In this paper, we study nonasymptotic Lp error bounds for interval length and conditional coverage in split conformalized quantile regression (CQR). Our bounds rely on local regularity conditions and accuracy guarantees for the estimated quantiles. We further instantiate our bounds for quantile regression with sparse ReLU neural networks. We also consider covariate shift, where the calibration and test covariates have different distributions, and derive nonasymptotic bounds for this setting. We obtain matching minimax upper and lower bounds in expectation for two constructed fixed-score calibration benchmarks under known covariate shift. The bounds match for every p∈[1,∞] in the scalar problem and for finite p in the K-threshold problem; for the latter, a high-probability minimax lower bound holds for every p∈[1,∞].
Conformal prediction endows arbitrary black-box predictors with finite-sample, distribution-free marginal coverage, yet marginal validity can hide severe covariate-specific miscalibration, while exact distribution-free conditional coverage is finite-sample unattainable. Randomly localized conformal prediction (RLCP) mitigates this gap by calibrating near the test point while preserving marginal coverage. Existing theory, however, lacks finite-sample guarantees for the realized localized set that jointly control conditional validity and oracle efficiency. We provide such guarantees. For any fixed score, under Hölder regularity of the conditional score CDF and standard density and kernel assumptions, we prove high-probability bounds, uniform over a realized localization neighbourhood, for the conditional-coverage gap and the length error relative to the oracle. The bounds decompose into an O(hβ) localization bias and a calibration term decreasing with calibration size, clarifying the bandwidth bias-variance tradeoff and when RLCP tracks the oracle. We also analyze data-split learned scores: when the score targets a pivotal score, as in conformalized quantile regression, uniform local guarantees decompose into fixed-score calibration and uniform score-estimation errors, showing that improved learning sharpens localized guarantees.
Regression problems with bounded continuous outcomes frequently arise in statistical and machine learning applications, such as the analysis of rates and proportions. A central challenge in this setting is predicting the response at a new covariate value. Most of the existing literature has focused either on point prediction or on interval prediction based on asymptotic approximations. We develop conformal prediction intervals for bounded outcomes within the framework of transformation regression models, encompassing widely used models such as beta regression and logit-normal regression. We construct non-conformity scores based on model-aligned residuals and identify a quantile-residual score that is particularly well suited to bounded outcomes, bridging normalized conformal prediction and distributional conformal prediction. This score accounts for both the heteroscedasticity inherent in such data and the asymmetry that emerges near the boundaries of the response space. We establish marginal validity and asymptotic conditional validity for both full and split conformal prediction, holding under model misspecification. A comprehensive simulation study confirms that both methods empirically attain valid finite-sample coverage, including cases under model misspecification. A real-data application demonstrates their practical performance against bootstrap-based alternatives.
Covariate shift often occurs because, in many real applications, the source and the target observations may be generated from different distributions. In this case, the standard metric under the source distribution is not appropriate. This paper considers deep neural network estimators for nonparametric quantile and Huber regression under covariate shift and from dependent observations. We deal with a generalized Bernstein-type inequality that is satisfied by many classical models, including i.i.d. observations, φ-mixing, strong mixing, and C-mixing processes. To perform the covariate shift phenomenon, we propose a sparse-penalized deep neural network (SPDNN) estimator that takes into account the discrepancy between the source and target distributions of the data. When the density ratio (between the source and target distributions of the covariate) is unknown, a two steps pre-training procedure is carried out: the first step is devoted to the construction of a least squares SPDNN estimator of the density ratio; which is used in the second step to perform a pre-training reweighted SPDNN estimator of the regression function. For both the quantile and the Huber regression, non-asymptotic error bounds of the proposed SPDNN estimators are established in the class of Hölder smooth functions. These estimators can adaptively attain (up to a logarithmic factor) the minimax optimal convergence rate from i.i.d. data as well as from several classical time series models.