Location Characteristics of Conditional Selective Confidence Intervals via Polyhedral Methods

TL;DR

This paper investigates the location properties of conditional selective confidence intervals derived via polyhedral methods, revealing how their behavior varies with the significance level and testing approach, especially in marginal cases.

Contribution

It provides a detailed analysis of the location characteristics of polyhedral-based confidence intervals, highlighting differences between highly significant and marginally significant cases.

Findings

Highly significant cases produce intervals similar to traditional confidence intervals.

Marginally significant cases can lead to intervals shifted far to the left, excluding plausible values.

Two-sided or randomized tests avoid the location problem.

Abstract

We examine the location properties of a conditional selective confidence interval constructed via the polyhedral method. The interval is derived from the distribution of a test statistic conditional on the event of statistical significance. For a one-sided test, its behavior depends on whether the parameter is highly or only marginally significant. In the highly significant case, the interval closely resembles the conventional confidence interval that ignores selection. By contrast, when the parameter is only marginally significant, the interval may shift far to the left of zero, potentially excluding all a priori plausible parameter values. This "location problem" does not arise if significance is determined by a two-sided test or by a one-sided test with randomized response (e.g., data carving).