Title: Can we trust the bootstrap? (for moderately difficult statistical problems)

Speaker: Noureddine El Karoui, Criteo Research and UC Berkeley

Abstract:
The bootstrap is an important and widely used tool for answering inferential questions in Statistics. It is particularly helpful in many analytically difficult situations. 

I will discuss the performance of the bootstrap for simple inferential problems in moderate and high-dimension. 

For instance, one can ask whether the bootstrap provides valid confidence intervals for individuals parameters in linear regression when the number of predictors is not infinitely small compared to the sample size. Similar questions related to Principal Component analysis are also natural from a practical standpoint. 

We will see that the answer to these questions is generally negative. 

Our assessment will be done through a mix of numerical and theoretical investigations. The theory will be developed under the assumptions that the ratio of number of predictors to number of observations is kept fixed in our asymptotics. This is a way to keep the ``statistical difficulty" of the problem fixed in the asymptotics. These asymptotic results tend to reflect the finite sample behavior of statistical methods better than traditional asymptotics. 
Interestingly, bootstrap methods that are thought to be perform equivalently well for inference - based on classical asymptotic arguments -  will be shown to have very different behavior numerically and in our theoretical framework. For instance, some are very conservative and some are very anti-conservative, while they are equally "intuitive". 

I will also discuss the behavior of other resampling plans, such as the jackknife, as well as ways to fix some of the problems we have identified. 

Based on joint papers with Elizabeth Purdom, UC Berkeley.