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Calibrated Bayesian Nonparametric Tolerance Intervals

Tony Pourmohamad, Robert Richardson, Bruno Sansó

Abstract

Tolerance intervals provide bounds that contain a specified proportion of a population with a given confidence level, yet their construction remains challenging when parametric assumptions fail or sample sizes are small. Traditional nonparametric methods, such as Wilks' intervals, lack flexibility and often require large samples to be valid. We propose a fully nonparametric approach for constructing one-sided and two-sided tolerance intervals using a calibrated Gibbs posterior. Leveraging the connection between tolerance limits and population quantiles, we employ a Gibbs posterior based on the asymmetric Laplace (check) loss function. A key feature of our method is the calibration of the learning rate, which ensures nominal frequentist coverage across diverse distributional shapes. Simulation studies show that the proposed approach often yields shorter intervals than classical nonparametric benchmarks while maintaining reliable coverage. The framework's practical utility is illustrated through applications in ecology, biopharmaceutical manufacturing, and environmental monitoring, demonstrating its flexibility and robustness across diverse applications.

Calibrated Bayesian Nonparametric Tolerance Intervals

Abstract

Tolerance intervals provide bounds that contain a specified proportion of a population with a given confidence level, yet their construction remains challenging when parametric assumptions fail or sample sizes are small. Traditional nonparametric methods, such as Wilks' intervals, lack flexibility and often require large samples to be valid. We propose a fully nonparametric approach for constructing one-sided and two-sided tolerance intervals using a calibrated Gibbs posterior. Leveraging the connection between tolerance limits and population quantiles, we employ a Gibbs posterior based on the asymmetric Laplace (check) loss function. A key feature of our method is the calibration of the learning rate, which ensures nominal frequentist coverage across diverse distributional shapes. Simulation studies show that the proposed approach often yields shorter intervals than classical nonparametric benchmarks while maintaining reliable coverage. The framework's practical utility is illustrated through applications in ecology, biopharmaceutical manufacturing, and environmental monitoring, demonstrating its flexibility and robustness across diverse applications.
Paper Structure (31 sections, 2 theorems, 37 equations, 3 figures, 8 tables)

This paper contains 31 sections, 2 theorems, 37 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Tolerance Intervals and Population Quantiles
  3. One-Sided Tolerance Intervals
  4. Two-Sided Tolerance Intervals
  5. Tolerance Intervals as Quantile Inference Problems
  6. Gibbs Posteriors for Quantile Inference
  7. Generalized Bayesian Posterior
  8. Check Loss and Quantile Functionals
  9. Construction of Tolerance Intervals
  10. Prior Specification and Posterior Computation
  11. Calibration of the Learning Rate $\eta$
  12. The Calibration Procedure
  13. Practical Considerations
  14. Quantile-Defined Calibration
  15. Content-Defined Calibration
  16. ...and 16 more sections

Key Result

Theorem 1

Under standard regularity conditions for quantile inference (continuity and positivity of $f$ at $Q_\tau$, locally positive prior, compact calibration bracket for $\eta$, and uniform consistency of the estimated coverage curve), the calibrated learning rate $\hat{\eta}_n$ converges in probability to In particular, for the quantile-calibration target used here,

Figures (3)

  • Figure 1: Joint Gibbs posterior draws of $(Q_{\tau_L},Q_{\tau_U})$ and the symmetry-based two-sided tolerance interval construction.
  • Figure 2: Empirical performance of upper one-sided tolerance intervals with content level $P=0.9$ and confidence level $1-\alpha=0.9$ as a function of sample size. Left panel: empirical coverage probability, with the dashed horizontal line indicating the nominal confidence level. Right panel: average upper tolerance bound. Results are shown for the Wilks nonparametric method, the YM method, and the calibrated Gibbs posterior approach.
  • Figure 3: Trajectories of $\eta_s$ across Normal and Pareto distributions. The grid contrasts the stable convergence of the Moderate regime against the volatility of the Aggressive regime and the slow inertia of the Conservative regime.

Theorems & Definitions (2)

  • Theorem 1: One-sided asymptotic validity of calibrated Gibbs bounds
  • Theorem 2: Two-sided asymptotic validity of calibrated Gibbs intervals