Tail allocation for conformal prediction intervals

2026-04-28

AI Summary

This work addresses the challenge of constructing a single predictive interval for regression tasks that achieves exact marginal coverage at a prescribed level. It proposes Tail-Allocation Conformal Quantile Regression (TA-CQR), which optimizes the central quantile-defined interval to determine an optimal tail allocation and integrates non-negative additive split conformal calibration to guarantee finite-sample exact marginal coverage under the exchangeability assumption. Theoretical analysis reveals the geometric structure of single-interval predictors, showing their equivalence to highest-density intervals under unimodal distributions, and establishes local recoverability, asymptotic negligibility of the calibration radius, and a finite-sample length oracle inequality. Experiments demonstrate that TA-CQR consistently attains superior coverage–length trade-offs, yielding substantially shorter prediction intervals while maintaining exact coverage on both simulated and real-world datasets.

Abstract

We study split-conformal prediction for regression when the reported prediction set must be a single interval, at target marginal coverage $1-α$, where $α$ is the nominal miscoverage level. Under this reporting constraint, the natural conditional target is the shortest interval with conditional mass at least $1-α$, rather than an equal-tailed interval or a possibly disconnected high-probability set. We parameterize this single-interval oracle by a lower-tail allocation, which determines how the nominal miscoverage $α$ is split between the two endpoints, and propose tail-allocation conformalized quantile regression (TA-CQR). TA-CQR estimates this allocation by searching over quantile-defined cores and then applies nonnegative additive split-conformal calibration, retaining exact finite-sample marginal coverage under exchangeability. The main contribution is theoretical. We characterize the oracle geometry, including its highest-density interpretation under unimodality and the positive connectedness cost induced by disconnected highest-density sets. We prove local recovery of the selected allocation and core, establish that calibration radii are asymptotically negligible under endpoint-density conditions, and give a finite-sample calibrated length oracle inequality with explicit grid, endpoint-quantile estimation, and calibration-sampling terms. Simulations and real-data examples report coverage and length jointly.

Problem

• conformal prediction
• prediction intervals
• tail allocation
• quantile regression
• coverage

Innovation

• conformal prediction
• tail allocation
• shortest prediction interval
• quantile regression
• finite-sample coverage