Nonparametric inference on monotone functions, with applications to observational studies
In this dissertation, we study general strategies for constructing nonparametric monotone function estimators in two broad statistical settings. In the first setting, a sensible initial estimator of the monotone function of interest is available, but may fail to be monotone. We study the correction of such an estimator obtained via projection onto the space of functions monotone over a finite grid in the domain. We demonstrate that this corrected estimator is always at least as good in supremum norm as the initial estimator, and provide conditions under which the two estimators are asymptotically equivalent. In the second setting we consider, a sensible estimator of the primitive of the function of interest is available. In this setting, estimators considered in the literature have often been of so-called Grenander type, being representable as the left derivative of the greatest convex minorant of the primitive estimator. We provide general conditions for consistency and pointwise convergence in distribution of a class of generalized Grenander-type estimators. This broad class allows the minorization operation to be performed on a data-dependent transformation of the domain. We use our general results from the second setting to perform a detailed study of generalized Grenander-type estimation of a monotone covariate-adjusted regression curve, which describes the effect of a continuous exposure on an outcome while adjusting for potential confounders using the G-formula. In particular, we show how our results can be used to conduct doubly-robust inference for this parameter.