Inference on function-valued parameters using a restricted score test.

Area of Science:

• Statistics
• Econometrics
• Machine Learning

Background:

• Inference on local parameters (e.g., density, regression functions) is crucial but challenging in nonparametric and semiparametric models.
• These complex estimands are often difficult to estimate at parametric rates, hindering calibrated inference.
• Many such estimands can be represented as minimizers of population risk functionals.

Purpose of the Study:

• To propose a general framework for nonparametric inference on infinite-dimensional risk minimizers.
• To extend the score test methodology to handle complex function estimation problems.
• To demonstrate the broad applicability of the proposed framework across various statistical challenges.

Main Methods:

• Leveraging the representation of estimands as minimizers of population risk functionals.
• Developing a nonparametric extension of the score test for inference on risk minimizers.
• Applying the framework to mean regression functions under nonparametric and partially additive models.

Main Results:

• The proposed framework is shown to be applicable to a wide range of problems.
• Analytic and computational examples illustrate the method's utility for mean regression.
• Simulations are used to evaluate the operating characteristics of the developed procedures.

Conclusions:

• The general framework provides a powerful tool for nonparametric inference on risk minimizers.
• The nonparametric score test extension offers a viable solution for calibrated inference in complex models.
• Potential applications include assessing effect heterogeneity, density inference, and conditional independence testing.