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...

. 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 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. 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...

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 mixed-effects 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...

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 leave-out-one 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 : :...

In this paper we study goodness-of-fit testing of single-index models. The large sample behavior of certain score-type test statistics is investigated. As a by-product, we obtain asymptotically distributionfree maximin tests for a large class of local alternatives. Furthermore, characteristic function based goodness-of-fit 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

This paper discusses the problem of testing the equality of two nonparametric regression curves against one-sided 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.

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 Kullback-Leibler (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 non-Gaussian situation. The spline fit uses a smoothing parameter obtained from the data via either the unbiased risk ...

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 data-driven 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 lack-of-fit 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...

A permutation test for the white noise hypothesis is described, offering power against a general class of smooth alternatives. Simulation results show that it performs well, as compared with similar tests available in the literature, in terms of power. An example demonstrates its use in a particular problem in which a test for randomness was sought without any specific alternative.

These Fortran-77 subroutines provide tools for penalized likelihood estimation and model checking for generalized linear models (GLMs) in which the model has a semi-parametric form. The routines build on GCVPACK (Bates et al., 1987) and are designed to use the generalized cross-validation criteria (Craven and Wahba. 1979) to determine the degree of data smoothing. 'Ihese problems include smoothed GLMs (O'Sullivan, YandeU and Raynor, 1986). iteratively reweighted least squares (Green, 1984), and general nonlinear problems. We present some of the problems PGLMPACK is designed for and describe the structure of the routines. General Problem: A variety of penalized nonlinear problems can be solved by an iterative scheme in which the inner step involves a linear model approximation. with J = bI...,y.)T the working values. 8 = (el,-.. the linearized model and E = (E~,..,E,,) ~ a random vector with zero mean and covariance w', which is often diagonal. m e matrix W is referred to as the working weights.) In many situation, a semiparametric model is appropriate, such as in which si is a c-vector of covariales with corresponding parameter vector a, xi is a d-vector of variates and f (.) is some "smooth " function. Smoothness can be enforced by a "roughness