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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 66 (11 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...
Generalized functional linear models
 Ann. Statist
, 2005
"... We propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function. A linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function, and the expe ..."
Abstract

Cited by 45 (6 self)
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We propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function. A linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function, and the expected value of the response is related to this linear predictor via a link function. If in addition a variance function is specified, this leads to a functional estimating equation which corresponds to maximizing a functional quasilikelihood. This general approach includes the special cases of the functional linear model, as well as functional Poisson regression and functional binomial regression. The latter leads to procedures for classification and discrimination of stochastic processes and functional data. We also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data, using a semiparametric quasilikelihood procedure. An essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated KarhunenLoève expansion. We develop asymptotic inference for the proposed class of generalized regression models. In the proposed asymptotic approach, the truncation parameter increases with sample size, and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics. We establish asymptotic normality for a properly scaled distance
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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