# Functional Linear Regression of Cumulative Distribution Functions

arXiv: https://arxiv.org/abs/2205.14545 · HTML: https://mlsanity.com/paper/2205.14545/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 52 references that are in the map.
- First posted on arXiv 28 May 2022.

## Builds on (most cited first, 7 of 7)

- [Cumulative Prospect Theory Meets Reinforcement Learning: Prediction and Control](https://mlsanity.com/paper/1506.02632/) (2015)
- [Off-Policy Risk Assessment in Contextual Bandits](https://mlsanity.com/paper/2104.08977/) (2021)
- [Localized Debiased Machine Learning: Efficient Inference on Quantile Treatment Effects and Beyond](https://mlsanity.com/paper/1912.12945/) (2019)
- [Generalized Risk-Aversion in Stochastic Multi-Armed Bandits](https://mlsanity.com/paper/1405.0833/) (2014)
- [Importance Weight Estimation and Generalization in Domain Adaptation under Label Shift](https://mlsanity.com/paper/2011.14251/) (2020)
- [Functional Linear Regression with Mixed Predictors](https://mlsanity.com/paper/2012.00460/) (2020)
- [hdm: High-Dimensional Metrics](https://mlsanity.com/paper/1608.00354/) (2016)

## Built on by (most cited first, 1 of 1)

- [A Model-Mediated Stacked Ensemble Approach for Depression Prediction Among Professionals](https://mlsanity.com/paper/2506.14459/) (2025)

## Abstract (arXiv)

The estimation of cumulative distribution functions (CDF) is an important learning task with a great variety of downstream applications, such as risk assessments in predictions and decision making. In this paper, we study functional regression of contextual CDFs where each data point is sampled from a linear combination of context dependent CDF basis functions. We propose functional ridge-regression-based estimation methods that estimate CDFs accurately everywhere. In particular, given n samples with d basis functions, we show estimation error upper bounds of widetilde O(√d/n) for fixed design, random design, and adversarial context cases. We also derive matching information theoretic lower bounds, establishing minimax optimality for CDF functional regression. Furthermore, we remove the burn-in time in the random design setting using an alternative penalized estimator. Then, we consider agnostic settings where there is a mismatch in the data generation process. We characterize the error of the proposed estimators in terms of the mismatched error, and show that the estimators are well-behaved under model mismatch. Moreover, to complete our study, we formalize infinite dimensional models where the parameter space is an infinite dimensional Hilbert space, and establish a self-normalized estimation error upper bound for this setting. Notably, the upper bound reduces to the widetilde O(√d/n) bound when the parameter space is constrained to be d-dimensional. Our comprehensive numerical experiments validate the efficacy of our estimation methods in both synthetic and practical settings.