  Uniformity testing is one of the most well-studied problems in property
testing, with many known test statistics, including ones based on counting
collisions, singletons, and the empirical TV distance. It is known that the
optimal sample complexity to distinguish the uniform distribution on $m$
elements from any $\epsilon$-far distribution with $1-\delta$ probability is $n
= \Theta\left(\frac{\sqrt{m \log (1/\delta)}}{\epsilon^2} + \frac{\log
(1/\delta)}{\epsilon^2}\right)$, which is achieved by the empirical TV tester.
Yet in simulation, these theoretical analyses are misleading: in many cases,
they do not correctly rank order the performance of existing testers, even in
an asymptotic regime of all parameters tending to $0$ or $\infty$.
  We explain this discrepancy by studying the \emph{constant factors} required
by the algorithms. We show that the collisions tester achieves a sharp maximal
constant in the number of standard deviations of separation between uniform and
non-uniform inputs. We then introduce a new tester based on the Huber loss, and
show that it not only matches this separation, but also has tails corresponding
to a Gaussian with this separation. This leads to a sample complexity of $(1 +
o(1))\frac{\sqrt{m \log (1/\delta)}}{\epsilon^2}$ in the regime where this term
is dominant, unlike all other existing testers.
