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

dimension reduction in a semiparametric regression model with errors in covariates we consider a semiparametric estimation method for general regression models when some of the predictors are measured with error the technique relies on a kernel regression of the true covariate on all the observed covariates and surrogates this requires a nonparametric regression in as many dimensions as there are covariates and surrogates the usual theory copes with such higher dimensional problems by using higher order kernels but this is unrealistic for most problems we show that the usual theory is essentially as good as one can do with this technique instead of regression with higher order kernels we propose the use of dimension reduction techniques we assume that the true covariate depends only on a linear combination of the observed covariates and surrogates if this linear combination were known we could apply the one dimensional versions of the semiparametric problem for which standard kernels are applicable we show that if one can estimate the linear directions at the root n rate then asymptotically the resulting estimator of the parameters in the main regression model behaves as if the linear combination were known simulations lend some credence to the asymptotic results

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

a semiparametric threshold model for censored longitudinal data analysis motivated by an investigation of the relationship between blood pressure change and progression of microalbuminuria ma among individuals with type i diabetes we propose a new semiparametric threshold model for censored longitudinal data analysis we also study a new semiparametric bayes information criterion type criterion for identifying the parametric component of the proposed model cluster effects in the model are implemented as unknown fixed effects asymptotic properties are established for the proposed estimators a quadratic approximation used to implement the estimation procedure makes the method very easy to implement by avoiding the computation of multiple integrals and the need for iterative algorithms simulation studies show that the proposed methods work well in practice an illustration using the wisconsin diabetes registry dataset suggests some interesting findings

local linear regression estimation under long range dependence strong consistency and rates we consider the nonparametric estimation of multivariate regression functions and their derivatives for a regression model with long range dependent errors we adopt a local linear fitting approach and establish strong consistency and rates for the estimators of the regression function and its derivatives the rates of convergence depend on the amount of smoothing relative to the strength of the long range dependence lrd resulting in distinct rates for small and large bandwidths moreover the conditions determining this dichotomy are different for the estimates of the regression function than for the estimates of its derivatives

semi continuity of the entropy of sinai probability measures for partially hyperbolic diffeomorphisms abstract we establish sufficient conditions for the upper semicontinuity and the continuity of the entropy of sinai probability measures invariant by partially hyperbolic diffeomorphisms and discuss their application in several examples

kernel estimation and model combination in a bandit problem with covariates multi armed bandit problem is an important optimization game that requires an exploration exploitation tradeoff to achieve optimal total reward motivated from industrial applications such as online advertising and clinical research we consider a setting where the rewards of bandit machines are associated with covariates and the accurate estimation of the corresponding mean reward functions plays an important role in the performance of allocation rules under a exible problem setup we establish asymptotic strong consistency and perform a finite time regret analysis for a sequential randomized allocation strategy based on kernel estimation in addition since many nonparametric and parametric methods in supervised learning may be applied to estimating the mean reward functions but guidance on how to choose among them is generally unavailable we propose a model combining allocation strategy for adaptive performance simulations and a real data evaluation are conducted to illustrate the performance of the proposed allocation strategy

dimension reduction and semiparametric estimation of survival models in this paper we propose a new dimension reduction method by introducing a nominal regression model with the hazard function as the conditional mean which naturally retrieves information from complete data and censored data as well moreover without requiring the linearity condition the new method can estimate the entire central subspace consistently and exhaustively the method also provides an alternative approach for the analysis of censored data assuming neither the link function nor the distribution hence it exhibits superior robustness properties numerical studies show that the method can indeed be readily used to efficiently estimate survival models explore the data structures and identify important variables

efficient statistical inference procedures for partially nonlinear models and their applications motivated by an analysis of a real data set in ecology we consider a class of partially nonlinear models where both a nonparametric component and a parametric component are present we develop two new estimation procedures to estimate the parameters in the parametric component consistency and asymptotic normality of the resulting estimators are established we further propose an estimation procedure and a generalized f test procedure for the nonparametric component in the partially nonlinear models asymptotic properties of the newly proposed estimation procedure and the test statistic are derived finite sample performance of the proposed inference procedures are assessed by monte carlo simulation studies an application in ecology is used to illustrate the proposed methods

a functional wavelet kernel approach for time series prediction summary we consider the prediction problem of a time series on a whole time interval in terms of its past the approach that we adopt is based on functional kernel nonparametric regression estimation techniques where observations are discrete recordings of segments of an underlying stochastic process considered as curves these curves are assumed to lie within the space of continuous functions and the discretized time series data set consists of a relatively small compared with the number of segments number of measurements made at regular times we estimate conditional expectations by using appropriate wavelet decompositions of the segmented sample paths a notion of similarity based on wavelet decompositions is used to calibrate the prediction asymptotic properties when the number of segments grows to 1 are investigated under mild conditions and a nonparametric resampling procedure is used to generate in a flexible way valid asymptotic pointwise prediction intervals for the trajectories predicted we illustrate the usefulness of the proposed functional wavelet kernel methodology in finite sample situations by means of a simulated example and two real life data sets and we compare the resulting predictions with those obtained by three other methods in the literature in particular with a smoothing spline method with an exponential smoothing procedure and with a seasonal autoregressive integrated moving average model

nonlinear and nonparametric regression and instrumental variables we consider regression when the predictor is measured with error and an instrumental variable iv is available the regression function can be modeled linearly nonlinearly or nonparametrically our major new result shows that the regression function and all parameters in the measurement error model are identified under relatively weak conditions much weaker than previously known to imply identifiability in addition we exploit a characterization of the iv estimator as a classical correction for attenuation method based on a particular estimate of the variance of the measurement error this estimate of the measurement error variance allows us to construct functional nonparametric regression estimators making no assumptions about the distribution of the unobserved predictor and structural estimators that use parametric assumptions about this distribution the functional estimators uses simulation extrapolation or deconvolution kernels and the structural method uses bayesian markov chain monte carlo

on a principal varying coefficient model we propose a novel varying coefficient model vcm called principal varying coefficient model pvcm by characterizing the varying coefficients through linear combinations of a few principal functions compared with the conventional vcm pvcm reduces the actual number of nonparametric functions and thus has better estimation efficiency compared with the semivarying coefficient model svcm pvcm is more flexible but with the same estimation efficiency when the number of principal functions in pvcm and the number of varying coefficients in svcm are the same model estimation and identification are investigated and the better estimation efficiency is justified theoretically incorporating the estimation with the l 1 penalty variables in the linear combinations can be selected automatically and hence the estimation efficiency can be further improved numerical experiments suggest that the model together with the estimation method is useful even when the number of covariates is large supplementary mat

semiparametric quasilikelihood and variance function estimation in measurement error models abstract we consider a quasilikelihood variance function model when a predictor x is measured with error and a surrogate w is observed when in addition to a primary data set containing y w a validation data set exists for which x w is observed we can i estimate the first and second moments of the response y given w by kernel regression ii use quasilikelihood and variance function techniques to estimate the regression parameters as well as variance structure parameters the estimators are shown to be asymptotically normally distributed with asymptotic variance depending on the size of the validation data set and not on the bandwith used in the kernel estimates a more refined analysis of the asymptotic covariance shows that the optimal bandwidth converges to zero at the rate n 1 3

robust estimation for the varying coefficient partially nonlinear models abstract in this paper we propose a robust estimation procedure based on the exponential squared loss esl function for the varying coefficient partially nonlinear model under some conditions the asymptotic properties of proposed estimators are established furthermore we develop a new minorization maximization mm algorithm to calculate the estimates for both non parametric and parametric parts and introduce a data driven procedure to select the tuning parameters simulation studies illustrate that the proposed method is more robust and efficient than the classical least squares technique when there are outliers in the dataset finally we apply the proposed methodology to analyze a real dataset the results reveal that the proposed has better the predictive ability

direct semiparametric estimation of single index models with discrete covariates abstract others have developed average derivative estimators of the parameter in the model e y x x g x where g is an unknown function and x is a random vector these estimators are noniterative and easy to compute but require that x be continuously distributed this article develops a noniterative easily computed estimator of for models in which some components of x are discrete the estimator is n consistent and asymptotically normal an application to data on product innovation by german manufacturers illustrates the estimator s usefulness

semiparametric group testing regression models group testing through the use of pooling has proven to be an efficient method of reducing the time and cost associated with screening for a binary characteristic of interest such as infection status a topic of key interest in the statistical literature involves the development of regression models that relate individual level covariates to testing responses observed from pooled specimens in this article we propose a general semiparametric framework that allows for the inclusion of multi dimensional covariates decoding information and imperfect testing the asymptotic properties of our estimators are presented and guidance on finite sample implementation is provided we illustrate the performance of our methods through simulation and by applying them to chlamydia and gonorrhea data collected by the nebraska public health laboratory as a part of the infertility prevention project

a fuzzy extended k nearest neighbors rule abstract the purpose of this paper is to describe an automatic fuzzy classification algorithm working in a partially supervised environment this algorithm is based on the fuzzy labeling of samples by means of a membership function as long as the membership function carries more information than the classical characteristic function the concepts of clustering and discrimination are extended to the fuzzy field first the learning set is fuzzily clustered the membership function is based on a weighted k nearest neighbors rule and does not require the optimization of any criterion second the test set is analyzed and a new membership function is proposed as the learning set is generally incomplete the creation of new fuzzy classes is studied the measure of fuzziness involves some analogies with statistical thermodynamics and the link between the membership function and the fermi dirac statistical function is discussed finally results on stimulated data are reported

a constructive approach to the estimation of dimension reduction directions in this paper we propose two new methods to estimate the dimension reduction directions of the central subspace cs by constructing a regression model such that the directions are all captured in the regression mean compared with the inverse regression estimation methods e g j amer statist assoc 86 1991 328 332 j amer statist assoc 86 1991 316 342 j amer statist assoc 87 1992 1025 1039 the new methods require no strong assumptions on the design of covariates or the functional relation between regressors and the response variable and have better performance than the inverse regression estimation methods for finite samples compared with the direct regression estimation methods e g j amer statist assoc 84 1989 986 995 ann statist 29 2001 1537 1566 j r stat soc ser b stat methodol 64 2002 363 410 which can only estimate the directions of cs in the regression mean the new methods can detect the directions of cs exhaustively consistency of the estimators and the convergence of corresponding algorithms are proved

statistical estimation in generalized multiparameter likelihood models multiparameter likelihood models mlms with multiple covariates have a wide range of applications however they encounter the curse of dimensionality problem when the dimension of the covariates is large we develop a generalized multiparameter likelihood model that copes with multiple covariates and adapts to dynamic structural changes well it includes some popular models such as the partially linear and varying coefficient models as special cases we present a simple effective two step method to estimate both the parametric and the nonparametric components when the model is fixed the proposed estimator of the parametric component has the n 1 2convergence rate and the estimator of the nonparametric component enjoys an adaptivity property we suggest a data driven procedure for selecting the bandwidths and propose an initial estimator in profile likelihood estimation of the parametric part to ensure stability of the approach in general settings we further develop an automatic procedure to ide

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

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

a law of the logarithm for kernel density estimators in this paper we derive a law of the logarithm for the maximal deviation between a kernel density estimator and the true underlying density function extensions to higher derivatives are included the results are applied to get optimal window widths with respect to almost sure uniform convergence

profile kernel likelihood inference with diverging number of parameters the generalized varying coefficient partially linear model with growing number of predictors arises in many contemporary scientific endeavor in this paper we set foot on both theoretical and practical sides of profile likelihood estimation and inference when the number of parameters grows with sample size the existence and asymptotic normality of the profile likelihood estimator are established under some regularity conditions profile likelihood ratio inference for the growing number of parameters is proposed and wilk s phenomenon is demonstrated a new algorithm called the accelerated profile kernel algorithm for computing profile kernel estimator is proposed and investigated simulation studies show that the resulting estimates are as efficient as the fully iterative profile kernel estimates for moderate sample sizes our proposed procedure saves much computational time over the fully iterative profile kernel one and gives stabler estimates a set of real data is analyzed using our proposed algorithm

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

kernel and probit estimates in quantal bioassay abstract a kernel method for the estimation of quantal dose response curves is considered in contrast to parametric modeling this local smoothing method does not require any assumptions beyond smoothness of the dose response curve and in this sense is nonparametric in finite sample situations the kernel estimate of the dose response curve is not necessarily monotone and therefore if the additional assumption of monotonicity of the dose response curve is made a monotonized version is discussed bias variance asymptotic normality and uniform consistency including rates of convergence of kernel estimates are derived and applied to establish consistency and limiting distribution of kernel estimates of the ed here ed is the effective dose at level that is the dose where 100 of the subjects show a response the properties of the kernel estimated ed are compared with the corresponding properties of the maximum likelihood estimator assuming the probit model practical application of the k

proprietes de convergence presque complete du predicteur a noyau let z n be a mixing process which is valued in mathbb e subset mathbb r q g be a real measurable function defined on mathbb e s and k be two positive integers we suppose the existence of a function r satisfying r e g z n s z n k 1 z n n n k and estimate the function r from a sequence z i i 1 n by r n with r n u frac sum limits i k n s g z i s k u z i k 1 cdots z i h n 1 sum limits i k n s k u z i k 1 cdots z i h n 1 forall u in mathbb e k where k is a kernel of kq and h n h n 0 we give various conditions on both the sequences h n and the sequence n which is associated with z n for the two following properties uniform complete convergence to r for the functional estimator r n and complete convergence to 0 for the real random variable r n z n k 1 z n r zn k 1 z n this last result concerns the predictor r n z n k 1 z n of g z n s from z i i 1 n when the process z n is stationary and markovian of order k

adjustment for missingness using auxiliary information in semiparametric regression in this article we study the estimation of mean response and regression coefficient in semiparametric regression problems when response variable is subject to nonrandom missingness when the missingness is independent of the response conditional on high dimensional auxiliary information the parametric approach may misspecify the relationship between covariates and response while the nonparametric approach is infeasible because of the curse of dimensionality to overcome this we study a model based approach to condense the auxiliary information and estimate the parameters of interest nonparametrically on the condensed covariate space our estimators possess the double robustness property i e they are consistent whenever the model for the response given auxiliary covariates or the model for the missingness given auxiliary covariate is correct we conduct a number of simulations to compare the numerical performance between our estimators and other existing estimators in the current missing data literature including the propensity score approach and the inverse probability weighted estimating equation a set of real data is used to illustrate our approach

asymptotic properties of integrated square error and cross validation for kernel estimation of a regression function we derive an asymptotic expansion of integrated square error in kernel type nonparametric regression a similar result is obtained for a cross validatory estimate of integrated square error together these expansions show that cross validation is asymptotically optimal in a certain sense

semiparametric regression for clustered data using generalized estimating equations we consider estimation in a semiparametric generalized linear model for clustered data using estimating equations our results apply to the case where the number of observations per cluster is finite whereas the number of clusters is large the mean of the outcome variable is of the form g xt t where g is a link function x and t are covariates is an unknown parameter vector and t is an unknown smooth function kernel estimating equations proposed previously in the literature are used to estimate the infinite dimensional nonparametric function t and a profile based estimating equation is used to estimate the finite dimensional parameter vector we show that for clustered data this conventional profile kernel method often fails to yield a n consistent estimator of along with appropriate inference unless working independence is assumed or t is artificially undersmoothed in which case asymptotic inference is possible to gain insight into these results we derive the se

efficiency loss caused by linearity condition in dimension reduction linearity sometimes jointly with constant variance is routinely assumed in the context of sufficient dimension reduction it is well understood that when these conditions do not hold blindly using them may lead to inconsistency in estimating the central subspace and the central mean subspace surprisingly we discover that even if these conditions do hold using them will bring efficiency loss this paradoxical phenomenon is illustrated through sliced inverse regression and principal hessian directions the efficiency loss also applies to other dimension reduction procedures we explain this empirical discovery by theoretical investigation

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

sinai bowen ruelle measures for certain henon maps we study maps t a b 2 2 defined by t a b left x y right left 1 a x 2 y bx right 0 0

nonparametric estimation of a generalized additive model with an unknown link function mean of a random scalar y conditional on a random vector x e rd let h x e yix x denote the conditional mean function this paper is concerned with estimating h under weak assumptions about its functional form when x is a multidimensional continuous random variable the paper describes a model that nests single index additive and multiplicative models and that can be estimated with a faster rate of convergence in probability than can a fully nonparametric model nonparametric estimation of h is usually unattractive when x is multidimensional the rate of convergence of nonparametric estimators decreases rapidly as the dimension of x increases and nonparametric estimates can be very imprecise with samples of practical size when x is multidimensional the curse of dimensionality this problem can be overcome through the use of dimension reduction methods they reduce the effective dimension of the estimation problem by making assumptions about the functional form of h that are stronger than those of a fully nonparametric estimator but weaker than those of a finite dimensional parametric model one method of dimension reduction is semiparametric single index modeling the best known and most easily estimated semiparametric single index model assumes that 1 1 h x g 3 x where 3 is an unknown d x 1 vector and g is an unknown function the vector 3 can be estimated with an n 1 2 rate of convergence in probability which is the rate that would be obtained if g were known han 1987 hardle and stoker 1989 horowitz and hardle 1996 ichimura 1993 klein and spady 1993 newey and stoker 1993 powell stock and stoker 1989 sherman 1993 stoker 1991 g can be estimated with the rate of convergence n r 2r 1 where r is an integer that depends on the smoothness of g but not on d dimension reduction can also be achieved by using a nonparametric generalized additive model in such a model

estimation in generalised varying coefficient models with unspecified link functions abstract in this paper we study the generalised varying coefficient models where the link function is unspecified and the response variable can be either continuous or discrete as the link function is unspecified the models under investigation become unidentifiable in this paper we derive an identification condition for the generalised varying coefficient models which is much weaker and more reasonable than that given by kuruwita et al 2011 whose model can be seen as a special case of our modelling framework under the identification condition we introduce a nonparametric iterative procedure to estimate the functional coefficient with its direction and norm as well as the unspecified link function and then establish the asymptotic properties of the resulting nonparametric estimators furthermore a weighted least squares based algorithm is provided to implement the iterative estimation procedure the simulation studies and empirical application show that our estimation methodology works quite well in both small and median sample cases

efficient semiparametric regression for longitudinal data with nonparametric covariance estimation summary for longitudinal data when the within subject covariance is misspecified the semiparametric regression estimator may be inefficient we propose a method that combines the efficient semiparametric estimator with nonparametric covariance estimation and is robust against misspecification of covariance models we show that kernel covariance estimation provides uniformly consistent estimators for the within subject covariance matrices and the semiparametric profile estimator with substituted nonparametric covariance is still semiparametrically efficient the finite sample performance of the proposed estimator is illustrated by simulation in an application to cd4 count data from an aids clinical trial we extend the proposed method to a functional analysis of the covariance model

nonparametric analysis of covariance in the problem of testing the equality of k regression curves from independent samples we discuss three methods using nonparametric estimators of the regression function the first test is based on a linear combination of estimators for the integrated variance function in the individual samples and in the combined sample the second approach transfers the classical one way analysis of variance to the situation of comparing non parametric curves while the third test compares the differences between the estimates of the individual regression functions by means of an l 2 distance we prove asymptotic normality of all considered statistics under the null hypothesis and local and fixed alternatives with different rates corresponding to the various cases additionally consistency of a wild bootstrap version of the tests is established in contrast to most of the procedures proposed in the literature the methods introduced in this paper are also applicable in the case of different design points in each sample and heteroscedastic errors a simulation study is conducted to investigate the finite sample properties of the new tests and a comparison with recently proposed and related procedures is performed

nonparametric transition based tests for jump diffusions we develop a specification test for the transition density of a discretely sampled continuous time jump diffusion process based on a comparison of a nonparametric estimate of the transition density or distribution function with their corresponding parametric counterparts assumed by the null hypothesis as a special case our method applies to pure diffusions we provide a direct comparison of the two densities for an arbitrary specification of the null parametric model using three different discrepancy measures between the null and alternative transition density and distribution functions we establish the asymptotic null distributions of proposed test statistics and compute their power functions we investigate the finite sample properties through simulations and compare them with those of other tests this article has supplementary material online

semilinear high dimensional model for normalization of microarray data a theoretical analysis and partial consistency normalization of microarray data is essential for removing experimental biases and revealing meaningful biological results motivated by a problem of normalizing microarray data a semilinear in slide model slim has been proposed to aggregate information from other arrays slim is generalized to account for across array information resulting in an even more dynamic semiparametric regression model this model can be used to normalize microarray data even when there is no replication within an array we demonstrate that this semiparametric model has a number of interesting features the parametric component and the nonparametric component that are of primary interest can be consistently estimated the former having a parametric rate and the latter having a nonparametric rate whereas the nuisance parameters cannot be consistently estimated this is an interesting extension of the partial consistent phenomena which itself is of theoretical interest the asymptotic normality for the parametric component

efficient semiparametric marginal estimation for longitudinal clustered data we consider marginal generalized semiparametric partially linear models for clustered data lin and carroll derived the semiparametric efficient score function for this problem in the multivariate gaussian case but they were unable to construct a semiparametric efficient estimator that actually achieved the semiparametric information bound here we propose such an estimator and generalize the work to marginal generalized partially linear models we investigate asymptotic relative efficiencies of the estimators that ignore the within cluster correlation structure either in nonparametric curve estimation or throughout we evaluate the finite sample performance of these estimators through simulations and illustrate it using a longitudinal cd4 cell count dataset both theoretical and numerical results indicate that properly taking into account the within subject correlation among the responses can substantially improve efficiency

estimation in partially linear models with missing covariates the partially linear model y xt z has been studied extensively when data are completely observed in this article we consider the case where the covariate x is sometimes missing with missingness probability depending on y z new methods are developed for estimating and our methods are shown to outperform asymptotically methods based only on the complete data asymptotic efficiency is discussed and the semiparametric efficient score function is derived justification of the use of the nonparametric bootstrap in this context is sketched the proposed estimators are extended to a working independence analysis of longitudinal clustered data and applied to analyze an aids clinical trial dataset the results of a simulation experiment are also given to illustrate our approach

asymptotic normality of the recursive kernel regression estimate under dependence conditions for i 1 2 ldots let x i and y i be mathbb r d valued d geq 1 integer and mathbb r valued respectively random variables and let x i y i i geq 1 be a strictly stationary and alpha mixing stochastic process set m x mathscr e y 1 mid x 1 x x in mathbb r d and let hat m n x be a certain recursive kernel estimate of m x under suitable regularity conditions and as n rightarrow infty it is shown that hat m n x properly normalized is asymptotically normal with mean 0 and a specified variance this result is established first under almost sure boundedness of the y i s and then by replacing boundedness by continuity of certain truncated moments it is also shown that for distinct points x 1 ldots x n in mathbb r d n geq 2 integer the joint distribution of the random vector hat m n x 1 ldots hat m n x n properly normalized is asymptotically n dimensional normal with mean vector 0 and a specified covariance function

generalized functional linear models with semiparametric single index interactions we introduce a new class of functional generalized linear models where the response is a scalar and some of the covariates are functional we assume that the response depends on multiple covariates a finite number of latent features in the functional predictor and interaction between the two to achieve parsimony the interaction between the multiple covariates and the functional predictor is modeled semiparametrically with a single index structure we propose a two step estimation procedure based on local estimating equations and investigate two situations a when the basis functions are predetermined e g fourier or wavelet basis functions and the functional features of interest are known and b when the basis functions are data driven such as with functional principal components asymptotic properties are developed notably we show that when the functional features are data driven the parameter estimates have an increased asymptotic variance due to the estimation error of the basis functio

shrinkage estimation of the varying coefficient model the varying coefficient model is a useful extension of the linear regression model nevertheless how to conduct variable selection for the varying coefficient model in a computationally efficient manner is poorly understood to solve the problem we propose here a novel method which combines the ideas of the local polynomial smoothing and the least absolute shrinkage and selection operator lasso the new method can do nonparametric estimation and variable selection simultaneously with a local constant estimator and the adaptive lasso penalty the new method can identify the true model consistently and that the resulting estimator can be as efficient as the oracle estimator numerical studies clearly confirm our theories extension to other shrinkage methods e g the scad i e the smoothly clipped absolute deviation and other smoothing methods is straightforward

nonparametric analysis of competing risks data with event category missing at random summary in competing risks setup the data for each subject consist of the event time censoring indicator and event category however sometimes the information about the event category can be missing as for example in a case when the date of death is known but the cause of death is not available in such situations treating subjects with missing event category as censored leads to the underestimation of the hazard functions we suggest nonparametric estimators for the cumulative cause specific hazards and the cumulative incidence functions which use the nadaraya watson estimator to obtain the contribution of an event with missing category to each of the cause specific hazards we derive the propertied of the proposed estimators optimal bandwidth is determined which minimizes the mean integrated squared errors of the proposed estimators over time the methodology is illustrated using data on lung infections in patients from the united states cystic fibrosis foundation patient registry

variable selection for partially linear models with measurement errors this article focuses on variable selection for partially linear models when the covariates are measured with additive errors we propose two classes of variable selection procedures penalized least squares and penalized quantile regression using the nonconvex penalized principle the first procedure corrects the bias in the loss function caused by the measurement error by applying the so called correction for attenuation approach whereas the second procedure corrects the bias by using orthogonal regression the sampling properties for the two procedures are investigated the rate of convergence and the asymptotic normality of the resulting estimates are established we further demonstrate that with proper choices of the penalty functions and the regularization parameter the resulting estimates perform asymptotically as well as an oracle procedure as proposed by fan and li choice of smoothing parameters is also discussed finite sample performance of the proposed variable selection procedures is asse

parameter estimation for differential equation models using a framework of measurement error in regression models differential equation de models are widely used in many scientific fields including engineering physics and biomedical sciences the so called forward problem the problem of simulations and predictions of state variables for given parameter values in the de models has been extensively studied by mathematicians physicists engineers and other scientists however the inverse problem the problem of parameter estimation based on the measurements of output variables has not been well explored using modern statistical methods although some least squares based approaches have been proposed and studied in this article we propose parameter estimation methods for ordinary differential equation ode models based on the local smoothing approach and a pseudo least squares psls principle under a framework of measurement error in regression models the asymptotic properties of the proposed psls estimator are established we also compare the psls method to the corresponding simulation extrapolation

robust check loss based inference of semiparametric models and its application in environmental data abstract composite quantile regression has been shown to be a more competitive method than the well known mean regression and quantile regression approaches since it can significantly improve the estimation efficiency for a wide class of errors in this paper we propose a profiling composite quantile regression for robust estimation of semiparametric varying coefficient partially linear model vcplm by combining the ideas of local linear composite quantile regression with the profile technique different from the step wise estimation method considered in kai et al 2011 an iterative algorithm is developed for the estimates of parametric and nonparametric functions under some mild regularity conditions we establish the asymptotic normalities of the obtained estimators and further briefly discuss the related asymptotic relative efficiency noteworthy an entirely different way from kai et al 2011 is presented to establish the asymptotic normality of the parametric component which is frequently of the primary interest in vcplm some monte carlo simulations with various errors and an environmental data application are conducted to evaluate the finite sample performance of the proposed methodology

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

cross validation in nonparametric regression with outliers a popular data driven method for choosing the bandwidth in standard kernel regression is cross validation even when there are outliers in the data robust kernel regression can be used to estimate the unknown regression curve robust and nonlinear time series analysis lecture notes in statist 1984 26 163 184 however under these circumstances standard cross validation is no longer a satisfactory bandwidth selector because it is unduly influenced by extreme prediction errors caused by the existence of these outliers a more robust method proposed here is a cross validation method that discounts the extreme prediction errors in large samples the robust method chooses consistent bandwidths and the consistency of the method is practically independent of the form in which extreme prediction errors are discounted additionally evaluation of the method s finite sample behavior in a simulation demonstrates that the proposed method performs favorably this method can also he applied to other problems for example model selection that require cross validation

some stochastics on monotone functions abstract a measure of departures of monotonicity of a given function the l r dip 1 r is introduced our analysis is performed to cover two different situations when the function is known our interest is related to its behavior in a stochastic model however in most cases the knowledge of the function is obtained through a preliminary estimation of the function in both situations the aim focuses in the obtainment of strong consistency results

conditions for existence of uniformly consistent classifiers we consider the statistical problem of binary classification which means attaching a random observation x from a separable metric space e to one of the two classes 0 or 1 we prove that the consistent estimation of conditional probability p x mathsf p y 1 mid x where y is the true class of x is equivalent to the consistency of a class of empirical classifiers we then investigate for what classes p there exist an estimate hat p that is consistent uniformly in p in p we show that this holds if and only if p is a totally bounded subset of l 1 e mu where mu is the distribution of x in the case where e is countable we give a complete characterization of classes pi allowing consistent estimation of p uniform in mu p in pi

new efficient estimation and variable selection methods for semiparametric varying coefficient partially linear models the complexity of semiparametric models poses new challenges to statistical inference and model selection that frequently arise from real applications in this work we propose new estimation and variable selection procedures for the semiparametric varying coefficient partially linear model we first study quantile regression estimates for the nonparametric varying coefficient functions and the parametric regression coefficients to achieve nice efficiency properties we further develop a semiparametric composite quantile regression procedure we establish the asymptotic normality of proposed estimators for both the parametric and nonparametric parts and show that the estimators achieve the best convergence rate moreover we show that the proposed method is much more efficient than the least squares based method for many non normal errors and that it only loses a small amount of efficiency for normal errors in addition it is shown that the loss in efficiency is at most 11 1 for estimating varying coefficient functions and is no greater than 13 6 for estimating parametric components to achieve sparsity with high dimensional covariates we propose adaptive penalization methods for variable selection in the semiparametric varying coefficient partially linear model and prove that the methods possess the oracle property extensive monte carlo simulation studies are conducted to examine the finite sample performance of the proposed procedures finally we apply the new methods to analyze the plasma beta carotene level data

pairwise curve synchronization for functional data summary data collected by scientists are increasingly in the form of trajectories or curves often these can be viewed as realizations of a composite process driven by both amplitude and time variation we consider the situation in which functional variation is dominated by time variation and develop a curve synchronization method that uses every trajectory in the sample as a reference to obtain pairwise warping functions in the first step these initial pairwise warping functions are then used to create improved estimators of the underlying individual warping functions in the second step a truncated averaging process is used to obtain robust estimation of individual warping functions the method compares well with other available time synchronization approaches and is illustrated with berkeley growth data and gene expression data for multiple sclerosis

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

local polynomial estimation for sensitivity analysis on models with correlated inputs sensitivity indexes when the inputs of a model are not independent are derived from local polynomial techniques two original estimators based on local polynomial smoothers are proposed both have good theoretical properties which are illustrated through analytical examples comparison with the bayesian approach developed by oakley and o hagan 2004 is also performed the two proposed estimators are used to carry out a sensitivity analysis on two real case models with correlated parameters

a loose pattern process approach to clustering fuzzy data sets a loose pattern process approach to clustering sets consists of three main computations loose pattern reject option tight pattern classifcation and loose pattern assigning classes the loose pattern rejection is implemented using a rule based on q nearest neighbors of each point two clustering methods glc and oupic are introduced as tight pattern clustering techniques the decisions of loose pattern assigning classes are related to a heuristic membership function the function and experiments with one set is discussed

minimum distance regression model checking with berkson measurement errors lack of fit testing of a regression model with berkson measurement error has not been discussed in the literature to date to fill this void we propose a class of tests based on minimized integrated square distances between a nonparametric regression function estimator and the parametric model being fitted we prove asymptotic normality of these test statistics under the null hypothesis and that of the corresponding minimum distance estimators under minimal conditions on the model being fitted we also prove consistency of the proposed tests against a class of fixed alternatives and obtain their asymptotic power against a class of local alternatives orthogonal to the null hypothesis these latter results are new even when there is no measurement error a simulation that is included shows very desirable finite sample behavior of the proposed inference procedures

nonparametric estimation of a regression function consider the regression model y i g x i e i i 1 2 n where x i s denote unordered design variables and g is an unknown function defined on the interval 0 1 assume e i are iid random variables with zero mean and finite variance priestley and chao 1972 and clark 1977 proposed estimators g 2n and g 3n respectively for g in this paper the asymptotic behavior of g 2n and g 3n is studied utilizing the properties of the new estimator g 1n it is shown that g 1n g 2n g 3n are asymptotically equivalent in various senses moreover consistency results are established and rates of uniform convergence obtained for example if e e 3 if g is lipschitz of order 1 and if n is any sequence of constants tending to as n then for all 0 a leqq b 1 n 1 mathord left vphantom 1 3 right kern nulldelimiterspace 3 mathord left vphantom n 1 mathord left vphantom 1 3 right kern nulldelimiterspace 3 beta n log n mathop sup limits a leqq x leqq b g 1n x g x xrightarrow w p 1 0 right kern nulldelimiterspace beta n log n mathop sup limits a leqq x leqq b g 1n x g x xrightarrow w p 1 0 as n finally when g is monotone a strong consistent isotonic estimator g n is considered

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

tuning parameter selectors for the smoothly clipped absolute deviation method summary the penalized least squares approach with smoothly clipped absolute deviation penalty has been consistently demonstrated to be an attractive regression shrinkage and selection method it not only automatically and consistently selects the important variables but also produces estimators which are as efficient as the oracle estimator however these attractive features depend on appropriate choice of the tuning parameter we show that the commonly used generalized crossvalidation cannot select the tuning parameter satisfactorily with a nonignorable overfitting effect in the resulting model in addition we propose a bic tuning parameter selector which is shown to be able to identify the true model consistently simulation studies are presented to support theoretical findings and an empirical example is given to illustrate its use in the female labor supply data

invariance principles for interval maps with an indifferent fixed point in this note we establish an almost sure invariance principle for a large class of interval maps with indifferent fixed points including the manneville pomeau map this implies a number of well known corollaries including the weak invariance principle and the law of the iterated logarithm

combining nonparametric and optimal linear time series predictions we introduce a semiparametric procedure for more efficient prediction of a strictly stationary process admitting an arma representation the procedure is based on the estimation of the arma representation followed by a nonparametric regression where the arma residuals are used as explanatory variables compared to standard nonparametric regression methods the number of explanatory variables can be reduced because our approach exploits the linear dependence of the process we establish consistency and asymptotic normality results for our estimator numerical experiments show that significant gains can be achieved with our approach all the supplemental materials used by this article are available online

estimation of the covariance matrix of random effects in longitudinal studies longitudinal studies are often conducted to explore the cohort and age effects in many scientific areas the within cluster correlation structure plays a very important role in longitudinal data analysis this is because not only can an estimator be improved by incorporating the within cluster correlation structure into the estimation procedure but also the within cluster correlation structure can sometimes provide valuable insights in practical problems for example it can reveal the correlation strengths among the impacts of various factors motivated by data typified by a set from bangladesh pertinent to the use of contraceptives we propose a random effect varying coefficient model and an estimation procedure for the within cluster correlation structure of the proposed model the estimation procedure is optimization free and the proposed estimators enjoy asymptotic normality under mild conditions simulations suggest that the proposed estimation is practicable for finite samples and resistent against mild forms of model m is specification finally we analyze the data mentioned above with the new random effect varying coefficient model together with the proposed estimation procedure which reveals some interesting sociological dynamics

a nonparametric calibration analysis in this paper we discuss a new approach to solve calibration problems in a nonparametric setting this approach is appealing because it yields estimates of the required quantities directly the method combines kernel and robust estimation techniques it relies on strong approximations of the estimating process and the extreme value theorem of bickel and rosenblatt using these results we first obtain robust pointwise estimates of the parameters of interest second we set up asymptotic simultaneous tolerance regions for many unknown values of the quantity to be calibrated the technique is illustrated on a radiocarbon dating problem the nonparametric calibration procedure proves to be of practical as well as theoretical interest moreover it is quick and simple to implement 1 introduction statistical calibration analyses provide a way to predict a quantity from the observation of another one by using a dose response type relationship the problem occurs in industry or the biological sciences when the quantity to be calibrated is hard or expensive to measure or is not observable in many situations the knowledge of the experimental process can hardly be translated into a parametric model for the dose response relationship a tempting alternative may be provided by a nonparametric modeling let z denote the random response variable we observe the unknown

laws of the iterated logarithm for nonparametric density estimators we establish a law of the iterated logarithm for a triangular array of independent random variables and apply it to obtain laws for a large class of nonparametric density estimators we consider the case of rosenblatt parzen kernel estimators trigonometric series estimators and orthogonal polynomial estimators in detail and point out that our technique has wider application

similarity of classes and fuzzy clustering abstract three aspects of current interest similarity of classes membership functions of patterns in classes and fuzzy clustering are examined in relation to multiple measures of classes three data sets are studied the results of pattern fuzziness in classes and pattern rejection from classes are obtained by a fuzzy clustering method

a semiparametric approach to dimension reduction we provide a novel and completely different approach to dimension reduction problems from the existing literature we cast the dimension reduction problem in a semiparametric estimation framework and derive estimating equations viewing this problem from the new angle allows us to derive a rich class of estimators and obtain the classical dimension reduction techniques as special cases in this class the semiparametric approach also reveals that in the inverse regression context while keeping the estimation structure intact the common assumption of linearity and or constant variance on the covariates can be removed at the cost of performing additional nonparametric regression the semiparametric estimators without these common assumptions are illustrated through simulation studies and a real data example this article has online supplementary material

nonparametric mixture of regression models motivated by an analysis of u s house price index hpi data we propose nonparametric finite mixture of regression models we study the identifiability issue of the proposed models and develop an estimation procedure by employing kernel regression we further systematically study the sampling properties of the proposed estimators and establish their asymptotic normality a modified em algorithm is proposed to carry out the estimation procedure we show that our algorithm preserves the ascent property of the em algorithm in an asymptotic sense monte carlo simulations are conducted to examine the finite sample performance of the proposed estimation procedure an empirical analysis of the u s hpi data is illustrated for the proposed methodology

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