Jo. Berger et A. Guglielmi, Bayesian and conditional frequentist testing of a parametric model versus nonparametric alternatives, J AM STAT A, 96(453), 2001, pp. 174-184
Testing the fit of data to a parametric model can be done by embedding the
parametric model in a nonparametric alternative and computing the Bayes fac
tor of the parametric model to the nonparametric alternative. Doing so by s
pecifying the nonparametric alternative via a Polya tree process is particu
larly attractive, from both theoretical and methodological perspectives. Am
ong the benefits is a degree of computational simplicity that even allows f
or robustness analyses to be implemented Default (nonsubjective) versions o
f this analysis are developed herein, in the sense that recommended choices
are provided for the (many) features of the Polya tree process that need t
o be specified. Considerable discussion of these features is also provided
to assist those who might be interested in subjective choices. A variety of
examples involving location-scale models are studied. Finally, it is shown
that the resulting procedure can we viewed as a conditional frequentist te
st, resulting in data-dependent reported error probabilities that have a re
al frequentist interpretation (as opposed to p values) in even small sample
situations.