Westling and Carone (2020) proposed a framework for studying the large sample distributional properties of generalized Grenander-type estimators, a versatile class of nonparametric estimators of monotone functions. The limiting distribution of those estimators is representable as the left derivative of the greatest convex minorant of a Gaussian process whose covariance kernel can be complicated and whose monomial mean can be of unknown order (when the degree of flatness of the function of interest is unknown). The standard nonparametric bootstrap is unable to consistently approximate the large sample distribution of the generalized Grenander-type estimators even if the monomial order of the mean is known, making statistical inference a challenging endeavour in applications. To address this inferential problem, we present a bootstrap-assisted inference procedure for generalized Grenander-type estimators. The procedure relies on a carefully crafted, yet automatic, transformation of the estimator. Moreover, our proposed method can be made ``flatness robust" in the sense that it can be made adaptive to the (possibly unknown) degree of flatness of the function of interest. The method requires only the consistent estimation of a single scalar quantity, for which we propose an automatic procedure based on numerical derivative estimation and the generalized jackknife. Under random sampling, our inference method can be implemented using a computationally attractive exchangeable bootstrap procedure. We illustrate our methods with examples and we also provide a small simulation study. The development of formal results is made possible by some technical results that may be of independent interest.

翻译：Westling和Carone（2020）提出了一个框架，用于研究广义Grenander型估计量的大样本分布性质。这是一类通用的单调函数非参数估计量。这类估计量的极限分布可表示为高斯过程最大凸包络的左导数，该高斯过程的协方差核可能较为复杂，且其单项式均值阶次可能未知（当目标函数的平坦程度未知时）。即使已知均值的单项式阶次，标准非参数自助法也无法一致地逼近广义Grenander型估计量的大样本分布，这使得统计推断在实际应用中面临挑战。为解决该推断问题，我们提出了一种适用于广义Grenander型估计量的自助法辅助推断程序。该程序依托于对估计量进行精心设计但仍具自动性的变换。此外，我们提出的方法具有"平坦性鲁棒"特性，即能自适应于目标函数（可能未知的）平坦程度。该方法仅需一致估计单个标量参数，我们据此提出了基于数值导数估计与广义刀切法的自动估计程序。在随机抽样条件下，我们的推断方法可通过计算高效的交换自助法实现。我们通过案例说明了方法的有效性，并进行了小规模模拟研究。本文形式化结果的推导得益于若干可能具有独立意义的技术性结论。