a geometrical method for removing edge effects from kernel type nonparametric regression estimators abstract we introduce a simple geometric method for removing edge effects from kernel type nonparametric regression estimators it involves reflecting the data set in two estimated points thereby generating a new data set with three times the range of the original data the usual kernel type estimator may be applied to the new enlarged data set without any danger of edge effects this technique is applicable generally to both regularly spaced and randomly spaced designs and admits a natural analog of leave one out cross validation the new cross validation algorithm may be extended to the very ends of the design interval unlike its more conventional counterpart which must be downweighted at the ends of the interval to avoid edge effects

boundary element collocation methods using splines with multiple knots we extend the theory of boundary element collocation methods by allowing reduced inter element smoothness or in other words by allowing trial functions that are splines with multiple knots our convergence analysis is based on a recurrence relation for the fourier coefficients of the numerical solution and so is restricted to uniform grids on smooth closed curves superconvergence is possible with special choices of the collocation points numerical experiments with a model problem confirm the convergence rates predicted by our theory

bias robust estimation in finite populations using nonparametric calibration a standard problem in sample survey inference is that of predicting the finite population total h of a function h y of a random variable y the model based approach to this problem first defines a working model for y and then predicts h by estimating its expectation under conditional on the sample values of y this approach leads to biased predictions if is incorrect we explore an automatic solution to this misspecification bias that uses nonparametric regression to define a robust but inefficient predictor of h and then calibrates this predictor for its bias under an application to prediction of the finite population distribution function of a population of australian beef farms is presented

testing lack of fit in multiple regression summary we study lack of fit tests based on orthogonal series estimators a common feature of these tests is that they are functions of score statistics that employ data driven model dimensions the criteria used to select the dimension are score based versions of aic and bic the tests can be applied in a wide variety of settings including both continuous and discrete data with two or more covariates a model sequence i e a path in the alternative models space has to be chosen critical points and p values of the lack of fit tests can be obtained via asymptotic distribution theory or by use of the bootstrap data examples and a simulation study illustrate the applicability of the tests

how far are automatically chosen regression smoothing parameters from their optimum abstract we address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation call the optimal bandwidth the minimizer of the average squared error we consider several automatically selected bandwidths that approximate the optimum how far are the automatically selected bandwidths from the optimum the answer is studied theoretically and through simulations the theoretical results include a central limit theorem that quantifies the convergence rate and gives the differences asymptotic distribution the convergence rate turns out to be excruciatingly slow this is not too disappointing because this rate is of the same order as the convergence rate of the difference between the minimizers of the average squared error and the mean average squared error in some simulations by john rice the selectors considered here performed quite differently from each other we anticipated that these differences would be reflected in differ

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

testing the parametric null model hypothesis in semiparametric partial and generalized spline models on etend les resultats de cox et koh 1986 aux modeles generalises de wahba 1985 et a divers autres modeles splines partiels on montre que la statistique de test a une relation etroite avec le comportement de la fonction de validation croisee generalisee

on the use of nonparametric regression for model checking summary the use of nonparametric regression is explored to check the fit of a parametric regression model the principal aim is to check the validity of the regression curve rather than necessarily to detect outliers a pseudo likelihood ratio test is developed to provide a global assessment of fit and simulation bands are used to indicate the nature of departures from the model the types of data considered include discrete response variables where standard diagnostic techniques are often not appropriate and first order autoregressive series several numerical examples are given nonparametric regression can be used in an informal graphical way to assess the relationship between a response and an explanatory variable in this paper we aim to develop more formal methods of assessing the assumptions of a parametric model in particular when regression diagnostics of the type developed for normal linear models are not readily available the principal aim is to check the validity of the systematic part of the model by comparing a nonparametric estimate of the regression curve with a parametric one such a comparison may also identify outliers although the distinction between outliers and model inadequacy is not always easy two techniques are used to assess the fit of a parametric model in 2 confidence bands are constructed around the fitted regression curve by simulation a comparison of these with the nonparametric curve gives an indication of the nature of any departures from the model in 3 a pseudo likelihood ratio test is developed this provides a quantitative global assessment of fit in applying these ideas special emphasis is given to discrete data and notably logistic regression because of the difficulty in applying standard residual based model checking techniques to this type of response variable a poisson regression example is discussed in 4 however the underlying ideas have wider applications autoregressive time series of order 1 are discussed in 6 sections 5 and 7 discuss general issues we first discuss the context of binary regression with a single covariate and the difficulties caused by the discreteness of the response variable the observed data are assumed to be of the form xi yi ni where xi is a covariate value and yi has a binomial

optimal bandwidth selection in nonparametric regression function estimation kernel estimators of an unknown multivariate regression function are investigated a bandwidth selection rule is considered which can be formulated in terms of cross validation under mild assumptions on the kernel and the unknown regression function it is seen that this rule is asymptotically optimal

testing for no effect when estimating a smooth function by nonparametric regression a randomization approach abstract when a linear regression function is estimated by ordinary least squares the null hypothesis of no relationship between the response and the design variable can be tested by the normal theory f test this article describes a new test for use in the case in which a smooth regression function is estimated by a nonparametric procedure such as kernel estimation or local regression the test is constructed as an approximation to an exact permutation test the test statistic is the ratio of sums of squares that are defined by analogy to the analysis of variance the null permutation distribution of the test statistic is approximated by matching its exact mean and variance to the moments of a gamma distribution a simulation study shows that the approximation is excellent for several regression procedures and for normal heavy tailed and skewed error distributions simulation results are also used to investigate the power of the test as a function of the smoothing parameter the test is applied to dat

model checking with nonparametric curves summary the use of local likelihood curve estimates in model checking is described test statistics are based on deviance differences and are interpreted using nonparametric bootstrap monte carlo some theoretical aspects are discussed methods are illustrated for regression with discrete data

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

semiparametric comparison of regression curves the comparison of nonparametric regression curves is considered it is assumed that there are parametric possibly nonlinear transformations of the axes which map one curve into the other estimation and testing of the parameters in the transformations are studied the rate of convergence is n 1 2 although the nonparametric components of the model typically have a rate slower than that a statistic is provided for testing the validity of a given completely parametric model

testing the goodness of fit of a linear model via nonparametric regression techniques abstract this article investigates the use of nonparametric regression methodology to test the adequacy of a parametric linear model the large sample properties of parametric goodness of fit tests for linearity are considered the inadequacies of such tests lead to the proposal of new tests that are constructed from nonparametric regression fits to the residuals from linear regression large sample theory is derived for two variants of this type of statistic the results demonstrate that such tests are consistent against all fixed smooth alternatives to linearity but are incapable of detecting local alternatives converging to a linear model at the parametric rate n 1 2 simulation experiments involving a test based on fitting cubic smoothing splines to residuals reveals that this test has good power properties against several reasonable alternatives

bootstrap test for difference between means in nonparametric regression abstract a bootstrap test is proposed for detecting a difference between two mean functions in the setting of nonparametric regression error distributions in the regression model are permitted to be arbitrary and unequal the test enjoys power properties akin to those in a parametric setting in the sense that it can distinguish between regression functions distant only n 1 2 apart where n is the sample size it has exceptional level accuracy with level error of only n 2 and uses a very accurate estimate of the critical point of an exact test being in error by only n 3 2 under the null hypothesis the test admits several generalizations for example to the case of testing for differences between several regression means this is a nonparametric regression analog of analysis of variance a simulation study using n as small as 15 corroborates the asymptotic result on level accuracy of the bootstrap test applications are illustrated with an example involving acid rain data

commonality of cusum von neumann and smoothing based goodness of fit tests summary recent papers by munson jernigan 1989 and buckley 1991 propose nonparametric tests for the hypothesis of no predictor effect in regression the munson jernigan test is similar to the von neumann 1941 test while that of buckley is based on a functional of cusums these tests are shown to be special cases of a wider class of tests based on nonparametric function estimation ideas fourier analysis is used to qualitatively compare the munson jernigan and buckley tests with two new tests constructed from nonparametric smoothers their relative powers are then studied by means of large sample analysis and simulation the cusum test is the most powerful for very smooth departures from the no effect hypothesis while the new tests based on smoothing ideas are clearly superior when the alternative is high frequency

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

multi layered stationary solutions for a spatially inhomogeneous allen cahn equation abstract we consider stationary solutions of a spatially inhomogeneous allen cahn type nonlinear diffusion equation in one space dimension the equation involves a small parameter e and its nonlinearity has the form h x 2 f u where h x represents the spatial inhomogeneity and f u is derived from a double well potential with equal well depth when e is very small stationary solutions develop transition layers we first show that those transition layers can appear only near the local minimum and local maximum points of the coefficient h x and that at most a single layer can appear near each local minimum point of h x we then discuss the stability of layered stationary solutions and prove that the morse index of a solution coincides with the total number of its layers that appear near the local maximum points of h x we also show the existence of stationary solutions having clustering layers at the local maximum points of h x

local polynomial estimation in multiparameter likelihood models abstract the nonparametric regression technique of local polynomial fitting is extended to multiparameter likelihood models some well known appealing features of local polynomial smoothers such as the behavior at the boundary are shown to carry over to the multiparameter case asymptotic consistency and normality of the resulting estimators are derived under suitable regularity conditions this work is motivated by the need for a nonparametric alternative to parametric dose response models for clustered binary data probability models for clustered binary response data include a success probability parameter and one or more correlation parameters the proposed local polynomial estimators can play an important role as a diagnostic tool or to suggest the form of the functional relationships in parametric likelihood models as an illustration it is shown how the local likelihood estimation procedure can be implemented for fitting a dose response curve based on the beta binomial model a data example and

the choice of weights in kernel regression estimation summary for kernel regression estimation a weighting scheme due to nadaraya and watson has been associated with random design and a convolution type weighting scheme with fixed design based on integrated mean square error none of the estimators is uniformly optimal in either design however the convolution type weights are minimax optimal further advantages of this estimator can be seen in the structure of the bias

kernel regression when the boundary region is large with an application to testing the adequacy of polynomial models abstract it is well known that kernel regression estimators are subject to so called boundary or edge effects a phenomenon in which the bias of an estimator increases near the endpoints of the estimation interval when the regression curve is linear or nearly linear the requisite amount of smoothing is so great that the boundary region is effectively the entire estimation interval special boundary kernels are proposed here to deal with such cases it is shown that the proposed kernel estimator has a property also enjoyed by cubic smoothing splines namely as the estimator s smoothing parameter becomes large the estimator tends to a straight line the limiting straight line is essentially the least squares line when the design points are equally spaced a simple generalization of ideas in the linear case leads to kernel estimates that are polynomials of any given degree for large bandwidths such estimates are an important component of a proposed test for the adequacy of a polynomial model the test s