Myerson's regularity condition of a distribution is a standard assumption in economics. In this paper, we study the complexity of describing a regular distribution within a small statistical distance. Our main result is that $\tilde{\Theta}{(\epsilon^{-0.5})}$ bits are necessary and sufficient to describe a regular distribution with support $[0,1]$ within $\epsilon$ Levy distance. We prove this by showing that we can learn the regular distribution approximately with $\tilde{O}(\epsilon^{-0.5})$ queries to the cumulative density function. As a corollary, we show that the pricing query complexity to learn the class of regular distribution with support $[0,1]$ within $\epsilon$ Levy distance is $\tilde{\Theta}{(\epsilon^{-2.5})}$. To learn the mixture of two regular distributions, $\tilde{\Theta}(\epsilon^{-3})$ pricing queries are required.

翻译：Myerson正则性条件是经济学中的标准假设。本文研究了在微小统计距离下描述正则分布的复杂度。主要结论是：在列维距离$\epsilon$下，描述支撑集为$[0,1]$的正则分布所需的比特数上下界均为$\tilde{\Theta}{(\epsilon^{-0.5})}$。我们通过证明仅需对累积分布函数进行$\tilde{O}(\epsilon^{-0.5})$次查询即可近似学习该正则分布来证实这一结论。作为推论，在列维距离$\epsilon$下学习支撑集为$[0,1]$的正则分布类所需的定价查询复杂度为$\tilde{\Theta}{(\epsilon^{-2.5})}$。对于两个正则分布的混合分布，则需要$\tilde{\Theta}(\epsilon^{-3})$次定价查询。