a simple smooth backfitting method for additive models in this paper a new smooth backfitting estimate is proposed for additive regression models the estimate has the simple structure of nadaraya watson smooth backfitting but at the same time achieves the oracle property of local linear smooth backfitting each component is estimated with the same asymptotic accuracy as if the other components were known

the existence and asymptotic properties of a backfitting projection algorithm under weak conditions we derive the asymptotic distribution of a new backfitting procedure for estimating the closest additive approximation to a nonparametric regression function the procedure employs a recent projection interpretation of popular kernel estimators provided by mammen marron turlach and wand 1997 and the asymptotic theory of our estimators is derived using the theory of additive projections reviewed in bickel klaassen ritov and wellner 1993 our procedure achieves the same bias and variance as the oracle estimator based on knowing the other components and in this sense improves on the method analysed in opsomer and ruppert 1997 we provide high level conditions independent of the sampling scheme we then verify that these conditions are satisfied in a regression and a time series autoregression under weak conditions

additive functional regression for densities as responses we propose and investigate additive density regression a novel additive functional regression model for situations where the responses are random distributions that can be viewed as random densiti

generalized additive and index models with shape constraints summary we study generalized additive models with shape restrictions e g monotonicity convexity and concavity imposed on each component of the additive prediction function we show that this framework facilitates a non parametric estimator of each additive component obtained by maximizing the likelihood the procedure is free of tuning parameters and under mild conditions is proved to be uniformly consistent on compact intervals more generally our methodology can be applied to generalized additive index models here again the procedure can be justified on theoretical grounds and like the original algorithm has highly competitive finite sample performance practical utility is illustrated through the use of these methods in the analysis of two real data sets our algorithms are publicly available in the r package scar short for shape constrained additive regression

local polynomial kernel regression for generalized linear models and quasi likelihood functions abstract we investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi likelihood contexts in the ordinary regression case local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity one noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators these properties are shown to carry over to generalized linear model and quasi likelihood settings we also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state of the art bandwidth selection methods

estimation of additive regression models with known links summary we consider estimation of an additive nonparametric regression model with known link function the asymptotic distribution of this regression estimate is given the practical performance is investigated via an application to the study of migration between east and west germany

backfitting and smooth backfitting for additive quantile models in this paper we study the ordinary backfitting and smooth backfitting as methods of fitting additive quantile models we show that these backfitting quantile estimators are asymptotically equivalent to the corresponding backfitting estimators of the additive components in a specially designed additive mean regression model this implies that the theoretical properties of the backfitting quantile estimators are not unlike those of backfitting mean regression estimators we also assess the finite sample properties of the two backfitting quantile estimators

nonparametric additive regression for repeatedly measured data summary we develop an easily computed smooth backfitting algorithm for additive model fitting in repeated measures problems our methodology easily copes with various settings such as when some covariates are the same over repeated response measurements we allow for a working covariance matrix for the regression errors showing that our method is most efficient when the correct covariance matrix is used the component functions achieve the known asymptotic variance lower bound for the scalar argument case smooth backfitting also leads directly to design independent biases in the local linear case simulations show our estimator has smaller variance than the usual kernel estimator this is also illustrated by an example from nutritional epidemiology

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

fitting a bivariate additive model by local polynomial regression while the additive model is a popular nonparametric regression method many of its theoretical properties are not well understood especially when the backfitting algorithm is used for computation of the estimators this article explores those properties when the additive model is fitted by local polynomial regression sufficient conditions guaranteeing the asymptotic existence of unique estimators for the bivariate additive model are given asymptotic approximations to the bias and the variance of a homoscedastic bivariate additive model with local polynomial terms of odd and even degree are computed this model is shown to have the same rate of convergence as that of univariate local polynomial regression

bandwidth selection for smooth backfitting in additive models the smooth backfitting introduced by mammen linton and nielsen ann statist 27 1999 1443 1490 is a promising technique to fit additive regression models and is known to achieve the oracle efficiency bound in this paper we propose and discuss three fully automated bandwidth selection methods for smooth backfitting in additive models the first one is a penalized least squares approach which is based on higher order stochastic expansions for the residual sums of squares of the smooth backfitting estimates the other two are plug in bandwidth selectors which rely on approximations of the average squared errors and whose utility is restricted to local linear fitting the large sample properties of these bandwidth selection methods are given their finite sample properties are also compared through simulation experiments

continuously additive models for nonlinear functional regression summary we introduce continuously additive models which can be viewed as extensions of additive regression models with vector predictors to the case of infinite dimensional predictors this approach produces a class of flexible functional nonlinear regression models where random predictor curves are coupled with scalar responses in continuously additive modelling integrals taken over a smooth surface along graphs of predictor functions relate the predictors to the responses in a nonlinear fashion we use tensor product basis expansions to fit the smooth regression surface that characterizes the model in a theoretical investigation we show that the predictions obtained from fitting continuously additive models are consistent and asymptotically normal we also consider extensions to generalized responses the proposed class of models outperforms existing functional regression models in simulations and real data examples

smooth backfitting in practice summary compared with the classical backfitting of buja hastie and tibshirani the smooth backfitting estimator sbe of mammen linton and nielsen not only provides complete asymptotic theory under weaker conditions but is also more efficient robust and easier to calculate however the original paper describing the sbe method is complex and the practical as well as the theoretical advantages of the method have still neither been recognized nor accepted by the statistical community we focus on a clear presentation of the idea the main theoretical results and practical aspects like implementation and simplification of the algorithm we introduce a feasible cross validation procedure and apply it to the problem of data driven bandwidth choice for the sbe by simulations it is shown that the sbe and our cross validation work very well indeed in particular the sbe is less affected by sparseness of data in high dimensional regression problems or strongly correlated designs the sbe has reasonable performance even in 100 dimensional additive regression problems

smooth backfitting in generalized additive models generalized additive models have been popular among statisticians and data analysts in multivariate nonparametric regression with non gaussian responses including binary and count data in this paper a new likelihood approach for fitting generalized additive models is proposed it aims to maximize a smoothed likelihood the additive functions are estimated by solving a system of nonlinear integral equations an iterative algorithm based on smooth backfitting is developed from the newton kantorovich theorem asymptotic properties of the estimator and convergence of the algorithm are discussed it is shown that our proposal based on local linear fit achieves the same bias and variance as the oracle estimator that uses knowledge of the other components numerical comparison with the recently proposed two stage estimator ann statist 32 2004 2412 2443 is also made

generalised structured models summary we present a general class of nonlinear regression and time series models that we call generalised structured models the class is a natural generalisation of generalised additive models and it includes generalised interaction models structured volatility models visual garch generalised autoregressive conditional heteroscedasticity models and varying coefficient models we discuss estimation principles including smoothing splines and a generalisation of the projection approach of mammen et al 1999 we finish the paper with some theoretical considerations about the asymptotic performance of the estimator for the general class of generalised structured models

rate optimal estimation for a general class of nonparametric regression models with unknown link functions this paper discusses a nonparametric regression model that naturally generalizes neural network models the model is based on a finite number of one dimensional transformations and can be estimated with a one dimensional rate of convergence the model contains the generalized additive model with unknown link function as a special case for this case it is shown that the additive components and link function can be estimated with the optimal rate by a smoothing spline that is the solution of a penalized least squares criterion

numerical simulation of free surface incompressible liquid flows surrounded by compressible gas a numerical model for the three dimensional simulation of liquid gas flows with free surfaces is presented the incompressible navier stokes equations are assumed to hold in the liquid domain in the gas domain the velocity is disregarded the pressure is supposed to be constant in each connected component of the gas domain and follows the ideal gas law the gas pressure is imposed as a normal force on the liquid gas interface an implicit splitting scheme is used to decouple the physical phenomena given the gas pressure on the interface the method described in j comput phys 155 1999 439 int j numer meth fluids 42 7 2003 697 is used to track the liquid domain and to compute the velocity and pressure fields in the liquid then the connected components of the gas domain are found using an original numbering algorithm finally the gas pressure is updated from the ideal gas law in each connected component of gas the implementation is validated in the frame of mould filling numerical results in two and three space dimensions show that the effect of pressure in the bubbles of gas trapped by the liquid cannot be neglected

nonparametric estimation of an additive model with a link function this paper describes an estimator of the additive components of a nonparametric additive model with a known link function when the additive components are twice continuously differentiable the estimator is asymptotically normally distributed with a rate of convergence in probability of 2 5 n this is true regardless of the finite dimension of the explanatory variable thus in contrast to the existing asymptotically normal estimator the new estimator has no curse of dimensionality moreover the asymptotic distribution of each additive component is the same as it would be if the other components were known with certainty

operational time and in sample density forecasting in this paper we consider a new structural model for in sample density forecasting in sample density forecasting is to estimate a structured density on a region where data are observed and then re use the estimated structured density on some region where data are not observed our structural assumption is that the density is a product of one dimensional functions with one function sitting on the scale of a transformed space of observations the transformation involves another unknown one dimensional function so that our model is formulated via a known smooth function of three underlying unknown one dimensional functions we present an innovative way of estimating the one dimensional functions and show that all the estimators of the three components achieve the optimal one dimensional rate of convergence we illustrate how one can use our approach by analyzing a real dataset and also verify the tractable finite sample performance of the method via a simulation study

smooth backfitting of proportional hazards with multiplicative components smooth backfitting has proven to have a number of theoretical and practical advantages in structured regression by projecting the data down onto the structured space of interest smooth backfitting

the dimensionality reduction principle for generalized additive models soit x y une paire de variables aleatoires telles que x x 1 x j s etende sur c 0 1 j la distribution conditionnelle de y etant donne x x est supposee appartenir a une famille exponentielle appropriee ayant pour parametre r f x denote la dependance de vis a vis de x et f l approximation additive de f ayant la log vraisemblance esperee maximum possible

entropic formulation of statistical mechanics we present an alternative formulation of equilibrium statistical mechanics which follows the method based on the maximum statistical entropy principle in information theory combined with the use of massieu planck functions the different statistical ensembles are obtained by a suitable restriction of the whole set of available microstates the main advantage is that all of the equations that relate the average values with derivatives of the partition function are formally identical in the different ensembles moreover einstein s fluctuation formula is also derived within the same framework this provides a suitable starting point for the calculation of fluctuations of extensive and intensive variables in any statistical ensemble