Many quantities of interest in modern statistical analysis are  
non-smooth functionals of the underlying generative distribution, the  
observed data, or both. Examples include the test error of a learned  
classifier, parameters indexing an estimated optimal decision policy,  
and coefficients in a regression model after model selection has been  
performed. This lack of smoothness can lead to non-regular asymptotics  
under likely real-life scenarios and thus invalidate standard  
statistical procedures like the bootstrap or series approximations.


The focus of this talk is the development of tools for conducting  
valid statistical inference for non-smooth functionals by means of  
smooth data-driven upper and lower bounds. More specifically, we  
sandwich the non-smooth functional between two smooth bounds and then  
approximate the distribution of the bounds using standard techniques.  
We then use estimated distributional features of the bounds, such as  
the quantiles, to make inference for the original non-smooth  
functional. We leverage the smoothness of the bounds to obtain  
consistent inference under both fixed and local alternatives. This  
consistency is important for high-quality performance in both large  
and small samples. Another important feature of these bounds is that  
they are adaptive to the underlying smoothness of the functional. That  
is, they are asymptotically tight in the case when the generative  
distribution happens to induce sufficient smoothness. A suite of  
empirical examples show that confidence intervals formed using the  
techniques presented in this talk perform favorably to competitors.