local asymptotics for polynomial spline regression in this paper we develop a general theory of local asymptotics for least squares estimates over polynomial spline spaces in a regression problem the polynomial spline spaces we consider include univariate splines tensor product splines and bivariate or multivariate splines on triangulations we establish asymptotic normality of the estimate and study the magnitude of the bias due to spline approximation the asymptotic normality holds uniformly over the points where the regression function is to be estimated and uniformly over a broad class of design densities error distributions and regression functions the bias is controlled by the minimum l norm of the error when the target regression function is approximated by a function in the polynomial spline space that is used to define the estimate the control of bias relies on the stability in l norm of l 2 projections onto polynomial spline spaces asymptotic normality of least squares estimates over polynomial or trigonometric polynomial spaces is also treated by the general theory in addition a preliminary analysis of additive models is provided

analysis of additive dependencies and concurvities using smallest additive principal components additive principal components are a nonlinear generalization of linear principal components their distinguishing feature is that linear forms i a i x i are replaced with additive functions i f i x i a considerable amount of flexibility for fitting data is gained when linear methods are replaced with additive ones our interest is in the smallest principal components which is somewhat uncommon smallest additive principal components amount to data descriptions in terms of approximate implicit equations i f i x i 0 estimation of such equations is a data analytic method in its own right competing in some cases with the more customary regression approaches it is also a diagnostic tool in additive regression for detection of so called concurvity this term describes degeneracies among predictor variables in additive regression models similar to collinearity in linear regression models concurvity may lead to statistically unstable contributions of variables to additive models as an example we show in a reanalysis of the ozone data from breiman and friedman that concurvity does indeed exist in this particular data a fact which may impact the interpretation of the additive fits in the second half of this paper we give some second order theory including the description of null situations and eigenexpansions derived from associated eigenproblems we show how ace and additive principal components are related and we outline some analytical methods for theoretical calculations of additive principal components lastly we consider methods of estimation and computation additive principal components have had a long tradition in psychometric research and correspondence analysis they have started receiving attention by statisticians only in recent years

projection estimation in multiple regression with application to functional anova models a general theory on rates of convergence of the least squares projection estimate in multiple regression is developed the theory is applied to the functional anova model where the multivariate regression function is modeled as a specified sum of a constant term main effects functions of one variable and selected interaction terms functions of two or more variables the least squares projection is onto an approximating space constructed from arbitrary linear spaces of functions and their tensor products respecting the assumed anova structure of the regression function the linear spaces that serve as building blocks can be any of the ones commonly used in practice polynomials trigonometric polynomials splines wavelets and finite elements the rate of convergence result that is obtained reinforces the intuition that low order anova modeling can achieve dimension reduction and thus overcome the curse of dimensionality moreover the components of the projection estimate in an appropriately defined anova decomposition provide consistent estimates of the corresponding components of the regression function when the regression function does not satisfy the assumed anova form the projection estimate converges to its best approximation of that form

minimax nonparametric classification ii model selection for adaptation for pt i see ibid vol 45 no 7 p 2271 84 1999 we study nonparametric estimation of a conditional probability for classification based on a collection of finite dimensional models for the sake of flexibility different types of models linear or nonlinear are allowed as long as each satisfies a dimensionality assumption we show that with a suitable model selection criterion the penalized maximum likelihood estimator has a risk bounded by an index of resolvability expressing a good tradeoff among approximation error estimation error and model complexity the bound does not require any assumption on the target conditional probability and can be used to demonstrate the adaptivity of estimators based on model selection examples are given with both splines and neural nets and problems of high dimensional estimation are considered the resulting adaptive estimator is shown to behave optimally or near optimally over sobolev classes with unknown orders of interaction and smoothness and classes of integrable fourier transform of gradient in terms of rates of convergence the performance is the same as if one knew which of them contains the true conditional probability in advance the corresponding classifier also converges optimally or nearly optimally simultaneously over these classes

improving point and interval estimators of monotone functions by rearrangement suppose that a target function is monotonic and an available original estimate of this target function is not monotonic rearrangements univariate and multivariate transform the original estimate to a monotonic estimate that always lies closer in common metrics to the target function furthermore suppose an original confidence interval which covers the target function with probability at least 1 is defined by an upper and lower endpoint functions that are not monotonic then the rearranged confidence interval defined by the rearranged upper and lower endpoint functions is monotonic shorter in length in common norms than the original interval and covers the target function with probability at least 1 we illustrate the results with a growth chart example

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

semi nonparametric estimation of the call option price surface under strike and time to expiry no arbitrage constraints abstract we suggest a semi nonparametric estimator for the call option price surface the estimator is a bivariate tensor product b spline to enforce no arbitrage constraints across strikes and expiry dates we establish sufficient no arbitrage conditions on the control net of the b spline surface the conditions are linear and therefore allow for an implementation of the estimator by means of standard quadratic programming techniques the consistency of the estimator is proved by means of simulations we explore the statistical efficiency benefits that are associated with estimating option price surfaces and state price densities under the full set of no arbitrage constraints we estimate a call option price surface families of first order strike derivatives and state price densities for s p 500 option data

experimental studies of separation on a two dimensional airfoil at low reynolds numbers the laminar separation transition and turbulent reattachment near the leading edge of a two dimensional naca 663 018 airfoil were investigated using a low speed smoke visualization wind tunnel lift and drag force measurements were made using an external strain gage balance for a chord reynolds number range of 40 goo400 000 an extensive flow visualization study was performed and correlated with the force measurements experiments were also conducted with distributed surface roughness at the leading edge and external acoustic excitation to influence the development of the airfoil boundary layer this study delineates the effects of angle of attack and chord reynolds number on the separation characteristics and airfoil performance nomenclature c model chord cd section profile drag coefficient uncorrected cf section lift coefficient uncorrected cp pressure coefficient acoustic frequency hz r reattachment location rc reynolds number based on chord length u civ s separation location t location of approximate end of transition freestream velocity x c nondimensional distance along chord a angle of attack v kinematic viscosity

separated shear layer development on an airfoil at low reynolds numbers flow transition in the separated shear layer on the upper surface of a naca 0025 airfoil at low reynolds numbers was investigated the study involved wind tunnel experiments and linear stability analysis detailed measurements were conducted for reynolds numbers of 100 000 and 150 000 at 0 5 and 10 degree angles of attack for all cases examined laminar boundary layer separation takes place on the upper surface of the airfoil the separated shear layer fails to reattach to the airfoil surface for the lower reynolds number but reattachment occurs for the higher reynolds number despite this difference in flow development experimental results show that a similar transition mechanism is attendant for both reynolds number flow regimes flow transition occurs due to the amplification of natural disturbances in the separated shear layer within a band of frequencies centered at some fundamental frequency the initial growth of disturbances centered at the fundamental frequency is followed by the growth of a subharmonic component eventually leading to flow transition the growing disturbances also cause shear layer roll up and the formation of roll up vortices the results show that inviscid stability theory can be employed to adequately estimate such salient characteristics as the frequency of the most amplified disturbances and their propagation speed this implies that the roll up vortices can be attributed to inviscid instability however the results suggest that viscous and nonparallel effects need to be accounted for to effectively model the convective growth of the disturbances in the separated shear layer

smooth function approximation using neural networks an algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented in this paper the approach is implemented for the approximation of smooth batch data containing the function s input output and possibly gradient information the training set is associated to the network adjustable parameters by nonlinear weight equations the cascade structure of these equations reveals that they can be treated as sets of linear systems hence the training process and the network approximation properties can be investigated via linear algebra four algorithms are developed to achieve exact or approximate matching of input output and or gradient based training sets their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques

using specially designed exponential families for density estimation i this paper have y being portions of the real line or of the plane but the methodology applies just as well to higher dimensionalities and to more complicated spaces estimates of g y are traditionally constructed in two quite different ways by maximum likelihood fitting within some parametric family such as the normal or by nonparametric methods such as kernel density estimation these two methods can be combined by putting an exponential family through a nonparametric estimator the resulting hybrid estimators are the specially designed exponential families of the title figure 1 shows a simple example of this methodology the y are pain i scores for n s 67 women each obtained by averaging the results from a questionnaire administered after an operation the scale runs from y s 0 s w x

finding supply function equilibria with asymmetric firms firms compete in supply functions when they offer a schedule of prices and quantities into a market for example this occurs in many wholesale electricity markets we study the equilibrium behaviour when firms differ both with regard to their costs and their capacities we characterize strong equilibrium solutions in which given the other players supply functions optimal profits are achieved for every demand realisation if the demand can be low enough for it to be met economically with supply from just one firm then the supply function equilibria are ordered in a natural way we consider equilibria in which for the highest levels of demand all but one of the firms have reached their capacity limit we show that there can be at most one supply function equilibrium with this property we also propose a new numerical method to find asymmetric supply function equilibria using piecewise linear approximations and a discretization of the demand distribution we show that this approach has good theoretical convergence behaviour finally we present numerical results from an implementation using gams to demonstrate that the approach is effective in practice

improving estimates of monotone functions by rearrangement suppose that a target function f0 r r is monotonic namely weakly increasing and an original estimate f of the target function is available which is not weakly increasing many common estimation methods used in statistics produce such estimates f we show that these estimates can always be improved with no harm using rearrangement techniques the rearrangement methods univariate and multivariate transform the original estimate to a monotonic estimate f and the resulting estimate is closer to the true curve f0 in common metrics than the original estimate f we illustrate the results with a computational example and an empirical example dealing with age height growth charts

estimation of a regression function by maxima of minima of linear functions in this paper estimation of a regression function from independent and identically distributed random variables is considered estimates are defined by minimization of the empirical l2 risk over a class of functions which are defined as maxima of minima of linear functions results concerning the rate of convergence of the estimates are derived in particular it is shown that for smooth regression functions satisfying the assumption of single index models the estimate is able to achieve up to some logarithmic factor the corresponding optimal one dimensional rate of convergence hence under these assumptions the estimate is able to circumvent the so called curse of dimensionality the small sample behavior of the estimates is illustrated by applying them to simulated data

a bayesian approach to nonparametric bivariate regression abstract this article outlines a general bayesian approach to estimating a bivariate regression function in a nonparametric manner it models the function using a bivariate regression spline basis with many terms binary indicator variables corresponding to these terms are introduced to explicitly model the uncertainty of whether or not the terms provide a significant contribution to the regression the regression function is estimated using an estimate of its posterior mean smoothing over the distribution of these binary indicator variables to make the computations tractable all estimates are obtained using markov chain monte carlo sampling extensive simulated comparisons are provided that demonstrate the competitive performance of this approach against other data driven bivariate surface estimators prominent in the literature it is then shown how the procedure can be extended to provide a general approach to nonparametric bivariate surface estimation in two difficult regression settings the first

distribution free estimation of heteroskedastic binary response models using probit logit criterion functions abstract in this paper estimators for distribution free heteroskedastic binary response models are proposed the estimation procedures are based on relationships between distribution free models with a conditional median restriction and parametric models such as probit logit exhibiting multiplicative heteroskedasticity the first proposed estimator is based on the observational equivalence between the two models and is a semiparametric sieve estimator see e g gallant and nychka 1987 ai and chen 2003 and chen et al 2005 for the regression coefficients based on maximizing standard logit probit criterion functions such as nlls and mle this procedure has the advantage that choice probabilities and regression coefficients are estimated simultaneously the second proposed procedure is based on the equivalence between existing semiparametric estimators for the conditional median model manski 1975 manski 1985 horowitz 1992 and the standard parametric probit logit nlls estimator this estimator has the advantage of being implementable with standard software packages such as stata distribution theory is developed for both estimators and a monte carlo study indicates they both perform well in finite samples

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

a robust weakly compressible sph method and its comparison with an incompressible sph summary this paper presents a comparative study for the weakly compressible wcsph and incompressible isph smoothed particle hydrodynamics methods by providing numerical solutions for fluid flows over an airfoil and a square obstacle improved wcsph and isph techniques are used to solve these two bluff body flow problems it is shown that both approaches can handle complex geometries using the multiple boundary tangents mbt method and eliminate particle clustering induced instabilities with the implementation of a particle fracture repair procedure as well as the corrected sph discretization scheme wcsph and isph simulation results are compared and validated with those of a finite element method fem the quantitative comparisons of wcsph isph and fem results in terms of strouhal number for the square obstacle test case and the pressure envelope surface traction forces and velocity gradients on the airfoil boundaries as well as the lift and drag values for the airfoil geometry indicate that the wcsph method with the suggested implementation produces numerical results as accurate and reliable as those of the isph and fem methods copyright 2011 john wiley sons ltd

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

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

nonparametric regression based on hierarchical interaction models in this paper we introduce the so called hierarchical interaction models where we assume that the computation of the value of a function m mathbb r d rightarrow mathbb r is done in several layers where in each layer a function of at most d inputs computed by the previous layer is evaluated we investigate two different regression estimates based on polynomial splines and on neural networks and show that if the regression function satisfies a hierarchical interaction model and all occurring functions in the model are smooth the rate of convergence of these estimates depends on d and not on d hence in this case the estimates can achieve good rate of convergence even for large d and are in this sense able to circumvent the so called curse of dimensionality