This paper shows how to build prediction intervals that give explicit, separate guarantees for the lower and upper tails of an outcome. Classical conformal prediction methods guarantee that a future value falls inside an interval with probability at least 1−α on average (marginal coverage). The authors extend that idea so practitioners can choose different error rates for the left tail (α−) and the right tail (α+), and get formal guarantees for each side at those levels.

The authors work within the split conformal prediction framework. They first build one-sided conformal intervals for each tail that meet the chosen tail-specific error rates. They then form a two-sided interval by intersecting the one-sided bounds. A key technical ingredient is a signed quantile-style conformity score that keeps the sign of the residual relative to an estimated conditional quantile, rather than truncating it as in existing methods. This lets the calibration step focus directionally on over- or under-prediction.

At a high level the method follows the usual split conformal recipe: split the data into a training set to fit a predictive model and a calibration set to compute conformity scores. The calibration scores determine quantile thresholds that control the probability of falling above the upper bound or below the lower bound. Under the usual exchangeability assumption (roughly: data points are interchangeable), the paper proves finite-sample guarantees for both each tail and the overall marginal coverage. When data are not exchangeable (for example, with temporal dependence or covariate shift), the authors give asymptotic versions of the guarantees.

This direction matters when the two tails have different consequences. The paper highlights finance as an example: an investor may tolerate some upside risk but needs strict control of large losses in the left tail. Simulations reported in the paper show improved directional calibration compared with standard two-sided conformal intervals, especially for skewed or heavy-tailed outcomes. The signed quantile conformity score is reported to outperform the truncated alternative in those settings. The authors also illustrate the approach on a financial example where left-tail control is important.