Shortest fixed-width confidence intervals for a bounded parameter: The Push algorithm

2025-11-17

AI Summary

This paper addresses the optimal construction of fixed-width confidence intervals under bounded parameter spaces, aiming to minimize interval length while ensuring non-decreasing coverage and a prespecified confidence level. We propose a generalized Push algorithm leveraging the monotone likelihood ratio (MLR) property, extending the Asparouhov–Lorden framework systematically to general bounded-parameter settings—including binomial, hypergeometric, and normal distributions—for the first time. We rigorously prove that the algorithm achieves minimal interval length under the fixed-width constraint and derive necessary and sufficient conditions for solution existence. The method integrates dynamic programming with a hybrid discrete–continuous search strategy and is efficiently implemented in R. Empirical evaluation on real-world data—including WHO tobacco use statistics—demonstrates substantial improvements in statistical inference accuracy and robustness over classical approaches, notably the normal z-interval.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Algorithm Configuration
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We present a method for computing optimal fixed-width confidence intervals for a single, bounded parameter, extending a method for the binomial due to Asparaouhov and Lorden, who called it the Push algorithm. The method produces the shortest possible non-decreasing confidence interval for a given confidence level, and if the Push interval does not exist for a given width and level, then no such interval exists. The method applies to any bounded parameter that is discrete, or is continuous and has the monotone likelihood ratio property. We demonstrate the method on the binomial, hypergeometric, and normal distributions with our available R package. In each of these distributions the proposed method outperforms the standard ones, and in the latter case even improves upon the $z$-interval. We apply the proposed method to World Health Organization (WHO) data on tobacco use.

Problem

• Computes optimal fixed-width confidence intervals for bounded parameters
• Extends Push algorithm method for binomial distributions
• Applies to discrete or continuous parameters with monotone likelihood ratio

Innovation

• Computes optimal fixed-width confidence intervals for bounded parameters
• Produces shortest non-decreasing intervals at given confidence levels
• Applies to discrete or continuous parameters with monotone likelihood ratios