Closed-form Tight Bounds and Approximations for the Median of a Gamma Distribution

TL;DR

This paper derives the tightest closed-form bounds and approximations for the median of a gamma distribution across all shape parameters, improving accuracy over previous methods.

Contribution

It introduces the first closed-form bounds and approximations for the gamma median valid for all positive shape parameters, with optimal tightness and interpolation methods.

Findings

Bounds stay between 48-50% and 50-55% percentiles.

Interpolated bounds provide tighter estimates.

A near-exact approximation at k=1 with minimal error.

Abstract

We show how to find upper and lower bounds to the median of a gamma distribution, over the entire range of shape parameter $k > 0$, that are the tightest possible bounds of the form $2^{-1/k} (A + Bk)$, with closed-form parameters $A$ and $B$. The lower bound of this form that is best at high $k$ stays between 48 and 50 percentile, while the uniquely best upper bound stays between 50 and 55 percentile. We show how to form even tighter bounds by interpolating between these bounds, yielding closed-form expressions that more tightly bound the median. Good closed-form approximations between the bounds are also found, including one that is exact at $k = 1$ and stays between 49.97 and 50.03 percentile.