Adaptive Confidence Intervals for Non-Smooth Functionals.

[摘要] Many quantities of interest in modern statistical analysis are non-smoothfunctionals of the underlying generative distribution, the observed data,or both.Examples include the test error of a learned classifier, parameters indexing an estimated optimal dynamic treatment regime, andthe coefficients in a regression model after model selection has been performed.This lack of smoothness can lead to non-regular asymptotics under many `real-life;; scenarios and thus invalidate standard statistical procedures like the bootstrap and series approximations.Statistical procedures that either ignore or assume away this non-regularity can performquite poorly, especially in small samples.The aim of this dissertation is (i) to illustrate the impact that non-regularitycan have on the performance of statistical inference procedures, especially in small samples, and (ii) the development of tools for conducting theoretically valid statistical inference for non-smooth functionals. In particular,we aim to develop confidence intervals that deliver asymptotically correct coverage under both fixed and local alternatives.To constructconfidence intervals we first derive smooth, data-dependent, upper andlower bounds on the functional of interest and then approximate the distribution of the bounds using standard techniques.We then use estimateddistributional features, such as the quantiles, to make inference for theoriginal non-smooth functional.We leverage the smoothness of thebounds to obtain consistent inference under both fixed an local alternatives.This consistency is instrumental in ensuring good performance in both in both large and small samples.An important feature of these bounds is that they are adaptive to the underlying non-smoothness of the functional.That is, they are asymptotically tight in the case when the generativedistribution happens to induce sufficient smoothness.

[发布机构] University of Michigan

[效力级别] Adaptive Confidence Intervals

[关键词] Nonregular Functionals;Adaptive Confidence Intervals;Statistics and Numeric Data;Science;Statistics