Functional Regression with Reconstructed Curves: Local Rank Stability and Optimized Confidence Certificates

Reconstructing functional predictors can change nearest-neighbor ranks and introduce regression bias. Our contribution is a set of deterministic and probabilistic results quantifying these effects for an existing rank-weighted smoother. We distinguish direct pointwise inference, same-sample equivalence to the exact-curve estimator, and deterministic prediction-perturbation certification. A joint assignment relaxation bounds fitted-value changes. Under local radial-density and polynomial small-ball assumptions, local crowding gives a sufficient rank-stability rate and resolution condition; this is not a general infinite-dimensional rate. We adapt established threshold weighting to certified intervals with finite-sample coverage only when smoothness, conditional noise, and reconstruction-error upper bounds are valid independently of the response errors used in the interval. The numerical study comprises 20,175 simulated datasets, including irregular grids, Laplace measurement errors, and population principal-component truncation, with paired Monte Carlo uncertainty and neighborhood sensitivity analyses. Certified intervals remain substantially conservative, including after radius optimization. Predictive rankings vary with target location, and PCR outperforms the tested neighbor methods on Tecator. The Tecator experiment audits controlled coarsening of measured spectra; it does not validate inference for an unobserved continuous trajectory or provide prospective reconstruction calibration.