# Geometry of tail allocation in conformal prediction intervals

arXiv: https://arxiv.org/abs/2604.25202 · HTML: https://mlsanity.com/paper/2604.25202/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 30 references that are in the map.
- First posted on arXiv 28 Apr 2026.

## Builds on (most cited first, 16 of 16)

- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Localized Conformal Prediction: A Generalized Inference Framework for Conformal Prediction](https://mlsanity.com/paper/2106.08460/) (2021)
- [Distributional conformal prediction](https://mlsanity.com/paper/1909.07889/) (2019)
- [Conformal Prediction using Conditional Histograms](https://mlsanity.com/paper/2105.08747/) (2021)
- [A comparison of some conformal quantile regression methods](https://mlsanity.com/paper/1909.05433/) (2019)
- [CD-split and HPD-split: efficient conformal regions in high dimensions](https://mlsanity.com/paper/2007.12778/) (2020)
- [Conformalized Survival Analysis](https://mlsanity.com/paper/2103.09763/) (2021)
- [Selection and Aggregation of Conformal Prediction Sets](https://mlsanity.com/paper/2104.13871/) (2021)
- [Length Optimization in Conformal Prediction](https://mlsanity.com/paper/2406.18814/) (2024)
- [Conformal Thresholded Intervals for Efficient Regression](https://mlsanity.com/paper/2407.14495/) (2024)
- [On Volume Minimization in Conformal Regression](https://mlsanity.com/paper/2502.09985/) (2025)
- [Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise](https://mlsanity.com/paper/2406.03258/) (2024)
- [Extreme Conformal Prediction: Reliable Intervals for High-Impact Events](https://mlsanity.com/paper/2505.08578/) (2025)
- [Fast Conformal Prediction using Conditional Interquantile Intervals](https://mlsanity.com/paper/2601.02769/) (2026)
- [Co-optimization for Adaptive Conformal Prediction](https://mlsanity.com/paper/2603.01719/) (2026)

## Abstract (arXiv)

Lower and upper errors of a two-sided conformal prediction interval can have different scientific consequences. The division of target miscoverage between the two endpoints determines the corresponding tail-specific guarantees and can alter interval length at first order when tail scales differ. We characterize this allocation-length relation after separate one-sided split calibration, which preserves the tail-specific guarantees and marginal coverage whenever the allocation is selected independently of the calibration sample. Tail-quantile response to proportional rescaling determines the resulting length geometry. For regularly varying tails, normalized length converges to gγ(c)=c-xi+γ(1-c)-xi, where c is the upper-tail allocation fraction, xi is the tail index, and γ is the lower-to-upper tail-scale ratio. A dominant tail produces a boundary optimum and makes the equal-tail interval asymptotically 2^xi times as long as the optimum. Comparable tails produce an interior optimum, with equal-tail allocation optimal only at matching scales. An empirical allocation rule attains the corresponding optimum without estimating tail parameters. In the de Haan class the effect moves to an additive scale. Calibration resolution determines whether ordinary ranks can realize these allocations. When calibration tail counts remain bounded, two-sided rank feasibility also constrains the allocation. Tail homogeneity transfers the length relation over covariates, while opposite dominant tails preclude one globally efficient allocation.