Scale Adaptation in Conformal Prediction: Efficiency and Coverage Allocation

Conformal prediction provides prediction intervals with finite-sample marginal coverage under exchangeability. However, valid intervals can still be unnecessarily wide when prediction errors are asymmetric or prediction difficulty varies across inputs. We study two related procedures to improve interval efficiency. Efficient Conformal Prediction with Flexible Quantile Allocation (ECP-FQ) chooses how to divide the error budget between the lower and upper residual tails. This adjusts interval placement, but gives every input the same width. ECP-FQ+ extends this approach by using estimated prediction-error magnitude to adjust interval centers and widths across inputs. Our main procedure keeps the center at the point prediction and selects one of five exponents to control how strongly width responds to this estimated error magnitude. Both methods select their interval shape on held-out data and use a separate calibration sample to preserve marginal coverage. Under an oracle location–scale model with varying scale and suitable regularity and integrability conditions, we show that slightly weakening proportional width adjustment reduces expected length even after finite-sample calibration. We also analyze imperfect scale estimates and characterize how scale adaptation affects interval length and conditional coverage. Empirically, compared with choosing between constant-width intervals and widths proportional to estimated prediction-error magnitude, allowing intermediate adjustments yields clear width reductions in 18 of 24 simulation settings with learned predictors. At 90% nominal coverage, the five-exponent selector reduces width by 1.8–12.5% in six of ten real dataset–learner settings, with pointwise paired 95% confidence intervals excluding zero; the remaining four settings show small, statistically inconclusive differences.