It is often of interest to assess whether a function-valued statistical parameter, such as a density function or a mean regression function, is equal to any function in a class of candidate null parameters. This can be framed as a statistical inference problem where the target estimand is a scalar measure of dissimilarity between the true function-valued parameter and the closest function among all candidate null values. These estimands are typically defined to be zero when the null holds and positive otherwise. While there is well-established theory and methodology for performing efficient inference when one assumes a parametric model for the function-valued parameter, methods for inference in the nonparametric setting are limited. When the null holds, and the target estimand resides at the boundary of the parameter space, existing nonparametric estimators either achieve a non-standard limiting distribution or a sub-optimal convergence rate, making inference challenging. In this work, we propose a strategy for constructing nonparametric estimators with improved asymptotic performance. Notably, our estimators converge at the parametric rate at the boundary of the parameter space and also achieve a tractable null limiting distribution. As illustrations, we discuss how this framework can be applied to perform inference in nonparametric regression problems, and also to perform nonparametric assessment of stochastic dependence.

翻译：评估函数值统计参数（如密度函数或均值回归函数）是否等于候选零参数类中的某个函数，通常是研究兴趣所在。这可以转化为一个统计推断问题，其中目标估计量是真实函数值参数与所有候选零值中最接近函数之间的标量差异度量。这些估计量通常在零假设成立时定义为零，否则为正。尽管在假设函数值参数服从参数模型时，已有完善的理论和方法进行高效推断，但非参数设置下的推断方法仍有限。当零假设成立且目标估计量位于参数空间边界时，现有的非参数估计量要么具有非标准极限分布，要么收敛速度次优，这使得推断具有挑战性。本文提出了一种构建具有改进渐近性能的非参数估计量的策略。值得注意的是，我们的估计量在参数空间边界处以参数速率收敛，并且具有易处理的零极限分布。作为示例，我们讨论了如何将该框架应用于非参数回归问题的推断以及随机依赖性的非参数评估。