Multiscale Sequential Auditing of Quantile Regression with Point Masses

Abstract

A correct conditional quantile at a point mass need not have an exact nominal hit probability, and pooled residual signs can conceal opposing conditional errors. We construct sequential audits of quantile-set membership using strict left-tail, tie, and strict right-tail outcomes. A closed-form information projection identifies the available sign information. Combining local continuous betting mixtures through context-tree weighting yields an anytime-valid audit and a finite-path oracle inequality over all prunings of a fixed covariate tree. A countable mixture over refining trees attains the full conditional information rate under iid alternatives with uniformly positive sign probabilities; no valid sign-based evidence sequence has a larger almost-sure exponential rate. We also bound the information lost at finite partition resolution and give an exact recurrence for evaluating the local mixtures. A closure argument identifies the limits of testing a prescribed quantile selection, and test inversion gives confidence sequences under conditional model correctness. Simulations examine localized errors, subgroup cancellation, and adaptive covariate allocation. A held-out count-response example illustrates why exact-hit calibration and quantile validity are different hypotheses.