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Consistent Specification Testing With Nuisance Parameters Present Only Under The Alternative
, 1995
"... . 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 ex ..."
Abstract

Cited by 57 (10 self)
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. 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. How and why the nuisance parameter approach works and how it can be extended bears closely on recent developments in artificial neural networks. Statistical content is provided by viewing specification tests with nuisance parameters as tests of hypotheses about Banachvalued 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. 1. Introduction In testing whether or not a parametric statistical model is correctly specified, there are a number of apparently distinct approaches one might take. T...
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"... Our purpose here is to unify the apparently disparate nonparametric and nuisance parameter approaches to testing models consistently for arbitrary misspecification (i.e., with power approach one asymptotically for all deviations from the null). The insight providing this unification is that, fundame ..."
Abstract
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Our purpose here is to unify the apparently disparate nonparametric and nuisance parameter approaches to testing models consistently for arbitrary misspecification (i.e., with power approach one asymptotically for all deviations from the null). The insight providing this unification is that, fundamentally, all the different tests, and in particular the two of direct interest to us, are based on estimates of topological &quot;distances &quot; between a restricted (e.g., parametric) model and an unrestricted model. In this context, the notion of weak denseness or weak denseness of a span in the space containing the object of interest plays the central role. Further, verifying weak denseness is often quite easy. As we shall see, the two forms of the tests are distinct because one estimates the topological distance directly in the nonparametric approach and indirectly in the nuisance parameter approach. By identifying the topological basis for a test and applying the notion of weak denseness appropriately, the fundamental relations between many of the different specifications testing approaches can be appreciated. As just one example, Eubank