For several important monotone parameters, such as the distribution function, monotone density function, and monotone regression function, sensible nonparametric estimators can be obtained by minimizing the empirical risk based on an appropriate loss function. For more complex monotone parameters, such as a monotone covariate-adjusted dose-response curve, or in the context of more complex data structures, this approach may not be possible and alternative approaches are needed.
We discuss general strategies for monotone function estimation in two important settings. In the first setting, the function of interest is considered to be a global summary of the data-generating mechanism, such as a distribution function. Specifically, the function is assumed to be a pathwise differentiable parameter. In such cases, an asymptotically linear estimator can be constructed for the evaluation of the function at each point in its domain. However, this pointwise estimator need not respect the monotonicity constraint. We consider the estimator obtained by projecting the pointwise estimator onto the constrained parameter space, and study its asymptotic properties. We apply our results to monotonicity-respecting estimation and inference on a counterfactual cumulative incidence curve in a secondary analysis of a clinical trial of an HIV vaccine.
In the second setting, the function of interest is a local summary of the data-generating mechanism such as a density or regression function. In this setting, the function is not pathwise differentiable, though its primitive function may be. We consider estimating the function by differentiating the greatest convex minorant of an estimator of this primitive. We provide sufficient conditions for consistency and pointwise convergence in distribution of the resulting estimator, both at a high level and in the important special case where the primitive estimator is asymptotically linear. In many important examples, this approach reduces to estimators previously proposed in the literature. Beyond unifying existing methods, this general framework can be used to tackle problems where the traditional loss-based approach may not apply. We use our results to study (i) estimation of a monotone hazard function using survival data subject to possibly informative censoring, and (ii) estimation of a monotone dose-response curve based on data subject to potential confounding.
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