Westling and Carone (2020) recently proposed a unified framework for studying the large sample distributional properties of a class of monotone function estimators, substantially expanding the scope of such estimators in statistics, biostatistics, machine learning, and econometrics. The generic limiting distribution of those estimators is characterized as the greatest convex minorant of a non-stationary Gaussian process, which depends on several nuisance functional parameters. Unfortunately, the standard nonparametric bootstrap is unable to consistently approximate the large sample distribution of the monotone function estimators, making statistical inference a challenging endeavour in applications. To solve this problem, we present a valid bootstrap-based inference procedure for the general class of monotone function estimators, which relies on a carefully crafted and automatic transformation of the estimator. Our proposed method only requires the consistent estimation of a single unknown quantity, for which we propose an automatic procedure based on numerical derivative estimation. Under random sampling, our inference method restores validity of the exchangeable bootstrap, and therefore of the standard nonparametric bootstrap in particular. We illustrate our methods with several examples, and we also provide a small simulation study.
Matias D. Cattaneo is a Professor of Operations Research and Financial Engineering (ORFE) at Princeton University, where he is also an Associated Faculty in the Department of Economics, the Center for Statistics and Machine Learning (CSML), and the Program in Latin American Studies (PLAS). His research spans over econometrics, statistics, data science and decision science, with particular interest in program evaluation and causal inference. Matias earned a Ph.D. in Economics in 2008 and an M.A. in Statistics in 2005 from the University of California at Berkeley. He also completed an M.A. in Economics at Universidad Torcuato Di Tella in 2003 and a B.A. in Economics at Universidad de Buenos Aires in 2000.