This work develops algorithms for non-parametric confidence regions for samples from a univariate distribution whose support is a discrete mesh bounded on the left. We generalize the theory of Learned-Miller to preorders over the sample space. In this context, we show that the lexicographic low and lexicographic high orders are in some way extremal in the class of monotone preorders. From this theory we derive several approximation algorithms: 1) Closed form approximations for the lexicographic low and high orders with error tending to zero in the mesh size; 2) A polynomial-time approximation scheme for quantile orders with error tending to zero in the mesh size; 3) Monte Carlo methods for calculating quantile and lexicographic low orders applicable to any mesh size.

翻译：本文针对支撑集为左有界离散网格的单变量分布样本，开发了非参数置信区域的算法。我们将Learned-Miller的理论推广到样本空间上的预序关系。在此框架下，我们证明了在单调预序类中，字典序低阶与字典序高阶在某种意义上是极端的。基于该理论，我们推导出多种近似算法：1）针对字典序低阶与高阶的闭式近似，其误差随网格尺寸趋于零；2）针对分位数阶的多项式时间近似方案，其误差随网格尺寸趋于零；3）适用于任意网格尺寸、用于计算分位数阶与字典序低阶的蒙特卡洛方法。