Conformal Prediction Intervals with Tail-Specific Guarantees

Abstract: This paper extends classical conformal frameworks for constructing prediction intervals with global marginal coverage $1-α$ to intervals that provide explicitly calibrated guarantees for the upper and lower tails separately. Focusing on split conformal prediction, we first construct lower and upper one-sided conformal intervals that achieve marginal validity, and then derive the induced two-sided interval by intersection. Theoretical results prove both tail-specific and global marginal coverage of the induced two-sided interval. Results are presented first for the exchangeable setting, where coverage has finite-sample guarantees, and then for non-exchangeable data, where guarantees are asymptotic. Simulation studies show that the proposed approach achieves improved directional calibration relative to classical two-sided intervals, especially relevant in skewed data. Finally, the benefit of the proposed framework is showcased in a financial application, where one aims for return maximization while seeking strict control on the left tail.

• The paper introduces decoupled one-sided conformal prediction intervals that enable explicit tail-specific risk control with strong finite-sample guarantees.
• It deploys adaptive extensions such as ACI and DtACI to maintain asymptotic coverage under non-exchangeable and shifting data regimes.
• Empirical studies, including financial VaR applications, demonstrate improved directional calibration over classical CP methods in skewed, nonstationary settings.

Conformal Prediction Intervals with Tail-Specific Guarantees: A Technical Overview

Motivation and Problem Formulation

Conformal prediction (CP) provides a robust, distribution-free framework for constructing prediction intervals with finite-sample marginal coverage under exchangeability. However, classical CP intervals only control the overall (global) probability of coverage failure and do not allow for separate control over lower and upper tails. This limitation is nontrivial in settings such as financial risk management or clinical monitoring, where the consequences of undercoverage in different tails are inherently asymmetric and application domains often demand explicit calibration for tail-specific risks (e.g., Value-at-Risk in finance, toxicity thresholds in clinical trials).

This paper proposes a systematic CP framework that supports explicit, decoupled control over coverage errors in each tail, rather than simply enforcing average coverage. The construction delivers finite-sample marginal and tail-specific coverage under exchangeability, and introduces adaptive extensions that guarantee asymptotic coverage under non-exchangeable, temporally dependent, or distribution-shifting regimes.

Methodological Contributions

The framework advances four primary directions:

1. Tail-specific CP intervals: It introduces decoupled lower and upper one-sided CP procedures, enabling practitioners to set arbitrary miscoverage rates for the lower $(\alpha^-)$ and upper $(\alpha^+)$ tails. The final two-sided interval is obtained via intersection, $$[L_{n+1}, U_{n+1}] = [L_{n+1}, +\infty) \cap (-\infty, U_{n+1}]$$. This enables direct, application-informed risk allocation, such as stricter control on the downside in financial forecasting.
2. Theoretical guarantees: Strong finite-sample coverage results are established under exchangeability: lower and upper one-sided intervals satisfy $P(Y_{n+1} < L_{n+1}) \leq \alpha^-$ and $P(Y_{n+1} > U_{n+1}) \leq \alpha^+$, while their intersection achieves global marginal coverage at level $1 - \alpha$, with $\alpha = \alpha^- + \alpha^+$. Finite-sample optimality bounds (controlled by calibration sample size $n_c$) are formally characterized. For non-exchangeable data, adaptive procedures (ACI, DtACI) are shown to track target miscoverage rates almost surely, with explicit finite-sample regret and calibration deviation bounds.
3. Empirical validation: Simulation studies systematically evaluate the construction across symmetric, heavy-tailed, and skewed distributions, with both IID and AR(1)-dependent data. Compared to standard CP intervals and parametric (CLT-based) benchmarks, the intersection approach achieves substantial improvements in directional calibration—correcting severe instances of lower-tail undercoverage and upper-tail overcoverage by classical methods in asymmetric (skewed) settings—at the expense of a modest increase in average interval width.
4. Practical deployment: The framework is evaluated in a financial VaR estimation context. By focusing coverage calibration on the left tail, the method delivers demonstrably superior alignment with risk requirements compared to both GARCH-based and classical CP baselines, especially under nonstationary and stress periods.

Technical Development

The proposed methodology is built on the following principles:

• Split CP with one-sided conformity scores: Lower-tail and upper-tail intervals are constructed independently using modified quantile or residual-based conformity scores, e.g., for the lower tail,

$s^{q,L}(x, y) = \max\{\hat{q}_{\alpha}(x) - y, 0\}$

and analogously for the upper tail. The final two-sided interval is obtained by intersecting these one-sided sets.

• Direction-specific miscoverage adaptation (ACI, DtACI): For non-exchangeable regimes, adaptive miscoverage tracking is performed separately for lower and upper tails. Online procedures update corresponding target levels, and DtACI further aggregates multiple learning rates to enhance robustness and regret minimization, capitalizing on theoretical guarantees from adaptive online learning.
• Conformity score variants: Both truncated (max-based) and signed (non-truncated) quantile residuals are considered. The empirical results highlight a critical limitation with the truncated variant in skewed settings: extreme overcoverage may occur in one tail, violating the almost-sure distinctness assumption required by standard finite-sample bounds.

Empirical Results

Simulation results (for IID and AR(1) Gaussian, Student-t, and skewed Student-t innovations) demonstrate that the intersection-based approach satisfies both marginal and tail-specific coverage nearly exactly, matching the nominal levels across diverse settings. In skewed distributions, classical CP and parametric intervals systematically fail to achieve nominal tail-specific coverage—specifically, yielding lower-tail undercoverage and upper-tail overcoverage—whereas the intersection-based CP intervals restore calibration for each tail with only minor loss in efficiency (modestly wider intervals).

A financial application to S&P500, leveraged Nasdaq-100, and Energy sector ETF returns over periods of high volatility and structural change (COVID-19 onset, commodity price shocks) reinforces these findings. Classical two-sided CP and GARCH-based VaR methods fail to attain the regulatory or practical violation rates in the presence of skewed, nonstationary data, while the one-sided adaptive CP procedures consistently track the desired tail-specific coverage levels through regime changes.

Theoretical Implications

The work demonstrates that, for unimodal (or quasi-convex) residual distributions, the intersection of independent one-sided CP intervals delivers strictly stronger (i.e., directionally decomposed) coverage guarantees than the classical two-sided interval. The modular approach enables targeting of asymmetric risks, and, unlike equal-tailed and highest density regions, offers calibrated, user-controlled directional reliability.

A notable implication is an inherent efficiency loss at the global interval level—a non-vanishing gap in bounds—relative to standard two-sided split CP. This is an unavoidable cost when enforcing independent control for each tail in the absence of distributional symmetry, and is formally quantified.

Practical and Future Directions

From a practical standpoint, this framework is directly applicable in any predictive scenario where the nature or cost of errors is tail-dependent, especially in finance, medicine, and safety-critical settings. The method’s distribution-free guarantees, adaptability to dependence/shift, and finite-sample validity offer robustness advantages in real-world deployment.

The limitations around efficiency and independence of tail calibration suggest natural directions for further research:

• Jointly-coupled tail calibration: Exploiting dependency between tails to potentially recover interval tightness without sacrificing directional coverage.
• Weighted and non-stationary extensions: Incorporating weighted conformal inference and data-dependent weighting (e.g., through importance weighting for covariate shift).
• Bayesian CP integration: Embedding tail-specific control within Bayesian conformal frameworks to blend prior knowledge with data-driven calibration, potentially enhancing informativeness.

Conclusion

This work establishes a theoretically sound and practically viable framework for constructing conformal prediction intervals with explicit, user-controlled tail-specific coverage guarantees. The method is empirically validated to correct systematic calibration failures of classical approaches in asymmetric or nonstationary environments. It is theoretically supported by finite-sample and asymptotic coverage, with regret bounds for adaptive procedures, and opens multiple avenues for methodological innovation in prediction uncertainty quantification.

Reference: "Conformal Prediction Intervals with Tail-Specific Guarantees" (2606.18199).