Some Genericity Analyses in Nonparametric Statistics‡

Many nonparametric estimators and tests are naturally set in infinite dimensional contexts. Prevalence is the infinite dimensional analogue of full Lebesgue measure, shyness the analogue of being a Lebesgue null set. A prevalent set of prior distributions lead to wildly inconsistent Bayesian updating when independent and identically distributed observations happen in class of infinite spaces that includes R and N. For any rate of convergence, no matter how slow, only a shy set of target functions can be approximated by consistent nonparametric regression schemes in a class that includes series approximations, kernels and other locally weighted regressions, splines, and artificial neural networks. When the instruments allow for the existence of an instrumental regression, the regression function only exists for a shy set of dependent variables. The instruments allow for existence in a counterintuitive dense set of cases, shyness is an open question. A prevalent set of integrated conditional moment (ICM) specification tests are consistent, a dense subset of the finitely parametrized ICM tests are consistent, prevalence is an open question. Date: December 10, 2002. ‡Many thanks Xiaohong Chen, Stephen Donald, and Haskell Rosenthal, Dan Slesnick, Hal White, and Paul Wilson for helpful conversations about this paper. Tom Wiseman and Jeff Ely helped with some difficult terminological issues, but I’m not sure they want to be thanked.