🤖 AI Summary
This study addresses the limitation of traditional quantile regression, which requires prespecifying endpoint quantiles and thus struggles to flexibly characterize optimal intervals with fixed probability content but variable location—such as equal-tailed or shortest continuous intervals. The authors propose a mean-tilting framework that relaxes the check loss of quantile regression to estimate unlabeled interval root functionals and explicitly links interval location to internal mean shifts. Building on this, they develop a loss-based generalized Bayesian inference approach, integrating pseudo-asymmetric Laplace normal-exponential augmentation, Nishimura–Suchard augmentation, and Gaussian prior samplers, thereby accommodating both static regression and dynamic state-space extensions. The standard relaxed quantile regression algorithm has been implemented and validated for root functional estimation; the non-zero tilting algorithm has been derived and awaits implementation and empirical verification.
📝 Abstract
Probability content does not by itself determine interval placement. We study the interval functional induced by relaxed quantile regression (RQR), whose residual-product check loss estimates two unlabeled raw roots without preassigning endpoint quantiles. Under explicit conditions, the unrestricted population minimizer is the unique contiguous content-c interval whose retained mean equals the population mean. A fixed mean tilt preserves content while shifting that retained mean to mu + delta. Interior admissible tilts index all finite-root interior content-c windows, with boundary members obtained as qualified one-sided limits. Equal-tailed and shortest-contiguous intervals therefore have distribution-specific recovery tilts. We construct loss-based generalized posteriors using a pseudo-asymmetric-Laplace normal-exponential augmentation. A fixed-rate mean-tilted sampler covers static regression under proper Gaussian priors, with ordinary RQR obtained exactly at zero tilt. The implemented ordinary branch also supports a conditional-Gaussian regularized-horseshoe adapter using the Nishimura-Suchard augmentation (RHS-NS), and a frozen deep echo-state-network feature matrix is a deterministic nonlinear-design specialization of the same static scan. A dynamic linear extension replaces coefficient blocks by alternating root-specific forward-filtering backward-sampling steps: the stacked state prior is Gaussian, but the joint augmented observation kernel is quartic. Current software and empirical evidence concern ordinary RQR; nonzero-tilt algorithms are derived but not yet implemented or validated. All updates concern interval-root functionals under a loss and prior, not a response likelihood or posterior-predictive responses.