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

on consistent nonparametric order determination and chaos summary we give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models we argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos setting the stochastic models within the framework of non linear autoregression we introduce the notion of a generalized partial autocorrelation and an order we approach the estimation of the embedding dimension via order determination of an unknown non linear autoregression by cross validation and give justification by proving its consistency under global boundedness as a by product we provide a theoretical justification of the final prediction error approach of auestad and tj0stheim some illustrations based on the henon map and several real data sets are given the bias of the residual sum of squares as essentially a noise variance estimator is quantified

nonparametric inferences for additive models additive models with backfitting algorithms are popular multivariate nonparametric fitting techniques however the inferences of the models have not been very well developed due partially to the complexity of the backfitting estimators there are few tools available to answer some important and frequently asked questions such as whether a specific additive component is significant or admits a certain parametric form in an attempt to address these issues we extend the generalized likelihood ratio glr tests to additive models using the backfitting estimator we demonstrate that under the null models the newly proposed glr statistics follow asymptotically rescaled chi squared distributions with the scaling constants and the degrees of freedom independent of the nuisance parameters this demonstrates that the wilks phenomenon continues to hold under a variety of smoothing techniques and more relaxed models with unspecified error distributions we further prove that the glr tests are asymptotically o

nonlinear modeling of time series using multivariate adaptive regression splines mars abstract multivariate adaptive regression splines mars is a new methodology due to friedman for nonlinear regression modeling mars can be conceptualized as a generalization of recursive partitioning that uses spline fitting in lieu of other simple fitting functions given a set of predictor variables mars fits a model in the form of an expansion in product spline basis functions of predictors chosen during a forward and backward recursive partitioning strategy mars produces continuous models for high dimensional data that can have multiple partitions and predictor variable interactions predictor variable contributions and interactions in a mars model may be analyzed using an anova style decomposition by letting the predictor variables in mars be lagged values of a time series one obtains a new method for nonlinear autoregressive threshold modeling of time series a significant feature of this extension of mars is its ability to produce models with limit cycles when modeling time series data that

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

identification and nonparametric estimation of a transformed additively separable model let r x z be a function that along with its derivatives can be consistently estimated nonparametrically this paper discusses the identification and consistent estimation of the unknown functions h m g and f where r x z h m x z m x z g x f z and h is strictly monotonic an estimation algorithm is proposed for each of the model s unknown components when r x z represents a conditional mean function the resulting estimators use marginal integration to separate the components g and f our estimators are shown to have a limiting normal distribution with a faster rate of convergence than unrestricted nonparametric alternatives their small sample performance is studied in a monte carlo experiment we apply our results to estimate generalized homothetic production functions for four industries in the chinese economy

erratum time reversed dynamical entropy and irreversibility in markovian random processes a concept of time reversed entropy per unit time is introduced in analogy with the entropy per unit time by shannon kolmogorov and sinai this time reversed entropy per unit time characterizes the dynamical randomness of a stochastic process backward in time while the standard entropy per unit time characterizes the dynamical randomness forward in time the difference between the time reversed and standard entropies per unit time is shown to give the entropy production of markovian processes in nonequilibrium steady states

a decomposed equation for local entropy and entropy production in volume preserving coarse grained systems abstract in this study an equation for the local entropy is derived based on the formulation of a master equation and is applied to volume preserving maps the equation consists of the following terms unsteady convection diffusion probability weighted phase space volume expansion rate nonnegative entropy production and residuals the decomposition makes it possible to evaluate entropy production in terms of microscopic dynamics and is expected to be applicable to many coarse grained systems on the phase space when it is applied to two volume preserving multibaker chain systems it is confirmed that the summation of the nonnegative entropy production on each site numerically coincides with the entropy production introduced by gilbert et al t gilbert j r dorfman p gaspard entropy production fractals and relaxation to equilibrium phys rev lett 85 2000 1606 1609 and the phenomenological expression both in nonequilibrium steady and unsteady states the coincidence is brought about by the fact that the residual terms vanish in the thermodynamic limit when they are integrated on each site it follows that the entropy production is dominated by the nonnegative entropy production term and becomes positive in nonequilibrium states

a law of order estimation and leading order terms for a family of averaged quantities on a multibaker chain system in this study a family of local quantities defined on each partition and its averaging on a macroscopic small region site are defined on a multibaker chain system on its averaged quantities a law of order estimation in the bulk system is proved making it possible to estimate the order of the quantities with respect to the representative partition scale parameter moreover the form of the leading order terms of the averaged quantities is obtained and the form enables us to have the macroscopic quantity in the continuum limit as 0 and to confirm its partitioning independency these deliverables fully explain the numerical results obtained by ishida consistent with the irreversible thermodynamics

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

local partitioned regression in this paper we introduce a kernel based estimation principle for nonparametric models named local partitioned regression lpr this principle is a nonparametric generalization of the familiar partition regression in linear models it has several key advantages first it generates estimators for a very large class of semi and nonparametric models a number of examples that are particularly relevant for economic applications will be discussed in this paper this class contains the additive partially linear and varying coefficient models as well as several other models that have not been discussed in the literature second lpr based estimators achieve optimality criteria they have optimal speed of convergence and are oracle efficient moreover they are simple in structure widely applicable and computationally inexpensive a monte carlo simulation highlights these advantages

burmann expansion and test for additivity summary we propose a lagrange multiplier test for additivity based on the btirmann expansion of a conditional mean function the asymptotic null distribution of the test is shown to be x2 under some regularity conditions in contrast the lagrange multiplier test proposed by chen et al 1995 is based on the volterra expansion of the conditional mean function we discuss some desirable advantages of the bt rmann expansion over the volterra expansion for nonlinear time series modelling we also reported an empirical study which shows that in terms of empirical power the lagrange multiplier test motivated by the biirmann expansion outperforms the test of chen et al 1995 for the cases for which the lagrange multiplier test is designed for other cases for which none of the tests is specifically designed the empirical powers of the two tests are comparable finally we illustrated the use of the lagrange multiplier test with a blowfly experimental system

conformal deformations of cat 0 spaces we show that the class of cat 0 spaces is closed under suitable conformal changes in particular any cat 0 space admits a large variety of non trivial deformations

the dirichlet problem for harmonic maps from riemannian polyhedra to spaces of upper bounded curvature this is a continuation of the cambridge tract harmonic maps between riemannian polyhedra by j eells and the present author the variational solution to the dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary and thereby uniquely determined the domain space is a compact admissible riemannian polyhedron with boundary while the target can be for example a simply connected complete geodesic space of nonpositive alexandrov curvature alternatively the target may have upper bounded curvature provided that the maps have a suitably small range essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain

data driven local bandwidth selection for additive models with missing data this paper deals in the nonparametric estimation of additive models in the presence of missing data in the response variable specifically in the case of additive models estimated by the backfitting algorithm with local polynomial smoothers 1 three estimators are presented one based on the available data and two based on a complete sample from imputation techniques we also develop a data driven local bandwidth selector based on a wild bootstrap approximation of the mean squared error of the estimators the performance of the estimators and the local bootstrap bandwidth selection method are explored through simulation experiments

entropy production from open volume preserving to dissipative systems we generalize gaspard s method for computing the e entropy production rate in hamiltonian systems to dissipative systems with attractors considered earlier by tel vollmer and breymann this approach leads to a natural definition of a coarse grained gibbs entropy which is extensive and which can be expressed in terms of the srb measures and volumes of the coarse graining sets which cover the attractor one can also study the entropy and entropy production as functions of the degree of resolution of the coarse graining process and examine the limit as the coarse graining size approaches zero we show that this definition of the gibbs entropy leads to a positive rate of irreversible entropy production for reversible dissipative systems we apply the method to the case of a two dimensional map based upon a model considered by vollmer tel and breymann that is a deterministic version of a biased random walk we treat both volume preserving and dissipative versions of the basic map and make a comparison between the two cases we discuss the e entropy production rate as a function of the size of the coarse graining cells for these biased random walks and for an open system with flux boundary conditions show regions of exponential growth and decay of the rate of entropy production as the size of the cells decreases this work describes in some detail the relation between the results of gaspard those of of tel vollmer and breymann and those of ruelle on entropy production in various systems described by anosov or anosov like maps

nonparametric tests of linearity for time series summary we introduce tests of linearity for time series based on nonparametric estimates of the conditional mean and the conditional variance the tests are compared to a number of parametric tests and to nonparametric tests based on the bispectrum asymptotic expressions give bad approximations and the null distribution under linearity is constructed using resampling of the best linear approximation the new tests perform well on the examples tested

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

spline backfitted kernel smoothing of nonlinear additive autoregression model application of nonparametric and semiparametric regression techniques to high dimensional time series data has been hampered due to the lack of effective tools to address the curse of dimensionality under rather weak conditions we propose spline backfitted kernel estimators of the component functions for the nonlinear additive time series data that are both computationally expedient so they are usable for analyzing very high dimensional time series and theoretically reliable so inference can be made on the component functions with confidence simulation experiments have provided strong evidence that corroborates the asymptotic theory

functional coefficient autoregressive models in this article we propose a new class of models for nonlinear time series analysis investigate properties of the proposed model and suggest a modeling procedure for building such a model the proposed modeling procedure makes use of ideas from both parametric and nonparametric statistics a consistency result is given to support the procedure for illustration we apply the proposed model and procedure to several data sets and show that the resulting models substantially improve postsample multi step ahead forecasts over other models

nonparametric identification of nonlinear time series selecting significant lags abstract in this article we suggest a nonparametric procedure for selecting significant lags in the model description of a general nonlinear stationary time series the procedure can be applied to both the conditional mean and the conditional variance and is valid for heteroscedastic series the procedure is illustrated by simulations and sunspot data lynx data and blowfly data are analyzed it is indicated that projectors can be used in conjunction with the procedure for selecting significant lags to check the adequacy of an additive time series model

application of a least absolute shrinkage and selection operator to aeroelastic flight test data identification of non linear systems involves estimating unknown parameters and model regressor selection selection of a subset of candidate terms that best describes the observed output model selection is an important step in black box modelling of any observed process this procedure is concerned with selecting a subset of parameters to give a parsimonious description of the system which may afford greater insight into the functionality of the system or a simpler controller design in this study a least absolute shrinkage and selection operator lasso technique is investigated for computing efficient model descriptions of non linear aeroelastic systems the lasso minimises the residual sum of squares by the addition of an l1 penalty term on the parameter vector of the traditional l2 minimisation problem its use for model selection is a natural extension of this constrained minimisation approach to pseudolinear regression problems which produces some model parameters that are exactly zero and the

fourier series approximation of separable models the approximation of a function affected by noise in several dimensions suffers from the so called curse of dimensionality in this paper a fourier series method based on regularization is developed both for uniform and random design when a restriction on the complexity of the curve such as additivity is considered in order to circumvent the problem optimal convergence theorems are stated and numerical experiments are shown on several test problems available in the literature together with comparisons with alternative methods

revaluation of the first order upwind difference scheme to solve coarse grained master equations this study addresses the initial boundary value problem of coarse grained probability measure on the state space in which a differentiable vector field v is given and as a consequence the differenced continuity equation using the first order upwind difference scheme uds based on the finite volume method appears as the physical substance on the coarse grained dynamics at first the uds is theoretically shown to be equivalent to a class of coarse grained master equations cgme brought by a principle that we cannot distinguish state points in the same partition with each other the principle is based on the formulation of non equilibrium statistical mechanics to resolve the macroscopic irreversibility moreover the entropy production evaluated by the uds is also shown to be in accord with the average volume contraction rate in the steady state this is essential for the non equilibrium statistical dynamics and was numerically confirmed under the principle of coarse graining the uds is very superior to the conventional monte carlo method in computer time and storage and is very useful to solve the cgme

model free variable selection computing power has encouraged the modelling of data sets of ever increasing size data mining applications in finance marketing and bioinformatics are obvious examples a limitation of nearly all existing variable selection methods is the need to specify the correct model before selection when the number of predictors is large model formulation and validation can be difficult or even infeasible on the basis of the theory of sufficient dimension reduction we propose a new class of model free variable selection approaches the methods proposed assume no model of any form require no nonparametric smoothing and allow for general predictor effects the efficacy of the methods proposed is demonstrated via simulation and an empirical example is given

search for significant variables in nonparametric additive regression summary nonparametric additive regression is studied under the assumption that only a subset of nonparametric components is nonzero each of these nonzero components depends on its own particular explanatory variable called a significant variable the search problem for significant variables is considered and an algorithm is proposed which guarantees exponentially decreasing error probabilities as the sample size grows we show that it is reasonable to use a rough bin estimator rather than to estimate the nonparametric components with the fastest possible rate

variable selection and function estimation in additive nonparametric regression using a data based prior abstract a hierarchical bayesian approach is proposed for variable selection and function estimation in additive nonparametric gaussian regression models and additive nonparametric binary regression models the prior for each component function is an integrated wiener process resulting in a posterior mean estimate that is a cubic smoothing spline each of the explanatory variables is allowed to be in or out of the model and the regression functions are estimated by model averaging to allow variable selection and model averaging data based priors are used for the smoothing parameter and the slope at 0 of each component function a two step markov chain monte carlo method is used to efficiently obtain the data based prior and to carry out variable selection and function estimation it is shown by simulation that significant improvements in the function estimators can be obtained over an approach that estimates all the unknown functions simultaneously the methodology is illustrated for a binary regressio

identification of nonlinear time series first order characterization and order determination summary we study the possibility of identifying nonlinear time series using nonparametric estimates of the conditional mean and conditional variance it is shown that most nonlinear models satisfy the assumptions needed to apply nonparametric asymptotic theory sampling variations of the conditional quantities are studied by simulation and explained by asymptotic arguments for a number of first order nonlinear autoregressive processes the conditional mean and variance can be used for identification purposes but one must be aware of bias and misspecification effects we also propose a criterion for determining the order of a general nonlinear model the criterion is justified in parts by heuristics but encouraging results are obtained from a limited set of simulation experiments several

testing and modeling threshold autoregressive processes abstract the threshold autoregressive model is one of the nonlinear time series models available in the literature it was first proposed by tong 1978 and discussed in detail by tong and lim 1980 and tong 1983 the major features of this class of models are limit cycles amplitude dependent frequencies and jump phenomena much of the original motivation of the model is concerned with limit cycles of a cyclical time series and indeed the model is capable of producing asymmetric limit cycles the threshold autoregressive model however has not received much attention in application this is due to a the lack of a suitable modeling procedure and b the inability to identify the threshold variable and estimate the threshold values the primary goal of this article therefore is to suggest a simple yet widely applicable model building procedure for threshold autoregressive models based on some predictive residuals a simple statistic is proposed to test for threshold nonlinearity and specify the

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

search for additive nonlinear time series causal models pointwise consistent feasible procedures for estimating contemporaneous linear causal structure from time series data have been developed using multiple conditional independence tests but no such procedures are available for non linear systems we describe a feasible procedure for learning a class of non linear time series structures which we call additive non linear time series we show that for data generated from stationary models of this type two classes of conditional independence relations among time series variables and their lags can be tested efficiently and consistently using tests based on additive model regression combining results of statistical tests for these two classes of conditional independence relations and the temporal structure of time series data a new consistent model specification procedure is able to extract relatively detailed causal information we investigate the finite sample behavior of the procedure through simulation and illustrate the application of this method through analysis of the possible causal connections among four ocean indices several variants of the procedure are also discussed

entropy production in open volume preserving systems we describe a mechanism leading to positive entropy production in volume preserving systems under nonequilibrium conditions we consider volume preserving systems sustaining a diffusion process like the multibaker map or the lorentz gas a continuous flux of particles is imposed across the system resulting in a steady gradient of concentration in the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient the invariant measure becomes singular for instance in the multibaker map the limit invariant measure has a cumulative function given in terms of the nondifferentiable takagi function because of this singularity of the invariant measure the entropy must be defined as a coarse grained entropy instead of the fined grained gibbs entropy which would require the existence of a regular measure with a density the coarse grained entropy production is then shown to be asymptotically positive and moreover given by the entropy production expected from irreversible thermodynamics

nonparametric matching and efficient estimators of homothetically separable functions for vectors z and w and scalar v let r v z w be a function that can be non parametrically estimated consistently and asymptotically normally such as a distribution density or conditional mean regression function we provide consistent asymptotically normal nonparametric estimators for the functions g and h where r v z w h vg z w and some related models this framework encompasses homothetic and homothetically separable functions and transformed partly additive models r v z w h v g z w for unknown functions g and h such models reduce the curse of dimensionality provide a natural generalization of linear index models and are widely used in utility production and cost function applications we also provide an estimator of g that is oracle efficient achieving the same performance as an estimator based on local least squares when h is known

prospects of nonparametric modeling this article brie y summarizes key developments in nonparametric function estimation over the past three decades it highlights some potentially fruitful areas of future research on nonparametric modeling and its applications

nonparametric identification of nonlinear time series projections abstract we study the possibility of identifying general linear and nonlinear time series models using nonparametric methods the kernel estimators of the conditional mean and variance are used as a basis and the properties of these quantities as model indicators are briefly discussed some drawbacks are pointed out and motivated by these we introduce projections as tools of identification the projections are especially useful for additive modeling expressions for the asymptotic bias and variance are obtained the projection of the conditional variance is suggested as a tool for identifying heteroscedastic time series the results are illustrated by simulations for both the estimators of the projections and the estimators of the conditional mean and variance

fluctuation theorems for entropy production in open systems we derive a fluctuation theorem to describe entropy fluctuations in steady states of systems with density gradients due to open boundaries the fluctuations are related to the growth rate of the phase space density instead of the phase space contraction rate explicit derivations are presented for a multibaker map but the arguments are rather general and should hold for a much wider class of dynamical systems a comparison with recent results for stochastic systems is also given

consistent specification tests for semiparametric nonparametric models based on series estimation methods abstract this paper considers the problem of consistent model specification tests using series estimation methods the null models we consider in this paper all contain some nonparametric components a leading case we consider is to test for an additive partially linear model the null distribution of the test statistic is derived using a central limit theorem for hilbert valued random arrays the test statistic is shown to be able to detect local alternatives that approach the null models at the order of op n 1 2 we show that the wild bootstrap method can be used to approximate the null distribution of the test statistic a small monte carlo simulation is reported to examine the finite sample performance of the proposed test we also show that the proposed test can be easily modified to obtain series based consistent tests for other semiparametric nonparametric models

nonparametric function estimation involving time series consider a stationary time series mathbf x t y t t 0 pm 1 ldots with mathbf x t being mathbb r d valued and y t real valued the conditional mean function is given by theta mathbf x 0 e y 0 mid mathbf x 0 under appropriate regularity conditions a local average estimator of this function based on a finite realization mathbf x 1 y 1 ldots mathbf x n y n can be chosen to achieve the optimal rate of convergence n 1 2 d both pointwise and in l 2 norms restricted to a compact and it can also be chosen to achieve the optimal rate of convergence n 1 log n 1 2 d in l infty norm restricted to a compact similar results hold for local median estimators of the conditional median function which is given by theta mathbf x 0 operatorname med y 0 mid mathbf x 0

derivation of nonlinear fokker planck equations by means of approximations to the master equation nonlinear fokker planck equations fpes are derived as approximations to the master equation in cases of transitions among both discrete and continuous sets of states the nonlinear effects introduced through the transition probabilities are argued to be relevant for many real phenomena within the class of anomalous diffusion problems the nonlinear fpes obtained appear to be more general than some previously proposed on a purely phenomenological basis ones in spite of this the same kind of solution applies i e it is shown that the time dependent tsallis s probability distribution is a solution of both equations obtained either from discrete or continuous sets of states and that the corresponding stationary solution is in the infinite time limit a stable solution

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

structural tests in additive regression we consider the component analysis problem for a regression model with an additive structure the problem is to test whether some of the additive components are of polynomial structure e g linear without specifying the structure of the remaining components a particular case is the problem of selecting the significant covariates the method that we present is based on the wavelet transform using the haar basis which allows for applications under mild conditions on the design and smoothness of the regression function the results demonstrate that each component of the model can be tested with the rate corresponding to the case if all of the remaining components were known the proposed procedure is also computationally straightforward simulation results and a real data example about female labor supply demonstrate the test s good performance

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

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

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

component identification and estimation in nonlinear high dimensional regression models by structural adaptation this article proposes a new method of analysis of a partially linear model whose nonlinear component is completely unknown the target of analysis is identification of the set of regressors that enter in a nonlinear way in the model function and complete estimation of the model including slope coefficients of the linear component and the link function of the nonlinear component the procedure also allows selection of the significant regression variables we also develop a test of linear hypothesis against a partially linear alternative or more generally a test that the nonlinear component is m dimensional for m 0 1 2 the approach proposed in this article is fully adaptive to the unknown model structure and applies under mild conditions on the model the only important assumption is that the dimensionality of nonlinear component is relatively small the theoretical results indicate that the procedure provides a prescribed level of the identification error and estimates the linear component with