Consistent specification testing with nuisance parameters present only under the alternative

The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly extend the nuisance parameter approach. Bow and why the nuisance parameter approach works and how it can be extended bear closely on recent developments in artificial neural networks. Statistical content is provided by viewing specification tests with nuisance parameters as tests of hypotheses about Banach-valued random elements and applying the Banach central limit theorem and law of iterated logarithm, leading to simple procedures that can be used as a guide to when computationally more elaborate procedures may be warranted.