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

arXiv: https://arxiv.org/abs/2105.06646 · HTML: https://mlsanity.com/paper/2105.06646/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 May 2021.

## Built on by (most cited first, 4 of 4)

- [Is this model reliable for everyone? Testing for strong calibration](https://mlsanity.com/paper/2307.15247/) (2023)
- [Inference on Variable Importance for Treatment Effect Heterogeneity: Shapley Values and Beyond](https://mlsanity.com/paper/2510.18843/) (2025)
- [Valid Inference for Machine Learning Model Parameters](https://mlsanity.com/paper/2302.10840/) (2023)
- [ScoreStop: Gradient-based early stopping using functional score tests](https://mlsanity.com/paper/2606.02740/) (2026)

## Abstract (arXiv)

It is often of interest to make inference on an unknown function that is a local parameter of the data-generating mechanism, such as a density or regression function. Such estimands can typically only be estimated at a slower-than-parametric rate in nonparametric and semiparametric models, and performing calibrated inference can be challenging. In many cases, these estimands can be expressed as the minimizer of a population risk functional. Here, we propose a general framework that leverages such representation and provides a nonparametric extension of the score test for inference on an infinite-dimensional risk minimizer. We demonstrate that our framework is applicable in a wide variety of problems. As both analytic and computational examples, we describe how to use our general approach for inference on a mean regression function under (i) nonparametric and (ii) partially additive models, and evaluate the operating characteristics of the resulting procedures via simulations.