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 $g(x) = x - 1/3 + c_1/x$, 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 $c_1 \le 8/405$ and an upper bound if $c_1 \ge 1/45$. 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 $α, β\ge 1$ 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 $1 \le α\le β$, 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 $α\ge 1$.