A UNIFIED STUDY OF NONPARAMETRIC INFERENCE FOR MONOTONE FUNCTIONS

Area of Science:

• Statistics
• Nonparametric Inference

Background:

• Nonparametric inference on monotone functions is a well-studied area with established Grenander-type estimators.
• Existing methods often involve greatest convex minorants or least concave majorants of primitive function estimators.

Purpose of the Study:

• To provide general conditions for consistency and pointwise convergence in distribution for a class of generalized Grenander-type estimators.
• To extend the applicability of these estimators to more challenging problems and data types.

Main Methods:

• Developing a broad class of generalized Grenander-type estimators.
• Introducing data-dependent transformations of the domain for minorization/majoratization.
• Deriving simpler conditions and distributional theory for asymptotically linear estimators.

Main Results:

• Established general conditions for consistency and pointwise convergence in distribution for the proposed estimators.
• Recovered classical results in various well-studied problems, demonstrating the generalizability of the approach.
• Extended inference capabilities to monotone density/hazard functions with right-censored data and covariate-marginalized conditional mean functions.

Conclusions:

• The generalized Grenander-type estimators provide a unified and flexible framework for nonparametric inference on monotone functions.
• The proposed methods successfully address complex statistical problems, including those requiring flexible learning strategies.
• This work advances the field by extending classical results and enabling new applications in statistical modeling.