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18
Smoothing Spline ANOVA for Exponential Families, with Application to the Wisconsin Epidemiological Study of Diabetic Retinopathy
 ANN. STATIST
, 1995
"... Let y i ; i = 1; \Delta \Delta \Delta ; n be independent observations with the density of y i of the form h(y i ; f i ) = exp[y i f i \Gammab(f i )+c(y i )], where b and c are given functions and b is twice continuously differentiable and bounded away from 0. Let f i = f(t(i)), where t = (t 1 ; \De ..."
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Cited by 83 (44 self)
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Let y i ; i = 1; \Delta \Delta \Delta ; n be independent observations with the density of y i of the form h(y i ; f i ) = exp[y i f i \Gammab(f i )+c(y i )], where b and c are given functions and b is twice continuously differentiable and bounded away from 0. Let f i = f(t(i)), where t = (t 1 ; \Delta \Delta \Delta ; t d ) 2 T (1)\Omega \Delta \Delta \Delta\Omega T (d) = T , the T (ff) are measureable spaces of rather general form, and f is an unknown function on T with some assumed `smoothness' properties. Given fy i ; t(i); i = 1; \Delta \Delta \Delta ; ng, it is desired to estimate f(t) for t in some region of interest contained in T . We develop the fitting of smoothing spline ANOVA models to this data of the form f(t) = C + P ff f ff (t ff ) + P ff!fi f fffi (t ff ; t fi ) + \Delta \Delta \Delta. The components of the decomposition satisfy side conditions which generalize the usual side conditions for parametric ANOVA. The estimate of f is obtained as the minimizer...
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 ..."
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Cited by 55 (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...
Smoothing Spline Models With Correlated Random Errors
 Journal of the American Statistical Association
, 1996
"... Spline smoothing is a popular method of estimating the functions in a nonparametric regression model. Its performance greatly depends on the choice of smoothing parameters. Many methods of selecting smoothing parameters such as CV, GCV, UBR and GML are developed under the assumption of independent o ..."
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Cited by 21 (7 self)
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Spline smoothing is a popular method of estimating the functions in a nonparametric regression model. Its performance greatly depends on the choice of smoothing parameters. Many methods of selecting smoothing parameters such as CV, GCV, UBR and GML are developed under the assumption of independent observations. They fail badly when data are correlated. In this paper, we assume observations are correlated and the correlation matrix depends on a parsimonious set of parameters. We extend the GML, GCV and UBR methods to estimate the smoothing parameters and the correlation parameters simultaneously. We also connect a smoothing spline model with three mixedeffects models. These connections show that the smoothing spline estimates evaluated at design points are BLUP estimates and the GML estimates of the smoothing parameters and the correlation parameters are REML estimates. These connections also suggest a way to fit a spline model with correlated errors using the existing SAS procedure pr...
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
Hypothesis testing in smoothing spline Models, Manuscript
 Journal of Statistical Computation and Simulation
, 2002
"... Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (C ..."
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Cited by 7 (3 self)
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Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMSNSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.
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. iii Contents Acknowledgements i Abstract ii 1 Introduction 1 1.1 Smoothing Spline for Generalized Linear Model : : : : : : : : : : : : : 2 1.2 The Problem : :...
Testing For Superiority Among Two Regression Curves
, 2003
"... This paper discusses the problem of testing the equality of two nonparametric regression curves against onesided alternatives in a two sample heteroscedastic setting in which design and error densities may dier between the two populations. The paper proposes a class of tests using covariate matchin ..."
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Cited by 3 (1 self)
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This paper discusses the problem of testing the equality of two nonparametric regression curves against onesided alternatives in a two sample heteroscedastic setting in which design and error densities may dier between the two populations. The paper proposes a class of tests using covariate matching and derives their asymptotic power for local alternatives. Using a semiparametric approach, an upper bound on the asymptotic power of all tests against a given local alternative is obtained. For a given local alternative, a member of the proposed class of tests is shown to achieve this upper bound.
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 the O ..."
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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 ...
Testing Polynomial Covariate Effects in Linear and Generalized Linear Mixed Models
, 802
"... Abstract: An important feature of linear mixed models and generalized linear mixed models is that the conditional mean of the response given the random effects, after transformed by a link function, is linearly related to the fixed covariate effects and random effects. Therefore, it is of practical ..."
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Cited by 1 (0 self)
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Abstract: An important feature of linear mixed models and generalized linear mixed models is that the conditional mean of the response given the random effects, after transformed by a link function, is linearly related to the fixed covariate effects and random effects. Therefore, it is of practical importance to test the adequacy of this assumption, particularly the assumption of linear covariate effects. In this paper, we review procedures that can be used for testing polynomial covariate effects in these popular models. Specifically, four types of hypothesis testing approaches are reviewed, i.e. R tests, likelihood ratio tests, score tests and residualbased tests. Derivation and performance of each testing procedure will be discussed, including a small simulation study for comparing the likelihood ratio tests with the score tests.
Testing Lack of Fit of Regression Models Under Heteroscedasticity
"... A test is proposed for assessing the lack of fit of heteroscedastic nonlinear regression models that is based on comparison of nonparametric kernel and parametric fits. A datadriven method is proposed for bandwidth selection using the parametric null model asymptotically optimal bandwidth which lea ..."
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A test is proposed for assessing the lack of fit of heteroscedastic nonlinear regression models that is based on comparison of nonparametric kernel and parametric fits. A datadriven method is proposed for bandwidth selection using the parametric null model asymptotically optimal bandwidth which leads to a test that has a limiting normal distribution under the null hypothesis and is consistent against any fixed alternative. The resulting test is applied to the problem of testing the lackoffit of a generalized linear model. R ESUM E L'auteur propose de verifier l'adequation de modeles de regression non lineaires heteroscedastiques au moyen d'un test comparant l'ajustement d'estimations parametrique et non parametrique (nucleaire) du vecteur moyenne. Il propose une methode de selection empirique de la fenetre deduite de la longueur de fenetre asymptotiquement optimale sous l'hypothese nulle. La statistique qui en decoule est asymptotiquement gaussienne sous l'hypothese nulle et converg...