limiting distributions for l sb 1 regression estimators under general conditions it is well known that l 1 estimators of regression parameters are asymptotically normal if the distribution function has a positive derivative at 0 in this paper we derive the asymptotic distributions under more general conditions on the behavior of the distribution function near 0

adapting for heteroscedasticity in linear models in a heteroscedastic linear model it is known that if the variances are a parametric function of the design then one can construct an estimate of the regression parameter which is asymptotically equivalent to the weighted least squares estimate with known variances we show that the same is true when the only thing known about the variances is that they are determined by an unknown but smooth function of the design or the mean response

generalized partially linear single index models abstract the typical generalized linear model for a regression of a response y on predictors x z has conditional mean function based on a linear combination of x z we generalize these models to have a nonparametric component replacing the linear combination t 0x t 0z by 0 t 0x t 0z where 0 is an unknown function we call these generalized partially linear single index models gplsim the models include the single index models which have 0 0 using local linear methods we propose estimates of the unknown parameters 0 0 and the unknown function 0 and obtain their asymptotic distributions examples illustrate the models and the proposed estimation methodology

local polynomial variance function estimation the conditional variance function in a heteroscedastic nonparametric regression model is estimated by linear smoothing of squared residuals attention is focused on local polynomial smoothers both the mean and variance functions are assumed to be smooth but neither is assumed to be in a parametric family the biasing effect of preliminary estimation of the mean is studied and a degrees of freedom correction of bias is proposed the corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known a proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions the proposal is illustrated with data from the lidar method of measuring atmospheric pollutants and from turbulence model computations

effect of mean on variance function estimation in nonparametric regression variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived we are particularly interested in the effect of the unknown mean on the estimation of the variance function our results indicate that contrary to the common practice it is not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean when the mean function is not smooth instead it is more desirable to use estimators of the mean with minimal bias on the other hand when the mean function is very smooth our numerical results show that the residual based method performs better but not substantial better than the first order difference based estimator in addition our asymptotic results also correct the optimal rate claimed in hall and carroll j roy statist soc ser b51 1989 3 14

on contour crossings in contour advective simulations part 1 algorithm for detection and quantification this is the first of two papers devoted to the analysis of contour crossing errors that occur in contour advective simulations of fluid motion here an algorithm is presented for quantifying the error due to contour crossings the first step is to determine the relative proximity of all possible pairs of contours a digital representation of each contour is produced to aid in the corresponding calculation simple analysis of functions is then used to find any crossings between contours deemed close to each other by the digital representation method next the area in error of a pair of crossing contours is calculated by identifying the polygon or polygons that approximately bound the erroneous region finally some preliminary results of analysis of contour crossings that occur in contour advective semi lagrangian casl simulations of single layer quasigeostrophic turbulence are presented it is shown that the error due to contour crossings is small in the simulations considered here

doubly robust and efficient estimators for heteroscedastic partially linear single index models allowing high dimensional covariates we study the heteroscedastic partially linear single index model with an unspecified error variance function which allows for high dimensional covariates in both the linear and the single index components of the mean function we propose a class of consistent estimators of the parameters by using a proper weighting strategy an interesting finding is that the linearity condition which is widely assumed in the dimension reduction literature is not necessary for methodological or theoretical development it contributes only to the simplification of non optimal consistent estimation we also find that the performance of the usual weighted least square type of estimators deteriorates when the non parametric component is badly estimated however estimators in our family automatically provide protection against such deterioration in that the consistency can be achieved even if the baseline non parametric function is completely misspecified we further show that the most efficient estimator is a member of this family and can be easily obtained by using non parametric estimation properties of the estimators proposed are presented through theoretical illustration and numerical simulations an example on gender discrimination is used to demonstrate and to compare the practical performance of the estimators

eigenstructure assignment for componentwise ultimate bound minimization in discrete time linear systems this technical note presents a feedback design approach to minimise state ultimate bounds ubs in linear time invariant discrete time systems subject to non vanishing bounded disturbances a target closed loop eigenstructure is first constructed to achieve the lowest possible ubs for one or more state components of the system structural conditions on the system matrices are then derived to guarantee that the target eigenstructure can be assigned by state feedback when the required structural conditions are satisfied an eigenvector assignment procedure can be applied to simultaneously minimise multiple ubs in systems with multiple inputs the results are illustrated by a numerical example drawn from a distributed voltage control problem in inverter based microgrids

reference values obtained by kernel based estimation of quantile regressions in this paper we study the problem of estimating non parametrically a quantile regression curve as it applies to computing reference values we propose an automatic procedure that uses a symmetrized nearest neighbor kernel estimator of conditional distributions we also discuss ways of measuring the dispersion of quantile regression estimator one is based on the asymptotic distribution of such quantiles while the other relies on the bootstrap method the results of a small simulation study show that the methods of the paper perform rather well even in a situation where a good parametric solution is available as an example we analyze a small part of a data set that was collected to establish reference values for blood velocity in different parts of the umbilical cord of human fetuse as they grow toward birth

on difference based variance estimation in nonparametric regression when the covariate is high dimensional summary we consider the problem of estimating the noise variance in homoscedastic nonparametric regression models for low dimensional covariates t d d 1 2 difference based estimators have been investigated in a series of papers for a given length of such an estimator difference schemes which minimize the asymptotic mean squared error can be computed for d 1 and d 2 however from numerical studies it is known that for finite sample sizes the performance of these estimators may be deficient owing to a large finite sample bias we provide theoretical support for these findings in particular we show that with increasing dimension d this becomes more drastic if d4 these estimators even fail to be consistent a different class of estimators is discussed which allow better control of the bias and remain consistent when d4 these estimators are compared numerically with kernel type estimators which are asymptotically efficient and some guidance is given about when their use becomes necessary

adaptive variance function estimation in heteroscedastic nonparametric regression we consider a wavelet thresholding approach to adaptive variance function estimation in heteroscedastic nonparametric regression a data driven estimator is constructed by applying wavelet thresholding to the squared first order differences of the observations we show that the variance function estimator is nearly optimally adaptive to the smoothness of both the mean and variance functions the estimator is shown to achieve the optimal adaptive rate of convergence under the pointwise squared error simultaneously over a range of smoothness classes the estimator is also adaptively within a logarithmic factor of the minimax risk under the global mean integrated squared error over a collection of spatially inhomogeneous function classes numerical implementation and simulation results are also discussed

assessing the adequacy of variance function in heteroscedastic regression models heteroscedastic data arise in many applications in heteroscedastic regression analysis the variance is often modeled as a parametric function of the covariates or the regression mean we propose a kernel smoothing type nonparametric test for checking the adequacy of a given parametric variance structure the test does not need to specify a parametric distribution for the random errors it is shown that the test statistic has an asymptotical normal distribution under the null hypothesis and is powerful against a large class of alternatives we suggest a simple bootstrap algorithm to approximate the distribution of the test statistic in finite sample size numerical simulations demonstrate the satisfactory performance of the proposed test we also illustrate the application by the analysis of a radioimmunoassay data set

estimation for a partial linear single index model in this paper we study the estimation for a partial linear single index model a two stage estimation procedure is proposed to estimate the link function for the single index and the parameters in the single index as well as the parameters in the linear component of the model asymptotic normality is established for both parametric components for the index a constrained estimating equation leads to an asymptotically more efficient estimator than existing estimators in the sense that it is of a smaller limiting variance the estimator of the nonparametric link function achieves optimal convergence rates and the structural error variance is obtained in addition the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters a simulation study is performed and an application to a real dataset is illustrated the extension to multiple indices is briefly sketched

estimating residual variance in nonparametric regression using least squares summary we propose a new estimator for the error variance in a nonparametric regression model we estimate the error variance as the intercept in a simple linear regression model with squared differences of paired observations as the dependent variable and squared distances between the paired covariates as the regressor for the special case of a one dimensional domain with equally spaced design points we show that our method reaches an asymptotic optimal rate which is not achieved by some existing methods we conduct extensive simulations to evaluate finite sample performance of our method and compare it with existing methods our method can be extended to nonparametric regression models with multivariate functions defined on arbitrary subsets of normed spaces possibly observed on unequally spaced or clustered designed points

estimating the variance in nonparametric regression what is a reasonable choice the exact mean squared error mse of estimators of the variance in nonparametric regression based on quadratic forms is investigated in particular two classes of estimators are compared hall kay and titterington s optimal difference based estimators and a class of ordinary difference based estimators which generalize methods proposed by rice and gasser sroka and jennen steinmetz for small sample sizes the mse of the first estimator is essentially increased by the magnitude of the integrated first two squared derivatives of the regression function it is shown that in many situations ordinary difference based estimators are more appropriate for estimating the variance because they control the bias much better and hence have a much better overall performance it is also demonstrated that rice s estimator does not always behave well data driven guidelines are given to select the estimator with the smallest mse

inverse optimality in robust stabilization the concept of a robust control lyapunov function bf rclf is introduced and it is shown that the existence of an bf rclf for a control affine system is equivalent to robust stabilizability via continuous state feedback this extends artstein s theorem on nonlinear stabilizability to systems with disturbances it is then shown that every bf rclf satisfies the steady state hamilton jacobi isaacs hji equation associated with a meaningful game and that every member of a class of pointwise min norm control laws is optimal for such a game these control laws have desirable properties of optimality and can be computed directly from the bf rclf without solving the hji equation for the upper value function

a remark on the stability of interconnected nonlinear systems we use a notion of generalized l sub 2 gain for nonlinear systems where the gain is considered as a function of the state instead of a global constant for characterizing stability properties of interconnected systems the obtained results generalize the well known small gain theorem

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

variance functions and the minimum detectable concentration in assays abstract assay data are often fit by a nonlinear regression model incorporating heterogeneity of variance as ion radioimmunoassay for example typically the standard deviation of the response is taken to be proportional to a power theta of the mean there is considerable empirical evidence suggesting that for assays of a reseasonable size how one estimates the parameter theta does not greatly affect how well one estimates the the mean regression function an additional component of assay analysis is the estimation of auxillary constructs such as the minimum detectable concentration for which many definitions exist we focus on one such definition the minimum detectable concentration depends both on theta and the mean regression function we compare three standard method of estimating the parameter theta due to rodbard 1978 raab 1981a and carroll and ruppert 1982b when duplicate counts are taken at each concentration the first method is only 20 efficient asymptotically in comparison to the third and the resulting estimate of the minimum detectable concentration is asymptotically 3 3 times more variable for first than the third less dramatic results obtain for the second estimator compared to the third this estimator is still not efficient however simulation results and an example are supportive of the asymptotic theory keywords least squares method

a partial differential inequality for dissipative nonlinear systems abstract in this paper we show that storage functions for dissipative systems can be characterised by a partial differential inequality in the weak or viscosity sense the storage functions need not be differentiable this characterisation is stable with respect to perturbations we formulate a version of the positive real lemma for passive systems using the weak sense framework

signal to noise ratios performance criteria and transformations for the analysis of designed experiments taguchi uses performance criteria that he calls signal to noise sn ratios three such criteria are here denoted by sn t sn l and sn s the criterion sn t was to be used in preference to the standard deviation for the problem of achieving for some quality characteristic y the smallest mean squared error about an operating target value leon shoemaker and kacker 1987 showed how sn t was appropriate to solve this problem only when y was proportional to y on that assumption the same result could be obtained more simply by conducting the analysis in terms of log y rather than y a more general transformation approach is here introduced for other commonly met kinds of dependence between y and y including no dependence and a lambda plot is presented that uses the data to suggest an appropriate transformation the criteria sn l and sn s were for problems in which the objective was to make the response as large or as small as possible it is arg

valid post selection inference it is common practice in statistical data analysis to perform data driven variable selection and derive statistical inference from the resulting model such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori we propose to produce valid post selection inference by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence and retention intervals simultaneity is required for all linear functions that arise as coefficient estimates in all submodels by purchasing simultaneity insurance for all possible submodels the resulting post selection inference is rendered universally valid under all possible model selection procedures this inference is therefore generally conservative for particular selection procedures but it is always less conservative than full scheffe protection importantly it does not depend on the truth of the selected submodel and hence it produces valid inference even in wrong models we describe the structure of the simultaneous inference problem and give some asymptotic results

efficient estimation of conditional variance functions in stochastic regression author s fan jianqing yao qiwei abstract conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data under a general setup which includes the nonlinear time series model as a special case we propose an e cient and adaptive method for estimating the conditional variance the basic idea is to apply a local linear regression to the squared residuals we demonstrate that without knowing the regression function we can estimate the conditional variance asymptotically as well as if the regression were given this asymptotic result established under the assumption that the observations are made from a strictly stationary and absolutely regular process is also veri ed via simulation further the asymptotic result paves the way for adapting an automatic bandwidth selection scheme an application with nancial data illustrates the usefulness of the proposed techniques

correlation curves as local measures of variance explained by regression abstract we call a model for an experiment heterocorrelatious if the strength of the relationship between a response variable y and a covariate x is different in different regions of the covariate space for such experiments we introduce a correlation curve that measures heter ocorrelaticity in terms of the variance explained by regression locally at each covariate value more precisely the squared correlation curve is obtained by first expressing the usual linear model variance explained to total variance formula in terms of the residual variance and the regression slope and then replacing these by the conditional residual variance depending on x and the slope of the conditional mean of y given x x the correlation curve x satisfies the invariance properties of correlation it reduces to the galton pearson correlation in linear models it is between 1 and 1 it is 0 when x and y are independent and it is 1 when y is a function of x we introduce estimates of the correlation curve based on

estimation and testing for partially linear single index models in partially linear single index models we obtain the semiparametrically efficient profile least squares estimators of regression coefficients we also employ the smoothly clipped absolute deviation penalty scad approach to simultaneously select variables and estimate regression coefficients we show that the resulting scad estimators are consistent and possess the oracle property subsequently we demonstrate that a proposed tuning parameter selector bic identifies the true model consistently finally we develop a linear hypothesis test for the parametric coefficients and a goodness of fit test for the nonparametric component respectively monte carlo studies are also presented

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

adaptive drift estimation for nonparametric diffusion model we consider a nonparametric diffusion process whose drift and diffusion coefficients are nonparametric functions of the state variable the goal is to estimate the unknown drift coefficient we apply a locally linear smoother with a data driven bandwidth choice the procedure is fully adaptive and nearly optimal up to a log log factor the results about the quality of estimation are nonasymptotic and do not require any ergodic or mixing properties of the observed process

the impact of unidentified location effects on dispersion effects identification from unreplicated factorial designs there have been several publications on identifying dispersion effects from an unreplicated fractional or full factorial experiment some of these methods implicitly assume that unknown location effects are first identified correctly from the data in practice however an empirical location effects identification procedure may leave some small to moderate location effects undetected it is shown in this study that such unidentified location effects can cumulatively impair subsequent dispersione effects identification numerical evidence based on a welding experiment and results from a simulation study are provided a mathematical explanation of the impact of unidentified location effects on dispersion effects identification is given the dispersion effects identification methods initially proposed for unreplicated data can be naturally extended to replicated experiments with the impact of unidentified location effects eliminated a fluorescent lamp experiment is analyzed to illustrate these methods

simultaneous h infty control for nonlinear systems this note develops a novel method for designing simultaneous h infin state feedback controllers for a collection of single input nonlinear systems based on the kalman yakubovich popov lemma necessary and sufficient conditions for the existence of simultaneous h infin controllers are derived by the control storage function approach a universal formula for constructing continuous time invariant simultaneous h infin state feedback controllers is presented

robust stabilization of the angular velocity of a rigid body the problem of robust control for the angular velocity of a rigid body subject to external disturbances is addressed it is shown that if the disturbances are matched there exists a control law attenuating the effect of the disturbances whereas in the case of non matched disturbances no such a feedback law generically exists hence a new concept of disturbance attenuation is introduced and it is proved that the aforementioned problem is solvable in this weaker sense explicit expressions of the control laws solving the proposed robust stabilization problems are given

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

brief practical l2 disturbance attenuation for nonlinear systems in this work a notion of generalized l 2 gain for nonlinear systems where the gain is considered as a function of the state instead of a global constant is presented this new notion seems to be adequate to characterize the gain properties of several nonlinear systems which do not possess a uniform l 2 gain property i e the l 2 gain depends on the operating point moreover a notion of practical l 2 gain attenuation which extends the standard definition and parallels mutatae mutandis the concepts of practical stability is also proposed

penalized spline estimation for partially linear single index models single index models are potentially important tools for multivariate nonparametric regression they generalize linear regression by replacing the linear combination 0tx with a nonparametric component 0 0tx where 0 is an unknown univariate link function by reducing the dimensionality from that of a general covariate vector x to a univariate index 0tx single index models avoid the so called curse of dimensionality we propose penalized spline p spline estimation of 0 in partially linear single index models where the mean function has the form 0 0tx 0tz the p spline approach offers a number of advantages over other fitting methods for single index models all parameters in the p spline single index model can be estimated simultaneously by penalized nonlinear least squares as a direct least squares fitting method our approach is rapid and computationally stable standard nonlinear least squares software can be used moreover joint inference for 0 0 and 0 is possible by

smoothing splines with varying smoothing parameter this paper considers the development of spatially adaptive smoothing splines for the estimation of a regression function with nonhomogeneous smoothness across the domain two challenging issues arising in this context are the evaluation of the equivalent kernel and the determination of a local penalty the penalty is a function of the design points in order to accommodate local behaviour of the regression function we show that the spatially adaptive smoothing spline estimator is approximately a kernel estimator and that the equivalent kernel is spatially dependent the equivalent kernels for traditional smoothing splines are a special case of this general solution with the aid of the green s function for a two point boundary value problem explicit forms of the asymptotic mean and variance are obtained for any interior point thus the optimal roughness penalty function is obtained by approximately minimizing the asymptotic integrated mean squared error simulation results and an application illustrate the performance of the proposed estimator copyright 2013 oxford university press

efficient estimation in sufficient dimension reduction we develop an efficient estimation procedure for identifying and estimating the central subspace using a new way of parameterization we convert the problem of identifying the central subspace to the problem of estimating a finite dimensional parameter in a semiparametric model this conversion allows us to derive an efficient estimator which reaches the optimal semiparametric efficiency bound the resulting efficient estimator can exhaustively estimate the central subspace without imposing any distributional assumptions our proposed efficient estimation also provides a possibility for making inference of parameters that uniquely identify the central subspace we conduct simulation studies and a real data analysis to demonstrate the finite sample performance in comparison with several existing methods

guaranteed robustness bounds for matched disturbance nonlinear systems this paper considers input affine nonlinear systems with matched disturbances and shows how to compute an a priori upper bound of the h attenuation level achieved by the optimal l 2 controller and the suboptimal h central controller the case where the disturbance contains a constant term is also discussed these bounds are shown to depend only on the function mapping the control input to the performance variable this result is used to derive a robust control design for a special but practically important class of non input affine nonlinear systems consisting of the series connection of a nonlinear state and input dependent map and of a nonlinear input affine dynamical system approximate inversion of the nonlinear static map leads to a robust control problem which fits into the framework the effectiveness of the theoretical results is shown by its use for the robust control design of a diesel engine test bench

estimating conditional proportion curves by regression residuals researchers often derive a categorical outcome from an observed continuous measurement y for example human obesity status can be defined by the body mass index they proceed to estimate the conditional proportion curve p x p y c x x versus the covariate x by regular categorical data analysis in doing so they fail to fully use the rich information in the original continuous measurement researchers also face additional difficulties caused by the uncertainty of the cutoff c x which defines the categories for example some cutoff may cause artificial discontinuity or non smoothness in the conditional proportion curve p x we propose a fully nonparametric empirical estimator using the standardized regression residuals from the general location scale model y m x v x e for the original continuous measurement we prove the consistency of the proposed estimator in regular settings the proposed estimator has several attractive features it readily provides estimates for alternative cutoff values and easily accommodates the discontinuity in p x caused by special cutoff numerical examples demonstrate that our method has better performance in mean squared error than some alternative approaches we also discuss a design adjusted version of the proposed estimators to accommodate survey designs the proposed methods are applied to data on obesity measurements among female hispanic americans this new approach can be applied to other epidemiological and socio economic research that needs to define population categories from underlying continuous measurements copyright 2010 john wiley sons ltd

variance function estimation in regression the effect of estimating the mean abstract the authors consider estimation of a variance function g in regression problems such estimation requires simultaneous estimation of the mean function f we obtain sharp results on the extent to which the smoothness of f influences best rates of convergence for estimating g for example in nonparametric regression with two derivatives on g classical rates of convergence are possible if and only if the unknown f satisfies a lipschitz condition of order 1 3 or more if a parametric model is known for g then g may be estimated n 1 2 consistently if and only if f is lipschitz of order 1 2 or more optimal rates of convergence are attained by kernel estimators

the casl algorithm for quasi geostrophic flow in a cylinder the contour advective semi lagrangian casl algorithm is a hybrid lagrangian eulerian numerical algorithm combining contour dynamics for the advection of a materially conserved dynamical tracer with spectral techniques for the computation of the associated flow field it benefits from the accuracy of contour dynamics and the efficiency of spectral methods and in applications to quasi geostrophic rotating stratified flows has no time step restriction for numerical stability it abandons the expensive aspect of contour dynamics the computation of contour integrals to obtain the flow field and the inaccurate and costly aspect of spectral methods the advection of materially conserved tracers requiring a time step restriction for numerical stability and numerical diffusion the two approaches are tied together by a fast method of converting tracer contours to gridded field values lagrangian eulerian and the bilinear interpolation of the flow field from the grid to points on the contours eulerian lagrangian here we extend this method originally developed for horizontally periodic flows to flows within a cylindrical domain among other applications this permits a comparison with experiments which are often conducted in a cylindrical domain we demonstrate the accuracy of the algorithm in a few test cases then examine the differences between cylindrical and periodic domains for both simple vortex interactions and complex quasi geostrophic turbulence

variance estimation in nonparametric regression via the difference sequence method consider a gaussian nonparametric regression problem having both an unknown mean function and unknown variance function this article presents a class of difference based kernel estimators for the variance function optimal convergence rates that are uniform over broad functional classes and bandwidths are fully characterized and asymptotic normality is also established we also show that for suitable asymptotic formulations our estimators achieve the minimax rate

output to state stability and detectability of nonlinear systems the notion of input to state stability iss has proved to be useful in nonlinear systems analysis this paper discusses a dual notion output to state stability oss a characterization is provided in terms of a dissipation inequality involving storage lyapunov functions combining iss and oss there results the notion of input output to state stability ioss which is also studied and related to the notion of detectability the existence of observers and output injection

the asymptotic variance of semiparametric estimators this paper derives a general formula for the asymptotic variance of semiparametric estimators that accounts for the presence of nonparametric estimators of functions the general formula is specialized to show invariance of the asymptotic variance to the type of nonparametric estimator and to obtain correction terms for estimation of densities and mean square projections including conditional expectations regularity conditions for the validity of the formula are also given including primitive conditions for asymptotic normality when series estimators are present new examples considered include a semiparametric panel probit estimator and a series estimator of the average derivative copyright 1994 by the econometric society

asymptotically efficient estimation of a nonlinear model of the heteroscedasticity and the calibration of measurement systems the measurement system calibration includes the estimation of the sensor error models in order to get an optimal estimation of the measured parameters the paper is devoted to the estimation of a nonlinear parametric model of the heteroscedasticity and its application to the calibration of measurement systems the heteroscedasticity occurs in regression when the measurement noise variance is nonconstant the maximum likelihood ml estimation of variance function parameters leads to a system of nonlinear equations the iterative solution of these nonlinear equations is based entirely on a successful choice of initial conditions which is intractable in practice to overcome this difficulty another linear quasi ml estimator is proposed it is strongly consistent asymptotically gaussian and only slightly less efficient than the cramer rao lower bound by using this estimator as an initial condition an asymptotically efficient estimation is obtained by using one step noniterative newton method the theoretical findings have been applied to the calibration of the egnos gps positioning algorithm in the sub urban environments

variance function estimation abstract heteroscedastic regression models are used in fields including economics engineering and the biological and physical sciences often the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters standard asymptotic theory implies that how one estimates the variance function in particular the structural parameters has no effect on the first order properties of the regression parameter estimates there is evidence however both in practice and higher order theory to suggest that how one estimates the variance function does matter further in some settings estimation of the variance function is of independent interest or plays an important role in estimation of other quantities in this article we study variance function estimation in a unified way focusing on common methods proposed in the statistical and other literature to make both general observations and compare different estimation schemes we show that there are significant di

variances are not always nuisance parameters summary in classical problems e g comparing two populations fitting a regression surface etc variability is a nuisance parameter the term nuisance parameter is meant here in both the technical and the practical sense however there are many instances where understanding the structure of variability is just as central as understanding the mean structure the purpose of this article is to review a few of these problems i focus in particular on two issues a the determination of the validity of an assay and b the issue of the power for detecting health effects from nutrient intakes when the latter are measured by food frequency questionnaires i will also briefly mention the problems of variance structure in generalized linear mixed models robust parameter design in quality technology and the signal in microarrays in these and other problems treating variance structure as a nuisance instead of a central part of the modeling effort not only leads to inefficient estimation of means but also to misleading conclusions

on contour crossings in contour advective simulations part 2 analysis of crossing errors and methods for their prevention this is the second of two papers devoted to the analysis of contour crossing errors that occur in contour advective simulations of fluid motion where either vorticity or potential vorticity is represented by contours we begin with a detailed discussion on some of the potential mechanisms for contour crossing past work has suggested that the formation of contour crossings is due to inadequate spatial resolution of contours 1 the implementation of two schemes for preventing contour crossings within the framework of the contour advective semi lagrangian casl algorithm is detailed here we then present an analysis of contour crossing errors in simulations of quasigeostrophic turbulence on the f plane and the quasigeostrophic motion of an initially circular vortex patch on the b plane using the algorithm detailed in part 1 we find that in general individual crossings occur at scales smaller than the inversion grid scale on which velocity is calculated but at scales larger than that of the surgical scale that defines the smallest resolved features vorticity of a flow if the resolution of a quasigeostrophic turbulence simulation on the f plane is increased by doubling the number of grid points in each coordinate direction used in the calculation of the velocity field then the total area in error due to contour crossings remains unchanged a smaller number of crossings introducing larger scale area errors is replaced by a greater number of smaller local errors uniformly increasing the density of nodes along all contours and placement of nodes at points of close approach on contours are both effective methods for limiting contour crossings

nonparametric estimation of optimal performance criteria in quality engineering abstract box 1987 and leon et al 1987 discuss the problem of closeness to target in quality engineering if the mean response f x z depends on x z the variance function is a permia if it is g z i e depends only on z the goal is to find x sub o z sub o which minimizes variance while achieving a target mean value the authors pose and answer the question for given smoothness assumptions about f and g how accurately can we estimate x sub o and z sub o as part of the investigation they also find optimal rates of convergence for estimating f g and their derivatives keywords nonparametric regression performance measure quality control variance function estimation quality engineering

variance function partially linear single index models we consider heteroscedastic regression models where the mean function is a partially linear single index model and the variance function depends upon a generalized partially linear single index model we do not insist that the variance function depend only upon the mean function as happens in the classical generalized partially linear single index model we develop efficient and practical estimation methods for the variance function and for the mean function asymptotic theory for the parametric and nonparametric parts of the model is developed simulations illustrate the results an empirical example involving ozone levels is used to further illustrate the results and is shown to be a case where the variance function does not depend upon the mean function

efficient semiparametric estimator for heteroscedastic partially linear models summary we study the heteroscedastic partially linear model with an unspecified partial baseline component and a nonparametric variance function an interesting finding is that the performance of a naive weighted version of the existing estimator could deteriorate when the smooth baseline component is badly estimated to avoid this we propose a family of consistent estimators and investigate their asymptotic properties we show that the optimal semiparametric efficiency bound can be reached by a semiparametric kernel estimator in this family building upon our theoretical findings and heuristic arguments about the equivalence between kernel and spline smoothing we conjecture that a weighted partial spline estimator could also be semiparametric efficient properties of the proposed estimators are presented through theoretical illustration and numerical simulations

weak and strong uniform consistency of kernel regression estimates we study the estimation of a regression function by the kernel method under mild conditions on the window the bandwidth and the underlying distribution of the bivariate observations x i y i we obtain the weak and strong uniform convergence rates on a bounded interval these results parallel those of silverman 1978 on density estimation and extend those of schuster and yakowitz 1979 and collomb 1979 on regression estimation

a comparison of the contour surgery and pseudo spectral methods abstract two very different numerical methods which have been used to simulate high reynolds number two dimensional flows are compared for the first time one method the pseudo spectral method is fundamentally based on the eulerian description of fluid flow and the other the contour surgery method is inherently lagrangian the former makes use of a continuous distribution of vorticity while the latter a theoretically inviscid method makes use of a discrete distribution a comparison is nevertheless attempted in a model problem wherein the initial vorticity distribution is continuous we examine the stripping of an initially circular vortex by applied adverse shear in doubly periodic geometry the constraining geometry causes the flow to become very complex placing great demands on both computational methods the surprise is that as few as eight discrete levels of vorticity in contour surgery five results which are quantitatively close to those obtained by the pseudo spectral method at high resolution advantages and shortcomings of both methods are noted

semiparametric least squares sls and weighted sls estimation of single index models abstract for the class of single index models i construct a semiparametric estimator of coefficients up to a multiplicative constant that exhibits 1 n consistency and asymptotic normality this class of models includes censored and truncated tobit models binary choice models and duration models with unobserved individual heterogeneity and random censoring i also investigate a weighting scheme that achieves the semiparametric efficiency bound

nonparametric frontier estimation via local linear regression abstract in this paper we propose a nonparametric regression frontier model that assumes no specific parametric family of densities for the unobserved stochastic component that represents efficiency in the model nonparametric estimation of the regression frontier is obtained using a local linear estimator that is shown to be consistent and nh n asymptotically normal under standard assumptions the estimator we propose envelops the data but is not inherently biased as free disposal hull fdh or data envelopment analysis dea estimators it is also more robust to extreme values than the aforementioned estimators a monte carlo study is performed to provide preliminary evidence on the estimator s finite sample properties and to compare its performance to a bias corrected fdh estimator

global l 2 gain design for a class of nonlinear systems abstract it is known that the so called h control problem of a nonlinear system is locally solvable if the corresponding problem for the linearized system can be solved by linear feedback in this paper we prove that this condition suffices to solve also a global h control problem for a fairly large class of nonlinear systems if one is free to choose a state dependent weight of the control input using a two way backward and forward recursive induction argument we simultaneously construct starting from a solution of the riccati algebraic equation a global solution of the hamilton jacobi isaacs partial differential equation arising in the nonlinear h control as well as a state feedback control law that achieves global disturbance attenuation with internal stability for the nonlinear systems

semiparametric efficiency in gmm models with auxiliary data we study semiparametric efficiency bounds and efficient estimation of parameters defined through general moment restrictions with missing data identification relies on auxiliary data containing information about the distribution of the missing variables conditional on proxy variables that are observed in both the primary and the auxiliary database when such distribution is common to the two data sets the auxiliary sample can be independent of the primary sample or can be a subset of it for both cases we derive bounds when the probability of missing data given the proxy variables is unknown or known or belongs to a correctly specified parametric family we find that the conditional probability is not ancillary when the two samples are independent for all cases we discuss efficient semiparametric estimators an estimator based on a conditional expectation projection is shown to require milder regularity conditions than one based on inverse probability weighting

efficient estimation and computation of parameters and nonparametric functions in generalized semi non parametric regression models abstract the efficiency of estimation for the parameters in semiparametric models has been widely studied in the literature bickel et al 1993 in this paper we study efficient estimators for both parameters and nonparametric functions in a class of generalized semi non parametric regression models which cover commonly used semiparametric models such as partially linear models partially linear single index models and two sample semiparametric models we propose a maximum likelihood principle combined with the local linear technique for estimating the parameters and nonparametric functions the proposed estimators of the parameters and a linear functional of the nonparametric functions are consistent and asymptotically normal and are further shown to be semiparametrically efficient an efficient computational algorithm to achieve the maximization is proposed extensive simulation experiments show the superiority of the proposed methods three real data examples are analyzed and presented as an illustration

a multiple index model and dimension reduction dimension reduction can be used as an initial step in statistical modeling further specification of model structure is imminent and important when the reduced dimension is still greater than 1 in this article we investigate one method of specification that involves separating the linear component from the nonlinear components leading to further dimension reduction in the unknown link function and thus better estimation and easier interpretation of the model the specified model includes the popular econometric multiple index model and the partially linear single index model as its special cases a criterion is developed to validate the model specification an algorithm is proposed to estimate the model directly asymptotic distributions for the estimators of the parameters and the nonparametric link function are derived air pollution data in chicago are used to illustrate the modeling procedure and to demonstrate its advantages over the existing dimension reduction approaches

the embedded finite difference method for the poisson equation in a domain with an irregular boundary and dirichlet boundary conditions the poisson equation subject to dirichlet boundary conditions on an irregular domain can be treated by embedding the region in a rectangular domain and solving using finite differences over the rectangle the crucial issue is the discretization of the boundaries of the irregular domain in the past both linear and quadratic boundary treatments have been used and error bounds have been derived in both cases showing that the linear case gives uniform second order accuracy whereas the quadratic case gives third order accuracy at the boundaries and second order accuracy internally thus it has been recommended that the linear boundary treatment be used as it is simpler gives rise to a symmetric matrix formulation and has uniform accuracy the present work shows that this argument is inadequate because the coefficients of the error terms also play an important role we demonstrate this in the 1 d case by determining explicit expressions for the error for both the linear and quadratic boundary treatments it is shown that for the linear case the coefficient of error is in general large enough to dominate the calculation and that therefore it is necessary to use a quadratic boundary treatment in order to obtain errors comparable with those obtained for a regular domain we go on to show that the 1 d expressions for error can be used to approximate the boundary error for 2 d problems and that for the linear treatment the boundary error again dominates

robust estimation in heteroscedastic linear models we consider a heteroscedastic linear model in which the variances are given by a parametric function of the mean responses and a parameter theta we propose robust estimates for the regression parameter beta and show that as long as a reasonable starting estimate of theta is available our estimates of beta are asymptotically equivalent to the natural estimate obtained with known variances a particular method for estimating theta is proposed and shown by monte carlo to work quite well especially in power and exponential models for the variances we also briefly discuss a feedback estimate of beta

inference concerning the number of factors in a multivariate nonparametric relationship this paper considers the problem of determining the number of nonparametric factors in a multivariate nonparametric relationship the definition given is broad enough to encompass a number of potential applications in econometrics including inferring the rank of demand testing whether for a given set of instruments it is possible to identify a linear model and testing arbitrage pricing theory the paper gives methods for testing hypotheses concerning the number of factors using series and kernel based nonparametric methods and also considers the problem of estimating the number of factors the methods are compared in a small simulation study and an application to determining the rank of demand systems is considered

likelihood based local linear estimation of the conditional variance function we consider estimation of mean and variance functions with kernel weighted local polynomial fitting in a heteroscedastic nonparametric regression model our preferred estimators are based on a localized normal likelihood using a standard local linear form for estimating the mean and a local log linear form for estimating the variance it is important to allow two bandwidths in this problem separate ones for mean and variance estimation we provide data based methods for choosing the bandwidths we also consider asymptotic results and study and use them the methodology is compared with a popular competitor and is seen to perform better for small and moderate sample sizes in simulations a brief example is provided

a pseudospectral approach for polar and spherical geometries the singularities that arise in polar and spherical coordinates have in the past caused significant difficulties in obtaining accurate solutions to convection diffusion type problems in many fields astrophysics geophysics meteorology etc viewing pseudospectral methods as a limiting case of finite difference methods rather than as based on expansions in terms of orthogonal functions leads naturally to very simple yet highly effective fast fourier transform fft based pseudospectral methods in such geometries

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

correlation curves measures of association as functions of covariate values for experiments where the strength of association between a response variable y and a covariate x is different over different regions of values for the covariate x we propose local nonparametric dependence functions which measure the strength of association between y and x as a function of x x our dependence functions are extensions of galton s idea of strength of co relation from the bivariate normal case to the nonparametric case in particular a dependence function is obtained by expressing the usual galton pearson correlation coefficient in terms of the regression line slope 3 and the residual variance o 2 and then replacing 3 and 72 by a nonparametric regression slope 13 x and a nonparametric residual variance o 2 x var yix respectively our local dependence functions are standardized nonparametric regression curves which provide universal scale free measures of the strength of the relationship between variables in nonlinear models they share most of the properties of the correlation coefficient and they reduce to the usual correlation coefficient in the bivariate normal case for this reason we call them correlation curves we show that in a certain sense they quantify lehmann s notion of regression dependence finally the correlation curve concept is illustrated using data from a study of the relationship between cholesterol levels x and triglyceride concentrations y of heart patients