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

simultaneous inferences on the contrast of two hazard functions with censored observations in survival analysis often times the pattern of instantaneous risk over time is more interesting than that of the cumulative risk for this case a nonparametric hazard function estimate is more appropriate for summarizing the risk experience of a group of patients than the corresponding kaplan meier estimate in comparing a new treatment with a standard therapy it is important to know if the treatment loses it potency during the follow up period and if it does one would like to know when it becomes ineffective unfortunately with a plot of the differences of two kaplan meier curves it is rather difficult to capture such temporal trends in this article we propose simple procedures for constructing confidence bands for the contrast of two hazard functions with censored data the simultaneous interval estimates are quite useful for identifying possible values of the contrast over time with a certain degree of confidence the new proposals are illustrated with an example and a small simulation study

asymptotic optimality of generalized cl cross validation and generalized cross validation in regression with heteroskedastic errors abstract the problem considered here is that of using a data driven procedure to select a good estimate from a class of linear estimates indexed by a discrete parameter in contrast to other papers on this subject we consider models with heteroskedastic errors the results apply to model selection problems in linear regression and to nonparametric regression estimation via series estimators nearest neighbor estimators and local regression estimators among others generalized c l gc l cross validation cv and generalized cross validation gcv procedures are analyzed the gc l and cv criteria are shown to be asymptotically optimal under general conditions a feasible version of gc l however is available only in some applications the gcv criterion is found to be asymptotically optimal only under a condition that is satisfied in some applications but not in others for example it is satisfied in the nearest neighbor estimation context but not in the series estimation local regression estimation or model selection contexts thus the cv criterion is the only feasible criterion of the three that is asymptotically optimal under general conditions the proofs rely heavily on results of li 1987

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

mixture of regression models with varying mixing proportions a semiparametric approach in this article we study a class of semiparametric mixtures of regression models in which the regression functions are linear functions of the predictors but the mixing proportions are smoothing functions of a covariate we propose a one step backfitting estimation procedure to achieve the optimal convergence rates for both regression parameters and the nonparametric functions of mixing proportions we derive the asymptotic bias and variance of the one step estimate and further establish its asymptotic normality a modified expectation maximization type em type estimation procedure is investigated we show that the modified em algorithms preserve the asymptotic ascent property numerical simulations are conducted to examine the finite sample performance of the estimation procedures the proposed methodology is further illustrated via an analysis of a real dataset

confidence bands for regression functions abstract a method for obtaining conservative confidence bands for regression functions in low dimensions is given of particular importance is the applicability of the method to nonparametric regression computations examples and comparisons indicate the effectiveness of the method

confidence bands in nonparametric regression abstract new bias corrected confidence bands are proposed for nonparametric kernal regression these bands are constructed using only a kernel estimator of the regression curve and its data selected bandwidth they are shown to have asymptotically correct coverage properties and to behave well in a small sample study one consequence of the large sample developments is that bonferroni type bands for the regression curve at the design points also have conservative asymptotic coverage behavior with no bias correction

implementing an efficient minimum capacity cut algorithm in this paper we present an efficient implementation of theo mn n2 logn time algorithm originally proposed by nagamochi and ibaraki 1992 for computing the minimum capacity cut of an undirected network to enhance computation various ideas are added so that it can contract as many edges as possible in each iteration to evaluate the performance of the resulting implementation we conducted extensive computational experiments and compared the results with that of padberg and rinaldi s algorithm 1990 which is currently known as one of the practically fastest programs for this problem the results indicate that our program is considerably faster than padberg and rinaldi s program and its running time is not significantly affected by the types of the networks being solved

asymptotic confidence regions for kernel smoothing of a varying coefficient model with longitudinal data abstract we consider the estimation of the k 1 dimensional nonparametric component t of the varying coefficient model y t x t t t e t based on longitudinal observations yij x i tij tij i 1 n j 1 ni where tij is the jth observed design time point t of the ith subject and yij and x i tij are the real valued outcome and r k 1 valued covariate vectors of the ith subject at tij the subjects are independently selected but the repeated measurements within subject are possibly correlated asymptotic distributions are established for a kernel estimate of t that minimizes a local least squares criterion these asymptotic distributions are used to construct a class of approximate pointwise and simultaneous confidence regions for t applying these methods to an epidemiological study we show that our procedures are useful for predicting cd4 t helper lymphocytes cell changes among hiv human immunodeficiency virus infected persons the finite sample properties of o

residual variance and residual pattern in nonlinear regression summary a nonparametric estimator of residual variance in nonlinear regression is proposed it is based on local linear fitting asymptotically the estimator has a small bias but a larger variance compared with the parametric estimator in linear regression finite sample properties are investigated in a simulation study including a comparison with other nonparametric estimators the method is also useful for spotting heteroscedasticity and outliers in the residuals at an early stage of the data analysis a further application is checking the fit of parametric models this is illustrated for longitudinal growth data

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

estimation of heteroscedasticity in regression analysis consider the regression model y i g t i varepsilon i 1 leq i leq n with nonrandom design variables t i and measurements y i for the unknown regression function g cdot we assume that the data are heteroscedastic i e e varepsilon 2 i sigma 2 i not equiv operatorname const and investigate how to estimate sigma 2 i if sigma 2 i sigma 2 t i with a smooth function sigma 2 cdot initial estimators tilde sigma 2 i can be improved by kernel smoothers and the resulting class of estimators is shown to be uniformly consistent these estimates can be used to improve the estimation of the regression function g itself in parametric and nonparametric models further applications are suggested

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

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

on variance estimation in nonparametric regression a simple explicit estimator of variance in nonparametric regression is suggested based on the mean square of a sequence of residuals the estimator is valid for general error distributions has elementary bias and variance properties and is optimal to both first and second orders

asymptotic analysis of penalized likelihood and related estimators a general approach to the first order asymptotic analysis ofpenalized likelihood and related estimators is described the method gives expansions for the systematic and random error asymptotic convergence rates in a family of spectral norms are obtained the theory applies to a broad range of function estimation prob erns including non paxametric dellsity hazard and generalized regression curve estimation some examples are provided ams 1980 subject classifications primary 62 g05 secondary 62j05 41 a35 41 a25 47 a53 45 llo 45 m05

the average posterior variance of a smoothing spline and a consistent estimate of the average squared error a smoothing spline estimator can be interpreted in two ways either as the solution to a variational problem or as the posterior mean when a particular gaussian prior is placed on the unknown regression function the estimate of the average posterior variance of a spline estimate proposed by wahba is shown to converge in probability to a quantity proportional to the expected average squared error this result is established by relating this statistic to a consistent risk estimate based on generalized cross validation

bayesian confidence intervals for smoothing splines abstract the frequency properties of wahba s bayesian confidence intervals for smoothing splines are investigated by a large sample approximation and by a simulation study when the coverage probabilities for these pointwise confidence intervals are averaged across the observation points the average coverage probability acp should be close to the nominal level from a frequency point of view this agreement occurs because the average posterior variance for the spline is similar to a consistent estimate of the average squared error and because the average squared bias is a modest fraction of the total average squared error these properties are independent of the bayesian assumptions used to derive this confidence procedure and they explain why the acp is accurate for functions that are much smoother than the sample paths prescribed by the prior this analysis accounts for the choice of the smoothing parameter bandwidth using cross validation in the case of natural splines an adaptive method for avo

bootstrapping in nonparametric regression local adaptive smoothing and confidence bands abstract the operation of the bootstrap in the context of nonparametric regression is considered bootstrap samples are taken from estimated residuals to study the distribution of a suitably recentered kernel estimator the application of this principle to the problem of local adaptive choice of bandwidth and to the construction of confidence bands is investigated and compared with a direct method based on asymptotic means and variances the technique of the bootstrap is to replace any occurrence of the unknown distribution in the definition of the statistical function of interest by the empirical distribution function of the observed errors in a regression context these errors are not directly observed although their role can be played by the residuals from the fitted model in this article the fitted model is a kernel nonparametric regression estimator since nonparametric smoothing is involved an additional difficulty is created by the bias incurred in smoothing this bias however can be estimated

bootstrap simultaneous error bars for nonparametric regression simultaneous error bars are constructed for nonparametric kernel estimates of regression functions the method is based on the bootstrap where resampling is done from a suitably estimated residual distribution the error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints applications to an economic problem are given and comparison to both pointwise and bonferroni type bars is presented through a simulation study

classification of solitons for pluriclosed flow on complex surfaces we give a classification of compact solitons for the pluriclosed flow on complex surfaces first by exploiting results from the kodaira classification of surfaces we show that the complex surface underlying a soliton must be kahler except for the possibility of steady solitons on minimal hopf surfaces then we construct steady solitons on all class 1 hopf surfaces by exploiting a symmetry ansatz

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

confidence bands in nonparametric time series regression we consider nonparametric estimation of mean regression and conditional variance or volatility functions in nonlinear stochastic regression models simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct the imposed dependence structure allows applications in many linear and nonlinear auto regressive processes the results are applied to the s p 500 index data

simultaneous inference of linear models with time varying coefficients summary the paper considers construction of simultaneous confidence tubes for time varying regression coefficients in functional linear models using a gaussian approximation result for non stationary multiple time series we show that the constructed simultaneous confidence tubes have asymptotically correct nominal coverage probabilities our results are applied to the problem of testing whether the regression coefficients are of certain parametric forms which is a fundamental problem in the inference of functional linear models as an application we analyse an environmental data set and study the association between levels of pollutants and hospital admissions