Adaptive inference after model selection

Post-selection inference adaptive inference uniform confidence sets

Cite as:

Eric Laber and Susan Murphy (2018). Adaptive inference after model selection . RESEARCHERS.ONE, https://www.researchers.one/article/2018-09-18.

Abstract:

Penalized maximum likelihood methods that perform automatic variable are now ubiquitous in statistical research. It is well-known, however, that these estimators are nonregular and consequently have limiting distributions that can be highly sensitive to small perturbations of the underlying generative model. This is the case even for the fixed “p” framework. Hence, the usual asymptotic methods for inference, like the bootstrap and series approximations, often perform poorly in small samples and require modification. Here, we develop locally asymptotically consistent confidence intervals for regression coefficients when estimation is done using the Adaptive LASSO (Zou, 2006) in the fixed “p” framework. We construct the confidence intervals by sandwiching the nonregular functional of interest between two smooth, data-driven, upper and lower bounds and then approximating the distribution of the bounds using the bootstrap. We leverage the smoothness of the bounds to obtain consistent inference for the nonregular functional under both fixed and local alternatives. The bounds are adaptive to the amount of underlying nonregularity in the sense that they deliver asymptotically exact coverage whenever the underlying generative model is such that the Adaptive LASSO estimators are consistent and asymptotically normal, and conservative otherwise. The resultant confidence intervals possess a certain tightness property among all regular bounds. Although we focus on the Adaptive LASSO, our approach generalizes to other penalized methods.  (Originally published as a technical report in 2014.) 

Published on : September 14 2018
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