Statistics Seminar: "Models as Approximations -- A Model-Free Theory of Parametric Regression"

Abstract: In this talk we rethink what we do when we apply regression to data. We will do so without assuming the correctness of a fitted regression model.  We will think of the parameters of the fitted model as statistical functionals, here called ``regression functionals,'' which apply to largely arbitrary (X,Y) distributions.  In this view a fitted model is an approximation, not a "data generating process.''  The natural question is whether such an assumption-lean framework lends itself to a useful statistical theory.  Indeed it does: It is possible to (1) define a notion of well-specification for regression functionals that replaces the notion of correct specification of models, (2) create a well-specification diagnostic for regression functionals based on reweighting the data, (3) prove insightful Central Limit Theorems, (4) clear up the misconception that "model bias'' generates biased estimates, (5) exhibit standard errors of the plug-in/sandwich type as limit cases of (X,Y) bootstrap estimators, and (6) provide theoretical heuristics to indicate that (X,Y) bootstrap standard errors may generally be more stable than sandwich estimators. Joint work with Larry Brown, Arun Kuchibhotla, Richard Berk, Linda Zhao, Ed George.

Host: Todd Kuffner