  We study the question of identity testing for structured distributions. More
precisely, given samples from a {\em structured} distribution $q$ over $[n]$
and an explicit distribution $p$ over $[n]$, we wish to distinguish whether
$q=p$ versus $q$ is at least $\epsilon$-far from $p$, in $L_1$ distance. In
this work, we present a unified approach that yields new, simple testers, with
sample complexity that is information-theoretically optimal, for broad classes
of structured distributions, including $t$-flat distributions, $t$-modal
distributions, log-concave distributions, monotone hazard rate (MHR)
distributions, and mixtures thereof.
