On the length of post-model-selection confidence intervals conditional on polyhedral constraints

TL;DR

This paper investigates the length of post-model-selection confidence intervals under polyhedral constraints, revealing that some intervals have infinite expected length and characterizing the distribution of their lengths.

Contribution

It provides a detailed analysis of the expected length of polyhedral method confidence intervals, including conditions for infinite expected length and distributional properties.

Findings

One confidence interval has always infinite expected length.

A necessary and sufficient condition for the other interval's infinite expected length is identified.

The distribution of interval lengths follows a 1/(1-κ) behavior near κ=1.

Abstract

Valid inference after model selection is currently a very active area of research. The polyhedral method, pioneered by Lee, et al. (2016), allows for valid inference after model selection if the model selection event can be described by polyhedral constraints. In that reference, the method is exemplified by constructing two valid confidence intervals when the Lasso estimator is used to select a model. We here study the length of these intervals. For one of these confidence intervals, which is easier to compute, we find that its expected length is always infinite. For the other of these confidence intervals, whose computation is more demanding, we give a necessary and sufficient condition for its expected length to be infinite. In simulations, we find that this sufficient condition is typically satisfied, unless the selected model includes almost all or almost none of the available regressors. For the distribution of confidence interval length, we find that the $\kappa$-quantiles behave like $1/(1-\kappa)$ for $\kappa$ close to $1$. Our results can also be used to analyze other confidence intervals that are based on the polyhedral method.