A simple improvement on Kerman's approximation to the median of the beta distribution
Abstract
Kerman's approximation to the median of the Beta(α,β) distribution is the ratio g(α)/(g(α) + g(β)) with g(x) = x − 1/3, using the first two terms of the asymptotic expansion of the median of the standard gamma distribution. We extend this elegant ratio form to the one-parameter family with g(x) = x − 1/3 + c1/x, in which c1 = 0 gives Kerman's. For large shape parameters along any ray β= rα with r > 1, the resulting approximation is eventually a lower bound on the median if c1 ≤ 8/405 and an upper bound if c1 ≥ 1/45. We recommend c1 = 1/36 for its simplicity: it makes g(x) = (x − 1/6)2/x a perfect square over x, and it is within 1% of the minimax value of c1 for α, β≥ 1 under several error measures, while the theoretically distinguished values 8/405 and 1/45 are less accurate. The approximation is then ((α− 1/6)2/α)/((α− 1/6)2/α+ (β− 1/6)2/β). Its relative error lies between -0.171% and +0.212% for all 1 ≤ α≤ β, compared with up to -3.8% for Kerman's. It is more accurate than Kerman's approximation wherever the latter is defined, with relative error never more than 13/32 of Kerman's, the ratio approached asymptotically. The same g(α) is a useful approximation to the gamma distribution median, overestimating it by at most 0.212% for shape parameter α≥ 1.
Related
- On the median of the beta distribution in relation to the median of the gamma distribution
- Approximation by simple structures
- Simple-regret rates and minimax optimality of fixed-prior expected improvement in Matérn and squared-exponential RKHSs
- CRANDALL-PAZY RIGIDITY FOR LOWER SEMICONTINUOUS CONVEX FUNCTIONS: A BALANCED MEDIAN-FLOW ARGUMENT
- CRANDALL-PAZY RIGIDITY FOR C 1 CONVEX FUNCTIONS: A FIRST-ORDER MEDIAN-FLOW ARGUMENT
BibTeX
@misc{lyon2026simple,
title = {{A simple improvement on Kerman's approximation to the median of the beta distribution}},
author = {Richard F. Lyon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33859},
eprint = {2610.33859},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33859},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - A simple improvement on Kerman's approximation to the median of the beta distribution
AU - Richard F. Lyon
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33859
UR - https://arxiv.org/abs/2610.33859
AB - Kerman's approximation to the median of the Beta(α,β) distribution is the ratio g(α)/(g(α) + g(β)) with g(x) = x − 1/3, using the first two terms of the asymptotic expansion of the median of the standard gamma distribution. We extend this elegant ratio form to the one-parameter family with g(x) = x − 1/3 + c_1/x, in which c_1 = 0 gives Kerman's. For large shape parameters along any ray β= rα with r > 1, the resulting approximation is eventually a lower bound on the median if c_1 ≤ 8/405 and an upper bound if c_1 ≥ 1/45. We recommend c_1 = 1/36 for its simplicity: it makes g(x) = (x − 1/6)^2/x a perfect square over x, and it is within 1% of the minimax value of c_1 for α, β≥ 1 under several error measures, while the theoretically distinguished values 8/405 and 1/45 are less accurate. The approximation is then ((α− 1/6)^2/α)/((α− 1/6)^2/α+ (β− 1/6)^2/β). Its relative error lies between -0.171% and +0.212% for all 1 ≤ α≤ β, compared with up to -3.8% for Kerman's. It is more accurate than Kerman's approximation wherever the latter is defined, with relative error never more than 13/32 of Kerman's, the ratio approached asymptotically. The same g(α) is a useful approximation to the gamma distribution median, overestimating it by at most 0.212% for shape parameter α≥ 1.
ER -