some aspects of the spline smoothing approach to non parametric regression curve fitting non parametric regression using cubic splines is an attractive flexible and widely applicable approach to curve estimation although the basic idea was formulated many years ago the method is not as widely known or adopted as perhaps it should be the topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology particular subjects covered include the basic principles of the method the relation with moving average and other smoothing methods the automatic choice of the amount of smoothing and the use of residuals for diagnostic checking and model adaptation the question of providing inference regions for curves and for relevant properties of curves is approached via a finite dimensional bayesian formulation

the discrete galerkin method for integral equations a general theory is given for discretized versions of the galerkin method for solving fredholm integral equations of the second kind the discretized galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the galerkin method the theoretical framework that is given parallels that of the regular galerkin method including the error analysis of the superconvergence of the iterated galerkin and discrete galerkin solutions in some cases the iterated discrete galerkin solution is shown to coincide with the nystrom solution with the same numerical integration method the paper concludes with applications to finite element galerkin methods

discrete legendre spectral projection methods for fredholm hammerstein integral equations in this paper we discuss the discrete legendre galerkin and discrete legendre collocation methods for fredholm hammerstein integral equations with smooth kernel using sufficiently accurate numerical quadrature rule we obtain optimal convergence rates for both discrete legendre galerkin and discrete legendre collocation solutions in both infinity and l 2 norm numerical examples are given to illustrate the theoretical results

testing for polynomial regression using nonparametric regression techniques abstract in regression analysis it is important to test the validity of the assumed model prior to making inferences regarding the population of interest in this investigation we utilize nonparametric regression techniques to test the validity of a kth order polynomial regression model the departures from the polynomial model are assumed to belong to a smooth class of functions a parametric form is not assumed a test based on nonparametric regression fits to the residuals from kth order polynomial regression is proposed it utilizes a smoothing spline fit of order 2k to the residuals from kth order polynomial regression a test statistic based on this estimator is formulated and its asymptotic distribution is derived under alternatives converging to the null at a rate of n k where is the smoothing parameter we note that this rate of convergence is slower than the parametric rate of n power investigations are conducted through a small scale simulation study

iterated discrete polynomially based galerkin methods golberg and bowman appl math comput 96 1998 237 have studied polynomially based discrete galerkin method for fredholm and singular integral equations in this paper we consider polynomially based iterated discrete galerkin method for solution of operator equations and for eigenvalue problem associated with an integral operator with a smooth kernel we show that the error in the infinity norm both for approximation of operator equation and of spectral subspace is of the order of n r where n is the degree of the polynomial approximation and r is the smoothness of the kernel thus the iterated discrete galerkin solution improves upon the discrete galerkin solution which was shown to be of order n r 1 by golberg and bowman appl math comput 96 1998 237 we also give a shorter proof of the result by golberg and bowman which states that the error in 2 norm in discrete galerkin method is of the order of n r

superconvergence of the iterated galerkin methods for hammerstein equations in this paper the well known iterated galerkin method and iterated galerkin kantorovich regularization method for approximating the solution of fredholm integral equations of the second kind are generalized to hammerstein equations with smooth and weakly singular kernels the order of convergence of the galerkin method and those of superconvergence of the iterated methods are analyzed numerical examples are presented to illustrate the superconvergence of the iterated galerkin approximation for hammerstein equations with weakly singular kernels

optimal convergence rates for some discrete projection methods using recent results of sloan on hyperinterpolation operators j approx theory 81 1995 238 we show how to obtain optimal convergence rates for a variety of discrete polynomially based galerkin and collocation methods for a variety of fredholm and singular integral equations these results improve on those given in golberg j integral equations appl 8 1996 307 and are analogous to those obtained by atkinson and bogomolny for spline based methods math comp 48 1987 596 these results are then applied to analyze double approximation methods where both numerical integration and data errors are considered here we show that the errors are dominated by those in the data hence it may not be possible in practice to obtain optimal convergence rates unless the data can be computed exactly

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

goodness of fit statistics for age specific reference intervals the age specific reference interval is a commonly used screening tool in medicine it involves estimation of extreme quantile curves such as the 5th and 95th centiles of a reference distribution of clinically normal individuals it is crucial that models used to estimate such intervals fit the data extremely well however few procedures to assess goodness of fit have been proposed in the literature and even fewer have been evaluated systematically here we consider procedures based on the distribution of the z scores standardized residuals from a model and on pearson chi 2 statistics for observed and expected counts in groups defined by age and the estimated reference centile curves two of the procedures q and grid tests are mainly inferential whereas the third permutation bands and b tests is essentially graphical we obtain approximations to the null distributions of several relevant test statistics and examine their size and power for a range of models based on real data sets we recommend q tests in all situations where z scores are available since they are general simple to calculate and usually have the highest power among the three classes of test considered for the cases considered the grid tests are always inferior to the q and b tests

legendre spectral projection methods for urysohn integral equations in this paper we consider the legendre spectral galerkin and legendre spectral collocation methods to approximate the solution of urysohn integral equation we prove that the approximated solutions of the legendre galerkin and legendre collocation methods converge to the exact solution with the same orders o n r in l 2 norm and o n 1 2 r in infinity norm and the iterated legendre galerkin solution converges with the order o n 2 r in both l 2 norm and infinity norm whereas the iterated legendre collocation solution converges with the order o n r in both l 2 norm and infinity norm n being the highest degree of the legendre polynomial employed in the approximation and r being the smoothness of the kernel we are able to obtain similar superconvergence rates for the iterated galerkin solution for urysohn integral equations with smooth kernel as in the case of piecewise polynomial basis functions

covariate matched one sided tests for the difference between functional means abstract when testing hypotheses about the effects of different treatments variation among covariates can become confounded with that between treatments unless the treatments are applied using paired covariates in the context of unpaired covariates we propose implicit covariate matching methods for testing the hypothesis that one treatment effect is greater than another the methods are founded on the assumption that the mean treatment effect conditional on the covariate is a smooth function of the covariate they are implemented using new interpolation techniques for nonparametric curve estimation bootstrap arguments are used to construct critical points we show that even when the covariate distributions are identical for both treatments covariate matching of the type that we propose produces tests of greater power than methods that do not attempt matching our techniques have application to two sided hypothesis testing

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

superconvergence of the iterated collocation methods for hammerstein equations in this paper we analyse the iterated collocation method for hammerstein equations with smooth and weakly singular kernels the paper expands the study which began in 16 concerning the superconvergence of the iterated galerkin method for hammerstein equations we obtain in this paper a similar superconvergence result for the iterated collocation method for hammerstein equations we also discuss the discrete collocation method for weakly singular hammerstein equations some discrete collocation methods for hammerstein equations with smooth kernels were given previously in 3 18

testing for cubic smoothing splines under dependent data in most research on smoothing splines the focus has been on estimation while inference especially hypothesis testing has received less attention by defining design matrices for fixed and random effects and the structure of the covariance matrices of random errors in an appropriate way the cubic smoothing spline admits a mixed model formulation which places this nonparametric smoother firmly in a parametric setting thus nonlinear curves can be included with random effects and random coefficients the smoothing parameter is the ratio of the random coefficient and error variances and tests for linear regression reduce to tests for zero random coefficient variances we propose an exact f test for the situation and investigate its performance in a real pine stem data set and by simulation experiments under certain conditions the suggested methods can also be applied when the data are dependent

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

weighted local regression and kernel methods for nonparametric curve fitting abstract weighted local regression a popular technique for smoothing scatterplots is shown to be asymptotically equivalent to certain kernel smoothers since both methods are local weighted averages of the data it is proved that in the fixed design regression model given a weighted local regression procedure with any weight function there is a corresponding kernel method such that the quotients of weights distributed by both methods tend uniformly to 1 as the number of observations increases to infinity it is demonstrated by examples that in some instances the weights are nearly the same for both methods even for small samples the asymptotic equivalence allows the derivation of the leading terms of the mean squared error and of the local limit distribution for weighted local regression further a close correspondence is found between the orders of the polynomial to be locally fitted in weighted local regression and the order number of vanishing moments of the kernel employed in the kernel smoot

projection and multi projection methods for nonlinear integral equations on the half line abstract in this article galerkin and collocation methods and their iterated versions are discussed to solve the nonlinear hammerstein type integral equation on the half line for both convolution and non convolution kernels using the space of piecewise polynomial subspace we prove that the approximate and iterated approximate solution in galerkin and collocation methods converges with the order o n r and o n 2 r respectively in the infinity norm where n 1 is the maximum norm of the partition and r denotes the order of the piecewise polynomial employed in the approximation we improve these results further by considering multi projection methods in fact we show that iterated approximate solution in multi galerkin multi collocation converges with the order o n 4 r numerical results are presented to confirm the theoretical results

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

the use of marginal likelihood for a diagnostic test for the goodness of fit of the simple linear regression model a novel test procedure is proposed to diagnose the goodness of fit of a simple regression model by applying a marginal likelihood under the assumption of smoothness of the alternative regression curve no specific form for the alternative regression function is assumed an advantage of the proposed test procedure is that it simultaneously provides an estimated regression curve spline some existing diagnostic procedures are explored in relation to the one proposed the procedure is applied to data of a spectrophotometric analysis of mixtures of two types of flour cornstarch and casein soybean

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

the numerical solution of nonlinear two dimensional volterra fredholm integral equations of the second kind based on the radial basis functions approximation with error analysis in this paper we present a numerical method for solving two dimensional nonlinear volterra fredholm integral equations of the second kind the method approximates the solution by the discrete collocation method based on radial basis functions rbfs constructed on a set of disordered data the proposed method is meshless since it does not require any background mesh or domain elements error analysis of this method is also investigated numerical examples which compare the proposed method with 2d tfs method 4 approve its supremacy in terms of accuracy and computational cost using various rbfs we have concluded that mq rbf is the best choice for the proposed method

interference between units in randomized experiments in a randomized experiment comparing two treatments there is interference between units if applying the treatment to one unit may affect other units interference implies that treatment effects are not comparisons of two potential responses that a unit may exhibit one under treatment and the other under control but instead are inherently more complex interference is common in social settings where people communicate compete or spread disease in studies that treat one part of an organism using a symmetrical part as control in studies that apply different treatments to the same organism at different times and in many other situations available statistical tools are limited for instance fisher s sharp null hypothesis of no treatment effect implicitly entails no interference and so his randomization test may be used to test no effect but conventional ways of inverting the test to obtain confidence intervals say for an additive effect are not applicable with interference another commonly used

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

nonparametric estimation of rate equations for nutrient uptake abstract knowledge of the rate of a biological process is important for characterizing the system and is necessary for gaining a deeper understanding of the process consider measurements y made over time on a system following the model y f t e where f is a smooth unknown function and e is measurement error although most statistical methodology has focused on estimating f t or f t in some applications what is of real biological interest is the relationship between f and f one example is the study of nitrogen absorption by plant roots through a solution depletion experiment in this case f t is the nitrate concentration of the solution surrounding the roots at time t and f t is the absorption rate of nitrate by plant roots at time t one is interested in the rate of nitrate absorption as a function of concentration that is one is interested in where f f knowledge of is important in quantifying the ability of a particular plant species to absorb nitrogen and in comparing

meromorphic exact solutions of two extended 3 1 dimensional jimbo miwa equations abstract in this work abundant meromorphic exact solutions of two kinds of extended 3 1 dimensional jimbo miwa equations are obtained by making use of the complex method which contain rational solutions exponential function solutions and elliptic function solutions the properties of the results are also shown by some figures

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

a new collocation type method for hammerstein integral equations on presente une methode du type de collocation pour la resolution numerique des equations integrales d hammerstein on etablit la convergence de l approximation desiree vers la solution exacte

a discrete collocation type method for hammerstein equations in recent papers we have considered the numerical solution of the hammerstein equation y t f t int a b k t s g s y s ds quad t in a b by a method that first applies the standard collocation procedure to an equivalent equation for z t g t y t and then obtains an approximation to y by use of the equation y t f t int a b k t s z s ds quad t in a b we study here a discretized version of the above method this arises when numerical quadrature is used to approximate the definite integrals occurring in the said method we consider the use of interpolatory quadrature rules and we seek the discrete collocation approximation to z in certain piecewise polynomial function spaces our principal result gives the precision required of a quadrature rule to guarantee the best possible super convergence rate for the discrete approximation to y in the case where the kernel k is sufficiently smooth this rate is of the same order as the rate for an approximation to y obtain

on the use of nonparametric regression for checking linear relationships summary the problem of checking the linearity of a regression relationship is addressed through the idea of smoothing of a residual plot a pseudolikelihood ratio test statistic which measures the distance between the nonparametric and the parametric models is derived as a ratio of quadratic forms the distribution of this statistic under the null hypothesis of linearity is calculated numerically by using johnson curves a power study shows the new statistic to be more sensitive to non linearity than the durbin watson statistic

testing for heterogeneity of evoked potential signals using an approximation to an exact permutation test in brain evoked potential research and various other fields a signal embedded in noise is observed in each of many repeated trials we propose a new test of the null hypothesis that the signal is the same in every trial the test has power against the alternative hypothesis that the signal is changing slowly from trial to trial the test statistic is the distance between the average of the trials and a parametric or nonparametric estimate of a slowly changing signal we derive the exact mean and variance of the test statistic under permutation of the trials and use these moments to approximate the exact permutation test we use simulated evoked potential data to evaluate the proposed test and to compare it to three previous tests

exact likelihood ratio tests for penalised splines summary penalised spline based additive models allow a simple mixed model representation where the variance components control departures from linear models the smoothing parameter is the ratio of the random coefficient and error variances and tests for linear regression reduce to tests for zero random coefficient variances we propose exact likelihood and restricted likelihood ratio tests for testing polynomial regression versus a general alternative modelled by penalised splines their spectral decompositions are used as the basis of fast simulation algorithms we derive the asymptotic local power properties of the tests under weak conditions in particular we characterise the local alternatives that are detected with asymptotic probability one confidence intervals for the smoothing parameter are obtained by inverting the tests for a fixed smoothing parameter versus a general alternative we discuss f and r tests and show that ignoring the variability in the smoothing parameter estimator can have a dramatic effect on their null distributions the powers of several known tests are investigated and a small set of tests with good power properties is identified the restricted likelihood ratio test is among the best in terms of power

smoothing noisy data with spline functions a procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m 1 fitted ton distinct not necessarily equally spaced or uniformly weighted data points is presented the procedure requires orderm 2 n operations and therefore permits efficient orderm 2 n calculation of statistics associated with a polynomial smoothing spline including the generalized cross validation the method is a significant improvement over an existing method which requires ordern 3 operations

testing for linearity in a semiparametric regression model abstract we propose a new exact test for nonlinearity in a regression model with one possibly nonlinear component the test is based on the stochastic interpretation of spline smoothing given in wahba 1978 and is a point optimal invariant test as defined in king 1988 based on this stochastic model we show that the power of the exact test compares favourably with several other exact tests for nonlinearity proposed in the literature and that its level and power are robust to long and short tailed symmetric error distributions we also present an o n algorithm based on a state space approach to compute exact p values for the point optimal invariant test whereas previous implementations required o n3 operations an example is presented to illustrate the theory

spline based tests in survival analysis this paper examines a method for testing hypotheses on covariate effects in a proportional hazards model and also on how effects change over time in regression analysis of survival data the technique used is very general and can be applied to testing many other aspects of parametric and semiparametric models the basic idea is to formulate a flexible parametric alternative using fixed knot splines together with a penalty function that penalizes noisy alternatives more than smooth ones to focus the power of the tests toward smooth alternatives the test statistics are the analogs of ordinary likelihood based statistics only computed from a penalized likelihood formed by subtracting the penalty function from the ordinary log likelihood large sample approximations to the distributions are found when the number of knots is held fixed as the sample size increases numerical results suggest these approximations may be adequate with moderate sized samples

polynomially based multi projection methods for fredholm integral equations of the second kind in this paper we propose a multi projection and iterated multi projection methods for fredholm integral equations of the second kind with a smooth kernel using polynomial bases we obtain super convergence rates for the approximate solutions more precisely we prove that in m galerkin and m collocation methods not only iterative solution u n approximates the exact solution u in the supremum norm with the order of convergence n 4 k but also the derivatives of u n approximate the corresponding derivatives of u in the supremum norm with the same order of convergence n being the degree of polynomial approximation and k being the smoothness of the kernel

on the use of the variogram in checking for independence in spatial data the variogram is a standard tool in the analysis of spatial data and its shape provides useful information on the form of spatial correlation that may be present however it is also useful to be able to assess the evidence for the presence of any spatial correlation a method of doing this based on an assessment of whether the true function underlying the variogram is constant is proposed nonparametric smoothing of the squared differences of the observed variables on a suitably transformed scale is used to estimate variogram shape a statistic based on a ratio of quadratic forms is proposed and the test is constructed by investigating the distributional properties of this statistic under the assumption of an independent gaussian process the power of the test is investigated reference bands are proposed as a graphical follow up an example is discussed

convergence analysis for derivative dependent fredholm hammerstein integral equations with green s kernel abstract in this article we consider a class of derivative dependent fredholm hammerstein integral equations i e the integral equation where the nonlinear function inside the integral sign is dependent on derivative and the kernel function is of green s type we propose the piecewise polynomial based galerkin and iterated galerkin methods to solve these type of derivative dependent fredholm hammerstein integral equations we discuss the convergence and error analysis of the proposed methods and also obtain the superconvergence results for iterated galerkin approximations some numerical results are given to illustrate this improvement