Multiscale Sequential Auditing of Quantile Regression with Point Masses
Abstract
For discrete outcomes, a valid conditional quantile can have a hit probability different from its nominal level. Errors in different covariate groups can also cancel when residual signs are pooled. We study sequential audits that address both problems by testing the two inequalities that define quantile-set membership. The audit records whether each response falls below, equals, or exceeds its forecast. We derive the information projection for this three-outcome experiment and use continuous betting mixtures within the cells of a covariate tree. Context-tree weighting combines the resulting audits over all prunings, with an explicit finite-path oracle bound. Under iid alternatives with uniformly positive sign probabilities, a countable mixture of refining trees attains the full conditional sign-information rate. A matching upper bound shows that no valid evidence sequence based on the same observations can attain a larger almost-sure exponential rate. We also bound the information lost through covariate coarsening, derive a recurrence for the local mixtures, and discuss quantile selection and confidence-sequence inversion. Simulations show how partition resolution affects detection, and a held-out count-data example illustrates the difference between quantile validity and exact-hit calibration.
Related
- Sequential sparse Gaussian process quantile regression
- Bet on Features: Anytime-Valid and Feature-Aware Auditing of Conditional Quantile Forecasters
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BibTeX
@misc{ullah2026multiscale,
title = {{Multiscale Sequential Auditing of Quantile Regression with Point Masses}},
author = {Asad Ullah},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16000},
eprint = {2610.16000},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16000},
}
RIS
TY - UNPB
TI - Multiscale Sequential Auditing of Quantile Regression with Point Masses
AU - Asad Ullah
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.16000
UR - https://arxiv.org/abs/2610.16000
AB - For discrete outcomes, a valid conditional quantile can have a hit probability different from its nominal level. Errors in different covariate groups can also cancel when residual signs are pooled. We study sequential audits that address both problems by testing the two inequalities that define quantile-set membership. The audit records whether each response falls below, equals, or exceeds its forecast. We derive the information projection for this three-outcome experiment and use continuous betting mixtures within the cells of a covariate tree. Context-tree weighting combines the resulting audits over all prunings, with an explicit finite-path oracle bound. Under iid alternatives with uniformly positive sign probabilities, a countable mixture of refining trees attains the full conditional sign-information rate. A matching upper bound shows that no valid evidence sequence based on the same observations can attain a larger almost-sure exponential rate. We also bound the information lost through covariate coarsening, derive a recurrence for the local mixtures, and discuss quantile selection and confidence-sequence inversion. Simulations show how partition resolution affects detection, and a held-out count-data example illustrates the difference between quantile validity and exact-hit calibration.
ER -