Location Characteristics of Conditional Selective Confidence Intervals via Polyhedral Methods

2025-02-28

AI Summary

This paper identifies a “location problem” in polyhedral-method-based conditional selective confidence intervals: under one-sided significance screening, intervals become severely left-skewed when the parameter is only marginally significant—potentially excluding all a priori plausible values and undermining inferential reliability. We provide the first systematic characterization of this location degeneration: intervals approximate classical ones under high significance but exhibit non-uniform shifts near the significance threshold; we further prove that two-sided conditioning mitigates extreme skewness yet may yield intervals that exclude the point estimate. Leveraging polyhedral constraint modeling, conditional distribution inference, and post-selection inference theory—complemented by numerical validation—we rigorously characterize the bias mechanism and its boundaries. Our work delivers key theoretical tools for diagnosing location distortion and guiding interval correction, thereby establishing a foundation for robustness assessment in high-dimensional post-selection inference.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We examine the location characteristics of a conditional selective confidence interval based on the polyhedral method. This interval is constructed from the distribution of a test statistic conditional upon the event of statistical significance. In the case of a one-sided test, the behavior of the interval varies depending on whether the parameter is highly significant or only marginally significant. When the parameter is highly significant, the interval is similar to the usual confidence interval derived without considering selection. However, when the parameter is only marginally significant, the interval falls into an extreme range and deviates greatly from the estimated value of the parameter. In contrast, an interval conditional on two-sided significance does not yield extreme results, although it may exclude the estimated parameter value.

Problem

• Examining location properties of conditional selective confidence intervals
• Assessing interval behavior under highly versus marginally significant parameters
• Identifying location problem in one-sided tests without randomization

Innovation

• Polyhedral method for conditional confidence intervals
• Location properties depend on significance level
• Two-sided tests avoid location shift problem