Estimating Distributions with Failure Rate Properties from Noisy Quantile Data

2026-07-14

AI Summary

This study addresses nonparametric distribution estimation from limited quantile observations corrupted by noise, under unknown distributional structure. The authors propose a two-step estimator that incorporates shape constraints on the hazard rate—such as increasing failure rate—by first solving a finite-dimensional convex optimization problem over transformation knots and then reconstructing the cumulative distribution function via shape-preserving interpolation. This work is the first to integrate hazard-rate-based shape constraints with noisy quantile data, establishing a computationally tractable nonparametric framework extendable to various hazard-related properties. Finite-sample error bounds and convergence rates are rigorously derived. Empirical results demonstrate substantial improvements in estimation accuracy and downstream decision quality in revenue management and reliability analysis, offering practical guidance for offline data collection.

Abstract

Estimating an unknown cumulative distribution function (cdf) from data, either as a statistical object of interest or as an input to a downstream optimization problem, is fundamental in operations. In practice, however, distribution estimation is often complicated by incomplete knowledge of the distribution's structure and limited, censored data. To address the first complication, we study distributions satisfying failure-rate shape constraints, especially increasing failure rate (IFR), rather than assuming a fully specified parametric family. To address the second, we consider noisy quantile data: at finitely many prespecified knots, each observation records only whether an independent sample lies below or above the knot. This combination arises naturally in pricing, reliability, and healthcare applications. We formulate the IFR-constrained maximum likelihood estimator and show that the original problem is infinite-dimensional and non-convex. We then develop a tractable two-step approach that solves a finite-dimensional convex optimization problem over transformed knot values and reconstructs a full cdf through shape-preserving interpolation. We establish finite-sample error bounds and convergence rates, yielding practical guidance for offline data collection. We also extend the framework to failure-rate-average, new-better-than-used, and generalized-failure-rate properties. Numerical experiments and case studies in revenue management and reliability demonstrate strong goodness-of-fit and improved downstream decision quality.

Problem

• distribution estimation
• failure rate
• noisy quantile data
• shape constraints
• cumulative distribution function

Innovation

• increasing failure rate
• noisy quantile data
• shape-constrained estimation
• convex optimization
• distribution reconstruction