On the Length of Post-Model-Selection Confidence Intervals Conditional on Polyhedral Constraints

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 κ-quantiles behave like 1/(1 − κ) for κ close to 1. Our results can also be used to analyze other confidence intervals that are based on the polyhedral method.

Selective Inference for Effect Modification: An Empirical Investigation

“…Such phenomenon was observed in the original article by Lee et al (2016). More recently Kivaranovic and Leeb (2018) has proven that the expected length of the selective confidence interval constructed this way is infinity.…”

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“…2 ) random variable, truncated to the set S Ĉ1 , Ĉ2 defined in ( 14), we have that F S Ĉ1 , Ĉ2 µ,σ 2 ||ν|| 2 2 (t) is a monotonically decreasing function of µ for each t ∈ S Ĉ1 , Ĉ2 (see, e.g., Lemma A.2. of Kivaranovic and Leeb [2020]). Since α 2 < 1 − α 2 , it follows that θ l (t) and θ u (t) defined in ( 18) are unique, and that θ l (t) < θ u (t).…”

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“…Consider the case θ ≪ 0 : in such a situation the selection probability P(X > 0) is vanishingly small, and the data outcome X = x o contains little information about the value of θ . Indeed, Kivaranovic and Leeb 17 show that such confidence intervals have infinite expected length under repeated sampling. Suppose, instead, we apply the randomisation idea, and provide inference on θ if and only if U = X + W > 0 , where W is random noise, independent of X, with distribution N (0, γ ) .…”