Inference for Monotone Functions Under Short- and Long-Range Dependence: Confidence Intervals and New Universal Limits

We introduce new point-wise confidence interval estimates for monotone functions observed with additive, dependent noise. Our methodology applies to both short- and long-range dependence regimes for the errors. The interval estimates are obtained via the method of inversion of certain discrepancy statistics. This approach avoids the estimation of nuisance parameters such as the derivative of the unknown function, which previous methods are forced to deal with. The resulting estimates are therefore more accurate, stable, and widely applicable in practice under minimal assumptions on the trend and error structure. The dependence of the errors especially long-range dependence leads to new phenomena, where new universal limits based on convex minorant functionals of drifted fractional Brownian motion emerge. Some extensions to uniform confidence bands are also developed. Supplementary materials for this article are available online.

本文提出了针对加性相依噪声观测下单调函数的新型逐点置信区间估计方法。该方法适用于误差项的短程与长程相依情形。此类区间估计通过对特定偏差统计量（discrepancy statistics）进行反演得到。此方法无需估计讨厌参数（如未知函数的导数），而此前的相关方法必须处理此类参数。因此，所得到的估计结果在仅需对趋势项与误差结构施加极弱假设的实践场景中，具备更高的准确性、稳定性与广泛适用性。误差项的相依性，尤其是长程相依性，会催生新的现象：基于带漂移的分数布朗运动（drifted fractional Brownian motion）的凸下确界泛函的新型通用极限应运而生。本文还针对一致置信带开展了相关拓展工作。本文的补充材料可在线获取。