Bias of MLE scales with $1/N$? I was reading this paper (link) and it gave me some confusion.
$P(r|\theta)$ is a distribution that generates sample $r$ based on some Poisson distribution, whose mean and variance are defined as some function $f(\theta)$, e.g. $\cos(\theta)$. Given some samples of $r$, we want to find MLE of $\theta$.
In "ML Estimation" section (page 2) of the paper, it says

... This is in contrast to the bias of the ML estimate,
which is of the order of 1/N.

I am not sure why this is the case. The mean estimate using MLE is unbiased, and I know the variance has a bias and it scales with $1/N$. But I am not sure why in this context, the bias of $\theta$ scales with $1/N$.

First, statements like this about order are typically one-sided. So they're saying the bias of the MLE is no more than a multiple of $1/N$. It's sometimes zero.
The statement is also a bit imprecise in other ways.  It's quite possible for the MLE not to have a mean and so not to have a finite bias. On example is the MLE of the odds ratio in a 2x2 table, which is infinite if one of the diagonal cells is zero. There's a non-zero probability of having a zero on the diagonal (very small, usually, but non-zero) so the MLE is infinite with non-zero probability and so has no finite mean.  It would be more precise to define an asymptotic bias and say that was of order $1/N$
