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Westling and Carone (2020) proposed a framework for studying the large sampledistributional properties of generalized Grenander-type estimators, a versatileclass of nonparametric estimators of monotone functions. The limitingdistribution of those estimators is representable as the left derivative of thegreatest convex minorant of a Gaussian process whose covariance kernel can becomplicated and whose monomial mean can be of unknown order (when the degree offlatness of the function of interest is unknown). The standard nonparametricbootstrap is unable to consistently approximate the large sample distributionof the generalized Grenander-type estimators even if the monomial order of themean is known, making statistical inference a challenging endeavour inapplications. To address this inferential problem, we present abootstrap-assisted inference procedure for generalized Grenander-typeestimators. 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 (possiblyunknown) degree of flatness of the function of interest. The method requiresonly the consistent estimation of a single scalar quantity, for which wepropose an automatic procedure based on numerical derivative estimation and thegeneralized jackknife. Under random sampling, our inference method can beimplemented using a computationally attractive exchangeable bootstrapprocedure. We illustrate our methods with examples and we also provide a smallsimulation study. The development of formal results is made possible by sometechnical results that may be of independent interest.

Bootstrap-Assisted Inference for Generalized Grenander-type Estimators