Wasserstein Kernel Regression with Dependent Distribution-Valued Predictors: Local Recurrence, Joint Inference, and Multivariate Gaussian Geometry
- 1Laboratory of Statistics and Stochastic Processes, Djillali Liabès University, Sidi Bel Abbes 22000, Algeria
- 2Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
Abstract
We study scalar nonparametric regression with a probability-measure-valued predictor observed along a dependent time series. A Nadaraya–Watson smoother localizes directly in the quadratic Wasserstein metric. The revised theory separates generic functional-kernel arguments from contributions generated by Wasserstein predictor geometry and local recurrence. We introduce an aggregate local-recurrence condition that is weaker than pairwise anti-clustering and is compatible with strongly overlapping empirical-distribution windows. For one-dimensional Gaussian location–scale predictors, we obtain an exact Euclidean small-ball expansion; we then extend the geometry to nonsingular multivariate Gaussian predictors through the Bures–Wasserstein Riemannian metric, obtaining an explicit local dimension and small-ball exponent. Under geometric strong mixing, local Hölder smoothness, and small-ball regularity, we establish strong consistency, an almost-sure rate, a variance expansion, and a triangular-array central limit theorem. We further derive a local long-run-variance CLT, a finite-dimensional joint CLT for several Wasserstein targets, and simultaneous confidence intervals. Feasible inference uses a direct Bartlett–HAC estimator whose consistency is now justified by an explicit fourth-order local cumulant condition rather than by geometric mixing alone. For tuning, we introduce an empirical small-ball admissibility screen and allow selection from a possibly growing grid on a separated chronological block. A truncation extension handles locally unbounded responses. Simulations satisfy the stated bounded-response assumptions and stress-test overlapping windows. The Mauna Loa CO2 illustration adds interpolation, window/horizon, and deterministic-overlap sensitivity, stationarity diagnostics, coefficient intervals, and a stronger AIC-selected autoregressive comparator. The latter can outperform the Wasserstein smoother, so the application is interpreted as an empirical illustration rather than evidence of universal predictive superiority.
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BibTeX
@article{bouabsa2026wasserstein,
title = {{Wasserstein Kernel Regression with Dependent Distribution-Valued Predictors: Local Recurrence, Joint Inference, and Multivariate Gaussian Geometry}},
author = {Wahiba Bouabsa and Fatimah Alshahrani},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203832},
url = {https://www.mdpi.com/2227-7390/14/20/3832},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Wasserstein Kernel Regression with Dependent Distribution-Valued Predictors: Local Recurrence, Joint Inference, and Multivariate Gaussian Geometry
AU - Wahiba Bouabsa
AU - Fatimah Alshahrani
PY - 2026
DA - 2026/10/21
JO - Mathematics
DO - 10.3390/math14203832
UR - https://www.mdpi.com/2227-7390/14/20/3832
AB - We study scalar nonparametric regression with a probability-measure-valued predictor observed along a dependent time series. A Nadaraya–Watson smoother localizes directly in the quadratic Wasserstein metric. The revised theory separates generic functional-kernel arguments from contributions generated by Wasserstein predictor geometry and local recurrence. We introduce an aggregate local-recurrence condition that is weaker than pairwise anti-clustering and is compatible with strongly overlapping empirical-distribution windows. For one-dimensional Gaussian location–scale predictors, we obtain an exact Euclidean small-ball expansion; we then extend the geometry to nonsingular multivariate Gaussian predictors through the Bures–Wasserstein Riemannian metric, obtaining an explicit local dimension and small-ball exponent. Under geometric strong mixing, local Hölder smoothness, and small-ball regularity, we establish strong consistency, an almost-sure rate, a variance expansion, and a triangular-array central limit theorem. We further derive a local long-run-variance CLT, a finite-dimensional joint CLT for several Wasserstein targets, and simultaneous confidence intervals. Feasible inference uses a direct Bartlett–HAC estimator whose consistency is now justified by an explicit fourth-order local cumulant condition rather than by geometric mixing alone. For tuning, we introduce an empirical small-ball admissibility screen and allow selection from a possibly growing grid on a separated chronological block. A truncation extension handles locally unbounded responses. Simulations satisfy the stated bounded-response assumptions and stress-test overlapping windows. The Mauna Loa CO2 illustration adds interpolation, window/horizon, and deterministic-overlap sensitivity, stationarity diagnostics, coefficient intervals, and a stronger AIC-selected autoregressive comparator. The latter can outperform the Wasserstein smoother, so the application is interpreted as an empirical illustration rather than evidence of universal predictive superiority.
SN - 2227-7390
ER -