Tail allocation for conformal prediction intervals

TL;DR

This paper introduces TA-CQR, a method for split-conformal prediction that optimally allocates tail probabilities to produce shortest prediction intervals with exact coverage.

Contribution

It characterizes the geometry of the optimal tail allocation, proposes a new method for estimating it, and provides theoretical guarantees and empirical validation.

Findings

TA-CQR achieves exact finite-sample marginal coverage.

The method produces shorter prediction intervals compared to traditional approaches.

Theoretical analysis includes oracle geometry and asymptotic properties.

Abstract

We study split-conformal prediction for regression when the reported prediction set must be a single interval, at target marginal coverage $1-\alpha$, where $\alpha$ is the nominal miscoverage level. Under this reporting constraint, the natural conditional target is the shortest interval with conditional mass at least $1-\alpha$, 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 $\alpha$ 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.