CRPS-Optimal Binning for Univariate Conformal Regression

We propose a method for non-parametric conditional distribution estimation based on partitioning covariate-sorted observations into contiguous bins and using the within-bin empirical CDF as the predictive distribution. Bin boundaries are chosen to minimise the total leave-one-out Continuous Ranked Probability Score (LOO-CRPS), which admits a closed-form cost function with O(n2 log n) precomputation and O(n2) storage; the globally optimal K-partition is recovered by a dynamic programme in O(n2 K) time. Minimisation of within-sample LOO-CRPS turns out to be inappropriate for selecting K as it results in in-sample optimism. We instead select K by K-fold cross-validation of test CRPS, which yields a U-shaped criterion with a well-defined minimum. Having selected K^* and fitted the full-data partition, we form two complementary predictive objects: the Venn prediction band and a conformal prediction set based on CRPS as the nonconformity score, which carries a finite-sample marginal coverage guarantee at any prescribed level varepsilon. The conformal prediction is transductive and data-efficient, as all observations are used for both partitioning and p-value calculation, with no need to reserve a hold-out set. On real benchmarks against split-conformal competitors (Gaussian split conformal, CQR, CQR-QRF, and conformalized isotonic distributional regression), the method produces substantially narrower prediction intervals while maintaining near-nominal coverage.