# The Importance of Being a Band: Finite-Sample Exact Distribution-Free Prediction Sets for Functional Data

arXiv: https://arxiv.org/abs/2102.06746 · HTML: https://mlsanity.com/paper/2102.06746/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Feb 2021.

## Built on by (most cited first, 16 of 16)

- [SAFE: Multitask Failure Detection for Vision-Language-Action Models](https://mlsanity.com/paper/2506.09937/) (2025)
- [Can We Detect Failures Without Failure Data? Uncertainty-Aware Runtime Failure Detection for Imitation Learning Policies](https://mlsanity.com/paper/2503.08558/) (2025)
- [Failure Prediction at Runtime for Generative Robot Policies](https://mlsanity.com/paper/2510.09459/) (2025)
- [Conformal Prediction: A Data Perspective](https://mlsanity.com/paper/2410.06494/) (2024)
- [Rectifying Conformity Scores for Better Conditional Coverage](https://mlsanity.com/paper/2502.16336/) (2025)
- [Hide-and-Seek in Trajectories: Discovering Failure Signals for VLA Runtime Monitoring](https://mlsanity.com/paper/2605.30834/) (2026)
- [Calibrated Physics-Informed Uncertainty Quantification](https://mlsanity.com/paper/2502.04406/) (2025)
- [Failure Identification in Imitation Learning Via Statistical and Semantic Filtering](https://mlsanity.com/paper/2604.13788/) (2026)
- [JANET: Joint Adaptive predictioN-region Estimation for Time-series](https://mlsanity.com/paper/2407.06390/) (2024)
- [Reliable uncertainty quantification for 2D/3D anatomical landmark localization using multi-output conformal prediction](https://mlsanity.com/paper/2503.14106/) (2025)
- [Colorful Pinball: Density-Weighted Quantile Regression for Conditional Guarantee of Conformal Prediction](https://mlsanity.com/paper/2512.24139/) (2025)
- [Tri-Info: Generalizable, Interpretable Failure Prediction for VLA Models via Information Theory](https://mlsanity.com/paper/2606.19998/) (2026)
- [SAFECAST: Robust Failure Detection for VLA Policies with Contrast-Set Training and Calibration](https://mlsanity.com/paper/2608.04246/) (2026)
- [Temporal Difference Calibration in Sequential Tasks: Application to Vision-Language-Action Models](https://mlsanity.com/paper/2604.20472/) (2026)
- [Joint Registration and Conformal Prediction for Partially Observed Functional Data](https://mlsanity.com/paper/2502.15000/) (2025)
- [FailureSpot: Label-Efficient Timestamp-Level Failure Detection for Vision-Language-Action Models](https://mlsanity.com/paper/2609.04277/) (2026)

## Abstract (arXiv)

Functional Data Analysis represents a field of growing interest in statistics. Despite several studies have been proposed leading to fundamental results, the problem of obtaining valid and efficient prediction sets has not been thoroughly covered. Indeed, the great majority of methods currently in the literature rely on strong distributional assumptions (e.g, Gaussianity), dimension reduction techniques and/or asymptotic arguments. In this work, we propose a new nonparametric approach in the field of Conformal Prediction based on a new family of nonconformity measures inducing conformal predictors able to create closed-form finite-sample valid or exact prediction sets under very minimal distributional assumptions. In addition, our proposal ensures that the prediction sets obtained are bands, an essential feature in the functional setting that allows the visualization and interpretation of such sets. The procedure is also fast, scalable, does not rely on functional dimension reduction techniques and allows the user to select different nonconformity measures depending on the problem at hand always obtaining valid bands. Within this family of measures, we propose also a specific measure leading to prediction bands asymptotically no less efficient than those with constant width.