  The order of smoothness chosen in nonparametric estimation problems is
critical. This choice balances the tradeoff between model parsimony and data
overfitting. The most common approach used in this context is cross-validation.
However, cross-validation is computationally time consuming and often precludes
valid post-selection inference without further considerations. With this in
mind, borrowing elements from the objective Bayesian variable selection
literature, we propose an approach to select the degree of a polynomial basis.
Although the method can be extended to most series-based smoothers, we focus on
estimates arising from Bernstein polynomials for the regression function, using
mixtures of g-priors on the model parameter space and a hierarchical
specification for the priors on the order of smoothness. We prove the
asymptotic predictive optimality for the method, and through simulation
experiments, demonstrate that, compared to cross-validation, our approach is
one or two orders of magnitude faster and yields comparable predictive
accuracy. Moreover, our method provides simultaneous quantification of model
uncertainty and parameter estimates. We illustrate the method with real
applications for continuous and binary responses.
