  We study the bias of the isotonic regression estimator. While there is
extensive work characterizing the mean squared error of the isotonic regression
estimator, relatively little is known about the bias. In this paper, we provide
a sharp characterization, proving that the bias scales as $O(n^{-\beta/3})$ up
to log factors, where $1 \leq \beta \leq 2$ is the exponent corresponding to
H{\"o}lder smoothness of the underlying mean. Importantly, this result only
requires a strictly monotone mean and that the noise distribution has
subexponential tails, without relying on symmetric noise or other restrictive
assumptions.
