The HulC: Confidence Regions from Convex Hulls

Abstract

We develop and analyze the HulC, an intuitive and general method for constructing confidence sets using the convex hull of estimates constructed from subsets of the data. Unlike classical methods which are based on estimating the (limiting) distribution of an estimator, the HulC is often simpler to use and effectively bypasses this step. In comparison to the bootstrap, the HulC requires fewer regularity conditions and succeeds in many examples where the bootstrap provably fails. Unlike subsampling, the HulC does not require knowledge of the rate of convergence of the estimators on which it is based. The validity of the HulC requires knowledge of the (asymptotic) median-bias of the estimators. We further analyze a variant of our basic method, called the Adaptive HulC, which is fully data-driven and estimates the median-bias using subsampling. We show that the Adaptive HulC retains the aforementioned strengths of the HulC. In certain cases where the underlying estimators are pathologically asymmetric the HulC and Adaptive HulC can fail to provide useful confidence sets. We propose a final variant, the Unimodal HulC, which can salvage the situation in cases where the distribution of the underlying estimator is (asymptotically) unimodal. We discuss these methods in the context of several challenging inferential problems which arise in parametric, semi-parametric, and non-parametric inference. Although our focus is on validity under weak regularity conditions, we also provide some general results on the width of the HulC confidence sets, showing that in many cases the HulC confidence sets have near-optimal width.

Median Regularity and Honest Inference

“…The method of proof is constructive, explicitly providing an estimator from a uniformly valid confidence interval and vice versa. The confidence interval obtained from a median regular estimator is constructed via the recently proposed HulC method (Kuchibhotla et al, 2021). As a simple implication of the results of this paper we can conclude that inference for a functional is possible if and only if the HulC methodology yields valid inference with some estimator.…”

“…There exist several settings in the literature where finite sample median unbiased estimators exist. We only mention Pfanzagl (1979) that presents the most general result we know for construction of finite sample median unbiased estimators and refer the reader to Section 3 of Kuchibhotla et al (2021) for further references.…”

“…To construct a confidence interval at arbitrary level α, Theorem 2 (and Remark 2.2) of Kuchibhotla et al (2021) using the estimator θ n , its finite sample bias as approximately γ/2, and B = B α,γ/2 yields a confidence interval CI α,n satisfying…”

“…The non-regular behavior of this estimator is well-known; see, for example, Robins (2004, Appendix 1). On the other hand, there are several ways to construct uniformly valid intervals based on the estimator τ n (for instance, via the HulC procedure of Kuchibhotla et al (2021)).…”