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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...
Penalized Regression with ModelBased Penalties
, 2000
"... Nonparametric regression techniques such as spline smoothing and local fitting depend implicitly on a parametric model. For instance, the cubic smoothing spline estimate of a regression function based on observations t i ,Y i is the minimizer of # {Y i  (t i )} 2 + # # ( ## ) 2 .Since ..."
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Cited by 14 (2 self)
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Nonparametric regression techniques such as spline smoothing and local fitting depend implicitly on a parametric model. For instance, the cubic smoothing spline estimate of a regression function based on observations t i ,Y i is the minimizer of # {Y i  (t i )} 2 + # # ( ## ) 2 .Since # ( ## ) 2 is zero when is a line, the cubic smoothing spline estimate favors the parametric model (t)=# 0+# 1 t. Here the authors consider replacing # ( ## ) 2 with the more general expression # (L) 2 where L is a linear di#erential operator with possibly nonconstant coe#cients. The resulting estimate of performs well, particularly if L is small. They present present a O(n) algorithm for the computation of . This algorithm is applicable to a wide class of L's. They also suggest a method for the estimation of L. They study our estimates via simulation and apply them to several data sets. R ESUM E Les techniques de regression non parametrique telles que l'ajustement local ou ...
Model selection using wavelet decomposition and applications
 BIOMETRIKA
, 1997
"... In this paper we discuss how to use wavelet decompositions to select a regression model. The methodology relies on a minimum description length criterion which is used to determine the number of nonzero coefficients in the vector of wavelet coefficients. Consistency properties of the selection rule ..."
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Cited by 10 (2 self)
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In this paper we discuss how to use wavelet decompositions to select a regression model. The methodology relies on a minimum description length criterion which is used to determine the number of nonzero coefficients in the vector of wavelet coefficients. Consistency properties of the selection rule are established and simulation studies reveal information on the distribution of the minimum description length selector. We then apply the selection rule to specific problems, including testing for pure white noise. The power of this test is investigated via simulation studies and the selection criterion is also applied to testing for no effect in nonparametric regression.
Nonparametric Checks For SingleIndex Models
 Ann. Statist
, 2005
"... In this paper we study goodnessoffit testing of singleindex models. The large sample behavior of certain scoretype test statistics is investigated. As a byproduct, we obtain asymptotically distributionfree maximin tests for a large class of local alternatives. Furthermore, characteristic functi ..."
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Cited by 8 (3 self)
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In this paper we study goodnessoffit testing of singleindex models. The large sample behavior of certain scoretype test statistics is investigated. As a byproduct, we obtain asymptotically distributionfree maximin tests for a large class of local alternatives. Furthermore, characteristic function based goodnessoffit tests are proposed which are omnibus and able to detect peak alternatives. Simulation results indicate that the approximation through the limit distribution is acceptable already for moderate sample sizes. Applications to two real data sets are illustrated. 1. Introduction. Suppose
Model Fitting and Testing for NonGaussian Data with Large Data Sets
, 1996
"... We consider the application of the smoothing spline to the generalized linear model in large data set situations. First we derive a Generalized Approximate Cross Validation function (GACV ), which is an approximate leaveoutone cross validation function used to choose smoothing parameters. In order ..."
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Cited by 5 (2 self)
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We consider the application of the smoothing spline to the generalized linear model in large data set situations. First we derive a Generalized Approximate Cross Validation function (GACV ), which is an approximate leaveoutone cross validation function used to choose smoothing parameters. In order to apply the GACV function to a large data set situation, we propose a corresponding randomized version of it. To reduce the computational intensity of calculating the smoothing spline estimate, we suggest an approximate solution and a clustering method to choose a subset of the basis functions. Combining randomized GACV with this approximate solution, we apply it to binary response data from the Wisconsin Epidemiological Study of Diabetic Retinopathy in order to establish the accuracy of the model when applied to a large data set.
Testing the Generalized Linear Model Null Hypothesis versus `Smooth' Alternatives
, 1995
"... We consider y i ; i = 1; :::n independent observations from an exponential family with canonical parameter j(x i ), where the predictor variable x is in some index set and j is a `smooth' function of x. The usual GLIM models suppose that j has a parametric form j(x) = P p =1 fi OE (x) where ..."
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Cited by 2 (1 self)
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We consider y i ; i = 1; :::n independent observations from an exponential family with canonical parameter j(x i ), where the predictor variable x is in some index set and j is a `smooth' function of x. The usual GLIM models suppose that j has a parametric form j(x) = P p =1 fi OE (x) where the OE are given. This paper is concerned with testing the hypothesis that j is in the span of a given (low dimensional) set of OE versus general `smooth' alternatives. In the Gaussian case, studied by Cox, Koh, Wahba and Yandell(1988), test statistics are available whose distributions are independent of the nuisance fi , whereas in general this is not the case. We propose a symmetrized KullbackLeibler (SKL) distance test statistic, based on comparing a smoothing spline (penalized likelihood) fit and a GLIM fit, for testing the hypothesis j `parametric' vs j `smooth', in the nonGaussian situation. The spline fit uses a smoothing parameter obtained from the data via either the unbiased risk ...
Printed in Great Britain Hypothesis testing in semiparametric additive mixed models
"... We consider testing whether the nonparametric function in a semiparametric additive mixed model is a simple fixed degree polynomial, for example, a simple linear function. This test provides a goodnessoffit test for checking parametric models against nonparametric models. It is based on the mixedmo ..."
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We consider testing whether the nonparametric function in a semiparametric additive mixed model is a simple fixed degree polynomial, for example, a simple linear function. This test provides a goodnessoffit test for checking parametric models against nonparametric models. It is based on the mixedmodel representation of the smoothing spline estimator of the nonparametric function and the variance component score test by treating the inverse of the smoothing parameter as an extra variance component. We also consider testing the equivalence of two nonparametric functions in semiparametric additive mixed models for two groups, such as treatment and placebo groups. The proposed tests are applied to data from an epidemiological study and a clinical trial and their performance is evaluated through simulations.
unknown title
"... 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 ..."
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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