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Conformal Prediction Intervals with Tail-Specific Guarantees

This paper introduces a split conformal prediction framework that extends classical marginal coverage guarantees to provide explicitly calibrated, finite-sample (exchangeable) and asymptotic (non-exchangeable) control over upper and lower prediction tails separately, offering improved directional accuracy for skewed data and applications like financial risk management.

Original authors: Simone Cuonzo, Nina Deliu

Published 2026-06-17

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CC BY 4.0

Original authors: Simone Cuonzo, Nina Deliu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a weather forecaster trying to predict tomorrow's temperature. You want to give people a range, like "It will be between 50°F and 70°F."

The Old Way (Standard Prediction Intervals)
Traditionally, statisticians use a method called "Conformal Prediction" to create these ranges. They aim for a simple rule: "90% of the time, the real temperature will fall inside our box."

Think of this like a safety net. If you throw a ball, the net catches it 90% of the time. But here's the catch: the net might be lopsided. It might catch the ball 99% of the time when the ball is thrown high (the upper tail), but only catch it 81% of the time when the ball is thrown low (the lower tail). As long as the total catch rate is 90%, the old method is happy.

The Problem
In the real world, being wrong in one direction is often much worse than being wrong in the other.

In Finance: If you are an investor, a "low" prediction (a crash) is terrifying and could bankrupt you. A "high" prediction (a massive gain) is just a missed opportunity. You need a safety net that is extra strong on the low side, even if it's slightly looser on the high side.
In Medicine: If a patient's blood pressure is too high, they might have a stroke. If it's too low, they might faint. But if the doctor is worried about a stroke, they need a guarantee that the pressure won't spike, regardless of what happens on the low side.

The old "90% total" rule doesn't care about this. It treats a missed high prediction the same as a missed low prediction.

The New Solution (Tail-Specific Guarantees)
The authors of this paper, Simone Cuonzo and Nina Deliu, propose a smarter way to build these safety nets. Instead of building one big net, they build two separate nets and snap them together.

The Lower Net: They build a specific safety net just for the "low" side. They say, "We promise this net will catch the low values 95% of the time."
The Upper Net: They build a separate net for the "high" side. They say, "We promise this net will catch the high values 95% of the time."
The Intersection: They take the area where these two nets overlap. This becomes their final prediction interval.

Why is this better?

Customized Safety: You can tell the computer, "I don't care about the high side as much, but I need 99% certainty on the low side." The math adjusts the nets to give you exactly that.
No More Lopsided Nets: In the old method, if the data was "skewed" (like a pile of sand leaning to one side), the safety net would often fail on the steep side. The new method fixes this by calibrating each side independently.
Financial Safety: The paper tests this on stock market data. Investors are terrified of "left tail" events (crashes). The new method successfully guarantees that the predicted "Value at Risk" (the maximum likely loss) is accurate, whereas the old method often failed to catch these crashes, leaving investors exposed.

The Trade-off
Is there a downside? Yes, slightly. Because you are building two separate nets and snapping them together, the final range might be a tiny bit wider (less precise) than the old method. However, the authors argue that for high-stakes decisions (like managing money or patient health), it is better to have a slightly wider range that you can trust on the dangerous side, rather than a tight range that might fail you when you need it most.

In a Nutshell
This paper introduces a new mathematical tool that lets you build prediction intervals with customized safety guarantees. Instead of hoping for a good average, you can explicitly demand, "I need 99% certainty that I won't lose money," and the math ensures that promise is kept, even when the data is messy or unpredictable.

Technical Summary: Conformal Prediction Intervals with Tail-Specific Guarantees

Problem Statement
Classical conformal prediction (CP) frameworks provide prediction intervals with guaranteed marginal coverage (i.e., the probability that the true response falls within the interval is at least $1-\alpha$ ). However, this global guarantee treats the upper and lower tails of the predictive distribution symmetrically. In many real-world decision-making contexts—such as financial portfolio management (where downside risk is more critical than upside potential) or clinical monitoring (where specific thresholds trigger distinct actions)—the consequences of errors in the two tails are asymmetric. Standard CP methods do not offer explicit control over individual tail probabilities, potentially leading to systematic under-coverage in the tail of greatest concern while over-covering the other.

Methodology
The authors propose a framework to construct prediction intervals that satisfy both global marginal coverage and explicitly calibrated tail-specific coverage guarantees. The methodology is built upon Split Conformal Prediction and extends to non-exchangeable data via Adaptive Conformal Inference (ACI) and Dynamically-tuned Adaptive Conformal Inference (DtACI).

The core procedure involves a two-step construction:

One-Sided Intervals: The authors construct separate lower-bounded ( $[L, +\infty)$ ) and upper-bounded ( $(-\infty, U]$ ) conformal intervals. This is achieved by defining specific one-sided conformity scores. Notably, they introduce a signed quantile score ( $s_{q\text{-sgn}}$ ) which retains the sign of the residual relative to the estimated quantile, contrasting with the truncated scores used in standard Conformalized Quantile Regression (CQR).
Intersection: The final two-sided prediction interval is obtained by intersecting the independently calibrated one-sided intervals: $C_{n+1}^\cap = [L_{n+1}, U_{n+1}]$ .

For non-exchangeable data (e.g., time series with temporal dependence or distribution shifts), the framework adapts the miscoverage levels ( $\alpha^-$ and $\alpha^+$ ) for the lower and upper tails independently using online learning recursions (ACI) or an ensemble of learners with exponential weighting (DtACI). This allows the method to track target violation rates for each tail separately even under non-stationarity.

Key Contributions
The paper outlines four primary contributions:

C1 (Tail-Specific Validity): Introduction of a stronger notion of marginal validity that decomposes the total miscoverage $\alpha$ into user-defined directional components ( $\alpha^-$ and $\alpha^+$ ), allowing practitioners to assign stricter coverage requirements to specific tails.
C2 (Theoretical Guarantees): Proof of finite-sample tail-specific and global marginal coverage guarantees under the assumption of exchangeability. Additionally, asymptotic guarantees and short-term regret bounds are established for non-exchangeable data under ACI and DtACI.
C3 (Empirical Performance): Demonstration through simulation studies that the proposed approach achieves improved directional calibration compared to classical two-sided CP intervals, particularly in skewed and heavy-tailed distributions. The study highlights that the signed quantile score avoids the extreme over-coverage issues seen in truncated scores under strong skewness.
C4 (Financial Application): Application of the framework to financial return forecasting, specifically for Value-at-Risk (VaR) estimation. The method provides a distribution-free, theoretically valid approach to controlling downside risk (left tail) while simultaneously characterizing upside potential.

Results

Simulations: In scenarios involving skewed Student-t distributions (both IID and AR(1)), standard two-sided CP intervals and benchmarks exhibited significant under-coverage in the lower tail and extreme over-coverage (up to 100%) in the upper tail. The proposed intersection method successfully restored the target coverage levels for both tails independently, albeit with a slight increase in interval width (efficiency loss) due to the non-vanishing gap between probability bounds.
Financial Application: Applied to ETFs (SPY, TQQQ, XLE) with varying risk profiles, the proposed DtACI-based tail-calibrated intervals maintained the target lower-tail coverage (VaR) across different market regimes. In contrast, classical two-sided CP intervals and parametric GARCH-based VaR forecasts systematically under-covered the left tail, failing to provide the strict risk control required for regulatory and management purposes.

Significance and Claims
The paper claims that its framework addresses a critical gap in uncertainty quantification: the inability of standard CP to handle asymmetric consequences of prediction errors. By decoupling the calibration of the two tails, the method offers a robust tool for risk management where specific directional guarantees are paramount. The authors emphasize that their approach provides finite-sample guarantees for exchangeable data and asymptotic guarantees with regret bounds for non-exchangeable data, all without requiring strong distributional assumptions. The work positions itself as a practical solution for domains like finance and medicine, where "one size fits all" coverage is insufficient for decision-making under asymmetric risk.

Technical Summary: Conformal Prediction Intervals with Tail-Specific Guarantees

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