local empirical likelihood inference for varying coefficient density ratio models based on case control data in this article we develop a varying coefficient density ratio model for case control studies the case and control samples come from two different distributions under the model assumption the ratio of the two densities is related to the linear combination of covariates with varying coefficients through a known function a special case is the exponential tilt model where the log ratio of the two densities is a linear function of covariates we propose a local empirical likelihood el approach to estimate the nonparametric coefficient functions under some regularity assumptions the proposed estimators are shown to be consistent and asymptotically normally distributed the sieve empirical likelihood ratio selr test statistic for detecting whether the varying coefficients are really constant and other related hypotheses is constructed and it follows approximately a chi squared distribution we introduce a modified bootstrap procedure to estimate the null distribution of the selr when sample size is s

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

on model diagnostics using varying coefficient models summary the paper motivates the use of varying coefficient models for diagnostics in regression models with continuous and factorial covariates the varying coefficient model which is restricted only by assumptions concerning the smoothness of the effects is considered as the alternative to a parametric generalised linear model estimation is based on local likelihood smoothing and asymptotic properties of the estimators are derived for the investigation of the discrepancy between the parametric model and the smooth alternative a simple graphical procedure is proposed moreover tests are derived which allow for overall and componentwise investigation the componentwise tests may be used to examine interaction effects of continuous and factorial regressors

nonparametric variance estimation in the analysis of microarray data a measurement error approach this article investigates the effects of measurement error on the estimation of nonparametric variance functions we show that either ignoring measurement error or direct application of the simulation extrapolation simex method leads to inconsistent estimators nevertheless the direct simex method can reduce bias relative to a naive estimator we further propose a permutation simex method which leads to consistent estimators in theory the performance of both simex methods depends on approximations to the exact extrapolants simulations show that both simex methods perform better than ignoring measurement error the methodology is illustrated using microarray data from colon cancer patients

partially linear hazard regression with varying coefficients for multivariate survival data summary the paper studies estimation of partially linear hazard regression models with varying coefficients for multivariate survival data a profile pseudo partial likelihood estimation method is proposed the estimation of the parameters of the linear part is accomplished via maximization of the profile pseudo partial likelihood whereas the varying coefficient functions are considered as nuisance parameters that are profiled out of the likelihood it is shown that the estimators of the parameters are root n consistent and the estimators of the non parametric coefficient func tions achieve optimal convergence rates asymptotic normality is obtained for the estimators of the finite parameters and varying coefficient functions consistent estimators of the asymptotic variances are derived and empirically tested which facilitate inference for the model we prove that the varying coefficient functions can be estimated as well as if the parametric components were known and the failure times within each subject were independent simulations are con ducted to demonstrate the performance of the estimators proposed a real data set is analysed to illustrate the methodology proposed

efficient estimation and inferences for varying coefficient models abstract this article deals with statistical inferences based on the varying coefficient models proposed by hastie and tibshirani local polynomial regression techniques are used to estimate coefficient functions and the asymptotic normality of the resulting estimators is established the standard error formulas for estimated coefficients are derived and are empirically tested a goodness of fit test technique based on a nonparametric maximum likelihood ratio type of test is also proposed to detect whether certain coefficient functions in a varying coefficient model are constant or whether any covariates are statistically significant in the model the null distribution of the test is estimated by a conditional bootstrap method our estimation techniques involve solving hundreds of local likelihood equations to reduce the computational burden a one step newton raphson estimator is proposed and implemented the resulting one step procedure is shown to save computational cost on an order of tens with no

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

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

fitting heteroscedastic regression models abstract in heteroscedastic regression models assumptions about the error distribution determine the method of consistent estimation of parameters for example consider the case where the model specifies the regression and dispersion functions for the data but robustness is of concern and one wishes to use least absolute error regressions except in certain special circumstances parameter estimates obtained in this way are inconsistent in this article we expand the heteroscedastic model so that all of the common methods yield consistent estimates of the major model parameters asymptotic theory shows the extent to which standard results on the effect of estimating regression and dispersion parameters carry over into this setting careful attention is given to the question of when one can adapt for heteroscedasticity when estimating the regression parameters we find that in many cases such adaption is not possible this complicates inference about the regression parameters but does not lead to intracta

compact labeling scheme for ancestor queries we consider the following problem given a rooted tree t label the nodes of t in the most compact way such that given the labels of two nodes u and v one can determine in constant time by looking only at the labels whether u is ancestor of v the best known labeling scheme is rather straightforward and uses labels of length at most 2 log 2 n bits each where n is the number of nodes in the tree our main result in this paper is a labeling scheme with maximum label length log 2 n oh sqrt log n our motivation for studying this problem is enhancing the performance of web search engines in the context of this application each indexed document is a tree and the labels of all trees are maintained in main memory therefore even small improvements in the maximum label length are important

on estimation of partially linear transformation models we study a general class of partially linear transformation models which extend linear transformation models by incorporating nonlinear covariate effects in survival data analysis a new martingale based estimating equation approach consisting of both global and kernel weighted local estimation equations is developed for estimating the parametric and nonparametric covariate effects in a unified manner we show that with a proper choice of the kernel bandwidth parameter one can obtain the consistent and asymptotically normal parameter estimates for the linear effects asymptotic properties of the estimated nonlinear effects are established as well we further suggest a simple resampling method to estimate the asymptotic variance of the linear estimates and show its effectiveness to facilitate the implementation of the new procedure an iterative algorithm is developed numerical examples are given to illustrate the finite sample performance of the procedure supplementary materials are available online

data driven bandwidth selection in local polynomial fitting variable bandwidth and spatial adaptation when estimating a mean regression function and its derivatives locally weighted least squares regression has proven to be a very attractive technique the present paper focuses on the important issue of how to select the smoothing parameter or bandwidth in the case of estimating curves with a complicated structure a variable bandwidth is desirable furthermore the bandwidth should be indicated by the data themselves recent developments in nonparametric smoothing techniques inspired us to propose such a data driven bandwidth selection procedure which can be used to select both constant and variable bandwidths the idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator the procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives the resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves this is illustrated via a large variety of testing examples including examples with a large spatial variability the results are also compared with wavelet thresholding techniques and it seems that our results are at least comparable i e local polynomial regression using our data driven variable bandwidth has spatial adaptation properties that are similar to wavelets

strong and weak lagrange galerkin spectral element methods for the shallow water equations the lagrange galerkin spectral element method for the two dimensional shallow water equations is presented the equations are written in conservation form and the domains are discretized using quadrilateral elements lagrangian methods integrate the governing equations along the characteristic curves thus being well suited for resolving the nonlinearities introduced by the advection operator of the fluid dynamics equations two types of lagrange galerkin methods are presented the strong and weak formulations the strong form relies mainly on interpolation to achieve high accuracy while the weak form relies pri marily on integration lagrange galerkin schemes offer an increased efficiency by virtue of their less stringent cfl condition the use of quadrilateral elements permits the construction of spectral type finite element methods that exhibit exponential convergence as in the conventional spectral method yet they are constructed locally as in the finite element method this is the spectral element method in this paper we show how to fuse the lagrange galerkin methods with the spectral element method and present results for two standard test cases in order to compare and contrast these two hybrid schemes 2003 published by elsevier science ltd

adaptive varying coefficient linear models varying coefficient linear models arise from multivariate nonparametric regression nonlinear time series modelling and forecasting functional data analysis longitudinal data analysis and others it has been a common practice to assume that the vary coefficients are functions of a given variable which is often called an index a frequently asked question is which variable should be used as the index in this paper we explore the class of the varying coefficient linear models in which the index is unknown and is estimated as a linear combination of regression and or other variables this will enlarge the modelling capacity substantially we search for the index such that the derived varying coefficient model provides the best approximation to the underlying unknown multi dimensional regression function in the least square sense the search is implemented through the newly proposed hybrid backfitting algorithm the core of the algorithm is the alternative iteration between estimating the index through a one step scheme and estimating coefficient functions through a one dimensional local linear smoothing the generalised cross validation method for choosing bandwidth is efficiently incorporated into the algorithm the locally significant variables are selected in terms of the combined use of t statistic and akaike information criterion we further extend the algorithm for the models with two indices simulation shows that the proposed methodology has appreciable flexibility to model complex multivariate nonlinear structure and is practically feasible with average modern computers the methods are further illustrated through the canadian mink muskrat data in 1925 1994 and the pound dollar exchange rates in 1974 1983

correction of logistic regression relative risk estimates and confidence intervals for systematic within person measurement error errors in the measurement of exposure that are independent of disease status tend to bias relative risk estimates and other measures of effect in epidemiologic studies toward the null value two methods are provided to correct relative risk estimates obtained from logistic regression models for measurement errors in continuous exposures within cohort studies that may be due to either random unbiased within person variation or to systematic errors for individual subjects these methods require a separate validation study to estimate the regression coefficient lambda relating the surrogate measure to true exposure in the linear approximation method the true logistic regression coefficient beta is estimated by beta lambda where beta is the observed logistic regression coefficient based on the surrogate measure in the likelihood approximation method a second order taylor series expansion is used to approximate the logistic function enabling closed form likelihood estimation of beta confidence intervals for the corrected relative risks are provided that include a component representing error in the estimation of lambda based on simulation studies both methods perform well for true odds ratios up to 3 0 for higher odds ratios the likelihood approximation method was superior with respect to both bias and coverage probability an example is provided based on data from a prospective study of dietary fat intake and risk of breast cancer and a validation study of the questionnaire used to assess dietary fat intake

the kernel estimate of a regression function in likelihood based models abstract smoothing splines have a penalized likelihood motivation good and gaskins 1971 allowing direct application to nonparametric regression in likelihood based models the notion of a weighted likelihood for the nonparametric kernel estimation of a regression function is proposed generalizing the local likelihood theory of tibshirani and hastie 1987 let the data be of the form xi yi i 1 n where xi 0 1 d are lattice points and the yi are independent random variables from a family of distributions with parameter i g xi with g having continuous partial derivatives of order k 2 the goal is to arrive at a nonparametric estimate o of o g xo for a fixed point xo 0 1 d we consider the estimator o that maximizes the weighted likelihood function w n i 1 w xo xi b log f yi with f the density of yi w a symmetric kernel with compact support and b the bandwidth that controls the degree of smoothing sufficient conditions for consistency and asympt

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

empirical bias bandwidths for local polynomial nonparametric regression and density estimation abstract a data based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials the estimator called the empirical bias bandwidth selector ebbs is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function ebbs minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term the variance term used is exact not asymptotic though it involves the conditional variance of the response given the predictors that must be estimated the bias term is estimated empirically not from an asymptotic expression thus ebbs is similar to the double smoothing approach of hardle hall and marron and a local bandwidth selector of schucany but is developed here for a far wider class of estimation problems than what those authors considered ebbs is tested on simulated data and its performance seems quite satisfactory local polynomial smoothing of a histogram is a high

varying coefficients model with measurement error we propose a semiparametric partially varying coefficient model to study the relationship between serum creatinine concentration and the glomerular filtration rate gfr among kidney donors and patients with chronic kidney disease a regression model is used to relate serum creatinine to gfr and demographic factors in which coefficient of gfr is expressed as a function of age to allow its effect to be age dependent gfr measurements obtained from the clearance of a radioactively labeled isotope are assumed to be a surrogate for the true gfr with the relationship between measured and true gfr expressed using an additive error model we use locally corrected score equations to estimate parameters and coefficient functions and propose an expected generalized cross validation egcv method to select the kernel bandwidth the performance of the proposed methods which avoid distributional assumptions on the true gfr and residuals is investigated by simulation accounting for measurement error using the proposed model reduced apparent inconsistencies in the relationship between serum creatinine and gfr among different clinical data sets derived from kidney donor and chronic kidney disease source populations

locally efficient estimators for semiparametric models with measurement error we derive constructive locally efficient estimators in semiparametric measurement error models the setting is one in which the likelihood function depends on variables measured with and without error where the variables measured without error can be modeled nonparametrically the algorithm is based on backfitting we show that if one adopts a parametric model for the latent variable measured with error and if this model is correct then the estimator is semiparametric efficient if the latent variable model is misspecified then our methods lead to a consistent and asymptotically normal estimator our method further produces an estimator of the nonparametric function that achieves the standard bias and variance property we extend the methodology to allow estimation of parameters in the measurement error model by additional data in the form of replicates or instrumental variables the methods are illustrated through a simulation study and a data example where the putative latent variable distribution i

semiparametric analysis of heterogeneous data using varying scale generalized linear models this article describes a class of heteroscedastic generalized linear regression models in which a subset of the regression parameters are rescaled nonparametrically and develops efficient semiparametric inferences for the parametric components of the models such models provide a means to adapt for heterogeneity in the data due to varying exposures varying levels of aggregation and so on the class of models considered includes generalized partially linear models and nonparametrically scaled link function models as special cases we present an algorithm to estimate the scale function nonparametrically and obtain asymptotic distribution theory for regression parameter estimates in particular we establish that the asymptotic covariance of the semiparametric estimator for the parametric part of the model achieves the semiparametric lower bound we also describe bootstrap based goodness of scale test we illustrate the methodology with simulations published data and data from collaborative research on ultrasound safety

quasi likelihood estimation in semiparametric models abstract suppose the expected value of a response variable y may be written h x t where x and t are covariates each of which may be vector valued is an unknown parameter vector is an unknown smooth function and h is a known function in this article we outline a method for estimating the parameter of this type of semiparametric model using a quasi likelihood function algorithms for computing the estimates are given and the asymptotic distribution theory for the estimators is developed the generalization of this approach to the case in which y is a multivariate response is also considered the methodology is illustrated on two data sets and the results of a small monte carlo study are presented

profile likelihood and conditionally parametric models in this paper we outline a general approach to estimating the parametric component of a semiparametric model for the case of a scalar parametric component the method is based on the idea of first estimating a one dimensional subproblem of the original problem that is least favorable in the sense of stein the likelihood function for the scalar parameter along this estimated subproblem may be viewed as a generalization of the profile likelihood for the problem the scalar parameter is then estimated by maximizing this generalized profile likelihood this method of estimation is applied to a particular class of semiparametric models where it is shown that the resulting estimator is asymptotically efficient

on partial local smoothing rules for curve estimation summary we compare the performances of local and global rules for smoothing parameter choice in terms of asymptotic mean squared errors of the resulting estimators in some instances there is surprisingly little to choose between local and global approaches our analysis identifies contexts where the differences are small or large this work motivates development of smoothing rules that form a half way house between local and global smoothing there interpolation provides a basis for partial local smoothing a key result shows that interpolation on even a coarse grid can produce a very good approximation to full local smoothing our theoretical and numerical results lead us to suggest linear interpolation of a bandwidth obtained by integral approximations on discrete intervals

adapting for the missing link we consider the fitting of generalized linear models in wich the link function is assumed to be unknown and propose the following computational method first estimate regression coefficients using the canonical link then estimate the link via a kernel smoother treating the direction in the predictor space determined by the regression coefficients as known then reestimate the direction using the estimated link and alternate between these two steps we show that under fairly general conditions n 1 2 consistent estimates of the direction are obtained a small monte carlo study is presented

semiparametric estimation of animal abundance using capture recapture data from open populations a semiparametric partially linear model for the size of an open population is proposed and inference is conducted using weighted martingale estimating equations this extends a previous nonparametric approach to modeling capture recapture data for open populations with frequent capture occasions analytic expressions for the large sample variances are derived and these are confirmed in a simulation study the method is illustrated on monthly penguin banding data collected over 6 years

a semiparametric transformation approach to estimating usual daily intake distributions abstract the distribution of usual intakes of dietary components is important to individuals formulating food policy and to persons designing nutrition education programs the usual intake of a dietary component for a person is the long run average of daily intakes of that component for that person because it is impossible to directly observe usual intake for an individual it is necessary to develop an estimator of the distribution of usual intakes based on a sample of individuals with a small number of daily observations on a subsample of the individuals daily intake data for individuals are nonnegative and often very skewed also there is large day to day variation relative to the individual to individual variation and the within individual variance is correlated with the individual means we suggest a methodology for estimating usual intake distributions that allows for varying degrees of departure from normality and recognizes the measurement error associated with one day dietary intakes the est

local linear regression for generalized linear models with missing data fan heckman and wand proposed locally weighted kernel polynomial regression methods for generalized linear models and quasilikelihood functions when the covariate variables are missing at random we propose a weighted estimator based on the inverse selection probability weights distribution theory is derived when the selection probabilities are estimated nonparametrically we show that the asymptotic variance of the resulting nonparametric estimator of the mean function in the main regression model is the same as that when the selection probabilities are known while the biases are generally different this is different from results in parametric problems where it is known that estimating weights actually decreases asymptotic variance to reconcile the difference between the parametric and nonparametric problems we obtain a second order variance result for the nonparametric case we generalize this result to local estimating equations finite sample performance is examined via simulation studies the proposed method is demonstrated via an analysis of data from a case control study

local estimating equations abstract estimating equations have found wide popularity recently in parametric problems yielding consistent estimators with asymptotically valid inferences obtained via the sandwich formula motivated by a problem in nutritional epidemiology we use estimating equations to derive nonparametric estimators of a parameter depending on a predictor the nonparametric component is estimated via local polynomials with loess or kernel weighting asymptotic theory is derived for the latter in keeping with the estimating equation paradigm variances of the nonparametric function estimate are estimated using the sandwich method in an automatic fashion without the need typical in the literature to derive asymptotic formulas and plug in an estimate of a density function the same philosophy is used in estimating the bias of the nonparametric function that is an empirical method is used without deriving asymptotic theory on a case by case basis the methods are applied to a series of examples the applicatio

local estimation of age dependent variance components from longitudinal twin data summary in the study of longitudinal twin and family data interest is often in the covariance structure of the data and the decomposition of this covariance structure into genetic and environmental components rather than in estimating the mean function various parametric models for covariance structures have been proposed but e g in studies of children where growth spurts occur at various ages it is difficult to a prioridetermine an appropriate parametric model for the covariance structure in particular there is a general lack of the visualization procedures such as lowess that are invaluable in the initial stages of constructing a parametric model for a mean function here we use kernel smoothing to modify a cross sectional approach based on the sample covariance matrices to obtain smoothed estimates of the genetic and environmental variances and correlations for longitudinal twin data the methods are proposed t o be exploratory as an aid to parametric modeling rather than inferential although approximate asymptotic standard errors are derived in the appendix

large sample confidence regions based on subsamples under minimal assumptions in this article the construction of confidence regions by approximating the sampling distribution of some statistic is studied the true sampling distribution is estimated by an appropriate normalization of the values of the statistic computed over subsamples of the data in the i i d context the method has been studied by wu in regular situations where the statistic is asymptotically normal the goal of the present work is to prove the method yields asymptotically valid confidence regions under minimal conditions essentially all that is required is that the statistic suitably normalized possesses a limit distribution under the true model unlike the bootstrap the convergence to the limit distribution need not be uniform in any sense the method is readily adapted to parameters of stationary time series or more generally homogeneous random fields for example an immediate application is the construction of a confidence interval for the spectral density function of a homogeneous random field

variable selection in semiparametric regression modeling in this paper we are concerned with how to select significant variables in semiparametric modeling variable selection for semiparametric regression models consists of two components model selection for nonparametric components and select significant variables for parametric portion thus it is much more challenging than that for parametric models such as linear models and generalized linear models because traditional variable selection procedures including stepwise regression and the best subset selection require model selection to nonparametric components for each submodel this leads to very heavy computational burden in this paper we propose a class of variable selection procedures for semiparametric regression models using nonconcave penalized likelihood the newly proposed procedures are distinguished from the traditional ones in that they delete insignificant variables and estimate the coefficients of significant variables simultaneously this allows us to establish the sampling properties of the resulting estimate we first establish the rate of convergence of the resulting estimate with proper choices of penalty functions and regularization parameters we then establish the asymptotic normality of the resulting estimate and further demonstrate that the proposed procedures perform as well as an oracle procedure semiparametric generalized likelihood ratio test is proposed to select significant variables in the nonparametric component we investigate the asymptotic behavior of the proposed test and demonstrate its limiting null distribution follows a chi squared distribution which is independent of the nuisance parameters extensive monte carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedures