Bootstrap-assisted inference for generalized Grenander-type estimators
针对广义Grenander型估计量（一类非参数单调函数估计器）的推断难题，提出一种基于自助法的推断程序，通过自动变换和数值导数估计，实现对函数平坦度未知情形的稳健推断。
Westling and Carone (Ann. Statist. 48 (2020) 1001–1024) 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 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.