Given samples from an unknown distribution $p$, is it possible to distinguish
whether $p$ belongs to some class of distributions $\mathcal{C}$ versus $p$
being far from every distribution in $\mathcal{C}$? This fundamental question
has received tremendous attention in statistics, focusing primarily on
asymptotic analysis, and more recently in information theory and theoretical
computer science, where the emphasis has been on small sample size and
computational complexity. Nevertheless, even for basic properties of
distributions such as monotonicity, log-concavity, unimodality, independence,
and monotone-hazard rate, the optimal sample complexity is unknown.
We provide a general approach via which we obtain sample-optimal and
computationally efficient testers for all these distribution families. At the
core of our approach is an algorithm which solves the following problem: Given
samples from an unknown distribution $p$, and a known distribution $q$, are $p$
and $q$ close in $\chi^2$-distance, or far in total variation distance?
The optimality of our testers is established by providing matching lower
bounds with respect to both $n$ and $\varepsilon$. Finally, a necessary
building block for our testers and an important byproduct of our work are the
first known computationally efficient proper learners for discrete log-concave
and monotone hazard rate distributions.

研究如何通过样本来鉴别未知分布是否属于某个分布类，提出了一种可获得最优样本和计算效率的通用方法，并为分布的基本属性提供了测试方法。