution.

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