Mean-Tilted Intervals: A Generalized-Bayes Approach to Fixed-Content and Tolerance Intervals

Abstract

Intervals with the same probability content can have different endpoint placements and widths. This ambiguity matters for regression and tolerance inference because equal-tailed, mean-preserving, and shortest-contiguous intervals answer different questions. We study the residual-product criterion introduced as Relaxed Quantile Regression and show that, under regularity conditions, its unrestricted regular minimizer is the unique fixed-content interval whose retained mean equals the population mean. We call this target the mean-preserving interval (MPI). Mean-tilted intervals (MTIs) generalize MPI by replacing zero retained-mean balance with a fixed retained-mean offset: \(δ=0\) recovers MPI, and nonzero tilts index other contiguous fixed-content windows, including equal-tailed and shortest-contiguous intervals through distribution-specific tilts. For estimation, we develop a loss-based generalized-Bayes update for the two interval endpoints. A pseudo-asymmetric-Laplace normal-exponential augmentation gives Gibbs computation with generalized-inverse-Gaussian latent-scale updates and conditionally Gaussian endpoint updates. Exact inverse-scale moments also give a deterministic expectation/conditional-maximization mode algorithm. The framework covers ridge-regularized static regression, frozen-feature deep echo state network readouts, and dynamic linear root states. The same geometry motivates calibrated minimum-width tolerance actions. The empirical action selects the shortest closed order-statistic interval at a calibrated retained count, while a Dirichlet-process response-distribution layer gives fixed-interval Beta content probabilities for Bayesian-constrained scans. Tolerance confidence comes from scan calibration; posterior credibility summarizes the fitted generalized posterior.

29 pages, 3 figures, 1 table

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