local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

the generalized estimating equation approach when data are not missing completely at random abstract we propose two methods for handling missing data in generalized estimating equation gee analyses mean imputation and multiple imputation each provides valid gee estimates when data are missing at random missing outcomes are imputed sequentially starting from the outcome nearest in time to the observed outcome the estimators from the two kinds of imputation are compared with the weighting method of robins et al we show that multiple imputation with an infinite number of replications is asymptotically equivalent to mean imputation the methods are applied to a stroke study in which neurological outcomes are measured over time after stroke but some outcomes are missing due to death or loss to follow up

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

the circulant operator in the banach algebra of matrices abstract we study an operator c which maps every n by n matrix a n to a circulant matrix c a n that minimizes the frobenius norm a n c n f over all n by n circulant matrices c n the circulant matrix c a n called the optimal circulant preconditioner has proved to be a good preconditioner for a general class of toeplitz systems in this paper we give different formulations of the operator discuss its algebraic and geometric properties and compute its operator norms in different banach algebras of matrices using these results we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of i c n 1 a n f over all nonsingular circulant matrices c n

weighted estimators for proportional hazards regression with missing covariates missing covariate data are common in epidemiologic studies and disease prevention trials in this article regression parameter estimation in the cox proportional hazards model is considered when certain covariates are observed for all study subjects and other covariate data are collected only for a subset the article presents both simple weighted and kernel assisted fully augmented weighted estimators that use the partially incomplete data nonparametrically we use nonparametric methods to estimate selection probabilities in the simple weighted estimating functions we also use nonparametric kernel smoothing techniques to estimate certain conditional expectations in fully augmented weighted estimating functions the proposed methods are nonparametric in the sense that they require neither a model for the missing data mechanism nor specification of the conditional distribution of missing covariates given observed covariates these estimators allow the missing data mechanism to depend on outcome variables

local multiple imputation dealing with missing data via parametric multiple imputation methods usually implies stating several strong assumptions both about the distribution of the data and about underlying regression relationships if such parametric assumptions do not hold the multiply imputed data are not appropriate and might produce inconsistent estimators and thus misleading results in this paper a fully nonparametric and a semiparametric imputation method are studied both based on local resampling principles it is shown that the final estimator based on these local imputations is consistent under fewer or no parametric assumptions asymptotic expressions for bias variance and mean squared error are derived showing the theoretical impact of the different smoothing parameters simulations illustrate the usefulness and applicability of the method

empirical likelihood and general estimating equations for some time so called empirical likelihoods have been used heuristically for purposes of nonparametric estimation owen showed that empirical likelihood ratio statistics for various parameters theta f of an unknown distribution f have limiting chi square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods our objective in this paper is twofold first to link estimating functions or equations and empirical likelihood second to develop methods of combining information about parameters we do this by assuming that information about f and theta is available in the form of unbiased estimating functions empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood efficiency results for estimates of both theta and f are obtained the methods are illustrated on several problems and areas for future investigation are noted

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

nonparametric estimation of mean functionals with data missing at random abstract this article considers a distribution free estimation procedure for a basic pattern of missing data that often arises from the wellknown double sampling in survey methodology without parametric modeling of the missing mechanism or the joint distribution kernel regression estimators are used to estimate mean functionals through empirical estimation of the missing pattern a generalization of the method of cheng and wei is verified under the assumption of missing at random asymptotic distributions are derived for estimating the mean of the incomplete data and for estimating the mean treatment difference in a nonrandomized observational study the nonparametric method is compared with a naive pairwise deletion method and a linear regression method via the asymptotic relative efficiencies and a simulation study the comparison shows that the proposed nonparametric estimators attain reliable performances in general

the mm me ml el ef and gmm approaches to estimation a synthesis abstract the 20th century began on an auspicious statistical note with the publication of karl pearson s philos mag ser 50 1900 157 goodness of fit test which is regarded as one of the most important scientific breakthroughs the basic motivation behind this test was to see whether an assumed probability model adequately described the data at hand pearson philos trans roy soc london ser a 185 1894 71 also introduced a formal approach to statistical estimation through his method of moments mm estimation ronald a fisher while he was a third year undergraduate at the gonville and caius college cambridge suggested the maximum likelihood estimation mle procedure as an alternative to pearson s mm approach in 1922 fisher published a monumental paper that introduced such basic concepts as consistency efficiency sufficiency and even the term parameter with its present meaning fisher philos trans roy soc london ser a 222 1922 309 provided the analytical foundation of mle and studied its efficiency relative to the mm estimator fisher j roy statist soc 87 1924a 442 established the asymptotic equivalence of minimum 2 and ml estimators and wrote in favor of using minimum 2 method rather than pearson s mm approach recently econometricians have found working under assumed likelihood functions restrictive and have suggested using a generalized version of pearson s mm approach commonly known as the gmm estimation procedure as advocated in hansen econometrica 50 1982 1029 earlier godambe ann math statist 31 1960 1208 and durbin j roy statist soc ser b 22 1960 139 developed the estimating function ef approach to estimation that has been proven very useful for many statistical models a fundamental result is that score is the optimum ef ferguson ann math statist 29 1958 1046 considered an approach very similar to gmm and showed that estimation based on the pearson 2 statistic is equivalent to efficient gmm golan et al maximum entropy econometrics robust estimation with limited data wiley new york 1996 developed entropy based formulation that allowed them to solve a wide range of estimation and inference problems in econometrics more recently imbens et al econometrica 66 1998 333 kitamura and stutzer econometrica 65 1997 861 and mittelhammer et al econometric foundations cambridge university press cambridge 2000 put gmm within the framework of empirical likelihood el and maximum entropy me estimation it can be shown that many of these estimation techniques can be obtained as special cases of minimizing cressie and read j roy statist soc ser b 46 1984 440 power divergence criterion that comes directly from the pearson 1900 2 statistic in this way we are able to assimilate a number of seemingly unrelated estimation techniques into a unified framework

adjusting for nonignorable drop out using semiparametric nonresponse models abstract consider a study whose design calls for the study subjects to be followed from enrollment time t 0 to time t t at which point a primary endpoint of interest y is to be measured the design of the study also calls for measurements on a vector v t of covariates to be made at one or more times t during the interval 0 t we are interested in making inferences about the marginal mean 0 of y when some subjects drop out of the study at random times q prior to the common fixed end of follow up time t the purpose of this article is to show how to make inferences about 0 when the continuous drop out time q is modeled semiparametrically and no restrictions are placed on the joint distribution of the outcome and other measured variables in particular we consider two models for the conditional hazard of drop out given v t y where v t denotes the history of the process v t through time t t 0 t in the first model we assume that q t v t y exp 0 y where 0 is a scalar paramet

the analysis of categorical data from complex sample surveys chi squared tests for goodness of fit and independence in two way tables abstract the effect of stratification and clustering on the asymptotic distributions of standard pearson chi squared test statistics for goodness of fit simple hypothesis and independence in a two way contingency table denoted as x 2 and xi 2 respectively is investigated it is shown that both x 2 and xi 2 are asymptotically distributed as weighted sums of independent 1 2 random variables the weights are then related to the familiar design effects deffs used by survey samplers a simple correction to x 2 which requires only the knowledge of variance estimates or deffs for individual cells in the goodness of fit problem is proposed and empirical results on the performance of corrected x 2 provided empirical work on xi 2 indicated that the distortion of nominal significance level is substantially smaller with xi 2 than with x 2 some results under simple models for clustering are also given

a semiparametric method of multiple imputation in this paper we describe how to use multiple imputation semiparametrically to obtain estimates of parameters and their standard errors when some individuals have missing data the methods given require the investigator to know or be able to estimate the process generating the missing data but requires no full distributional form for the data the method is especially useful for non standard problems such as estimating the median when data are missing

weighted semiparametric estimation in regression analysis with missing covariate data abstract this article investigates estimation of the regression coefficients in an assumed mean function when covariates on some subjects are missing we examine the performance of a horvitz and thompson 1952 type weighted estimator by using different estimates of the selection probabilities which may be treated as nuisance parameters or a nuisance function in particular we investigate the properties of the estimate of the regression parameters when the selection probabilities are estimated by kernel smoothers we present large sample theory for the new estimator and conduct simulation studies comparing the proposed estimator to the maximum likelihood estimator and multiple imputation under various model assumptions and different missingness mechanisms in addition we provide two real examples that motivate this investigation

superoptimal preconditioned conjugate gradient iteration for image deblurring we study the superoptimal frobenius operators in the two level circulant algebra we consider two specific viewpoints 1 the regularizing properties in imaging and 2 the computational effort in connection with the preconditioned conjugate gradient method some numerical experiments illustrating the effectiveness of the proposed technique are given and discussed

on the bootstrap and confidence intervals we derive an explicit formula for the first term in an unconditional edgeworth type expansion of coverage probability for the nonparametric bootstrap technique applied to a very broad class of studentized statistics the class includes sample mean k sample mean sample correlation coefficient maximum likelihood estimators expressible as functions of vector means etc we suggest that the bootstrap is really an empiric one term edgeworth inversion with the bootstrap simulations implicitly estimating the first term in an edgeworth expansion this view of the bootstrap is reinforced by our discussion of the iterated bootstrap which inverts an edgeworth expansion to arbitrary order by simulating simulations

estimating equations inference with missing data there is a large and growing body of literature on estimating equation ee as an estimation approach one basic property of ee that has been universally adopted in practice is that of unbiasedness and there are deep conceptual reasons why unbiasedness is a desirable ee characteristic this article deals with inference from ees when data are missing at random the investigation is motivated by the observation that direct imputation of missing data in ees generally leads to ees that are biased and thus violates a basic assumption of the ee approach the main contribution of this article is that it goes beyond existing imputation methods and proposes a procedure whereby one mitigates the effects of missing data through a reformulation of ees imputed through a kernel regression method these modified ees then constitute a basis for inference by the generalized method of moments gmm and empirical likelihood el asymptotic properties of the gmm and el estimators of the unknown parameters are derived a

which circulant preconditioner is better the eigenvalue clustering of matrices s n 1 a n and c n 1 a n is experimentally studied where a n s n and c n respectively are toeplitz matrices strang and optimal circulant preconditioners generated by the fourier expansion of a function f x some illustrations are given to show how the clustering depends on the smoothness of f x and which preconditioner is preferable an original technique for experimental exploration of the clustering rate is presented this technique is based on the bisection idea and on the toeplitz decomposition of a three matrix product cac where a is a toeplitz matrix and c is a circulant in particular it is proved that the toeplitz displacement rank of cac is not greater than 4 provided that c and a are symmetric

recent developments in computer assisted analysis of mixtures this paper reviews recent developments in the area of computer assisted analysis of mixture distributions c a man given a biometric situation of interest in which under homogeneity assumptions a certain parametric density occurs such as the poisson the binomial the geometric the normal and so forth then it is argued that this situation can easily be enlarged to allow a variation of the scalar parameter in the population this situation is called unobserved heterogeneity this naturally leads to a specific form of nonparametric mixture distribution that can then be assumed to be the standard model in the biometric application of interest since it also incorporates the homogeneous situations as a special case besides developments in theory and algorithms the work focuses on developments in biometric applications such as meta analysis fertility studies estimation of prevalence under clustering and estimation of the distribution function of survival time under interval censoring the approach is nonparametric for the mixing distribution including leaving the number of components subpopulations of the mixing distribution unknown

missing at random likelihood ignorability and model completeness this paper provides further insight into the key concept of missing at random mar in incomplete data analysis following the usual selection modelling approach we envisage two models with separable parameters a model for the response of interest and a model for the missing data mechanism mdm if the response model is given by a complete density family then frequentist inference from the likelihood function ignoring the mdm is valid if and only if the mdm is mar this necessary and sufficient condition also holds more generally for models for coarse data such as censoring examples are given to show the necessity of the completeness of the underlying model for this equivalence to hold

expected estimating equations for missing data measurement error and misclassification with application to longitudinal nonignorable missing data missing data measurement error and misclassification are three important problems in many research fields such as epidemiological studies it is well known that missing data and measurement error in covariates may lead to biased estimation misclassification may be considered as a special type of measurement error for categorical data nevertheless we treat misclassification as a different problem from measurement error because statistical models for them are different indeed in the literature methods for these three problems were generally proposed separately given that statistical modeling for them are very different the problem is more challenging in a longitudinal study with nonignorable missing data in this article we consider estimation in generalized linear models under these three incomplete data models we propose a general approach based on expected estimating equations eees to solve these three incomplete data problems in a unified fashion this eee approach can be easily implemented and its asymptotic covariance can be obtained by sandwich estimation intensive simulation studies are performed under various incomplete data settings the proposed method is applied to a longitudinal study of oral bone density in relation to body bone density

handling estimating equation with nonignorably missing data based on sir algorithm abstract in this paper the estimation of parameters in estimating equation with nonignorably missing data is considered based on a logistic regression model for the response mechanism and an assumed parametric model for the distribution of respondents we propose to use the sampling importance resampling sir algorithm to calculate the conditional expectation for non respondents the proposed method can avoid the difficulty of high dimensional kernel smoothing on calculating the conditional expectation under nonignorably missing data based on the sir algorithm we derive a modified estimating equation to facilitate comparison we apply the empirical likelihood method to derive the parameters estimator some numerical simulation results based on linear and nonlinear models show that the proposed method is robust and performs the best among the compared existing methods so we extend the estimation method of e g y using sir algorithm with nonignorably missing response to the estimation of parameters in generalized estimating equation and obtain a new and more robust estimation method

extended generalized estimating equations for binary familial data with incomplete families dans ce papier nous evaluons les performances de deux methodes standard mais naives pour la prise en compte de donnees familiales incompletes dans les analyses utilisant les gee2 dans le contexte d une variable expliquee binaire nous proposons egalement une nouvelle methode pour analyser de telles donnees avec les gee2 quand les variables explicatives sont discretes a la difference des approches naives cette nouvelle methode ne necessite pas que le processus responsable des donnees manquantes soit aleatoire nous illustrons notre approche a l aide d un exemple qui examine l agregation familiale de l obesite

a finite element frequency domain method for 2d photonic crystals in this paper we propose a new finite element frequency domain fefd method to compute the band spectra of 2d photonic crystals without impurities exploiting periodicity to identify discretization points differing by a period it increases the effectiveness of the algorithm and reduces significantly its computational complexity the results of an extensive experimentation indicate that our method offers an effective alternative to the most quoted methods in the literature

assessing reproducibility by the within subject coefficient of variation with random effects models in this paper we consider the use of within subject coefficient of variation wcv for assessing the reproducibility or reliability of a measurement application to assessing reproducibility of biochemical markers for measuring bone turnover is described and the comparison with intraclass correlation is discussed both maximum likelihood and moment confidence intervals of wcv are obtained through their corresponding asymptotic distributions normal and log normal cases are considered in general wcv is preferred when the measurement scale bears intrinsic meaning and is not subject to arbitrary shifting the intraclass correlation may be preferred when a fixed population of subjects can be well identified

empirical likelihood based inference under imputation for missing response data inference under kernel regression imputation for missing response data is considered an adjusted empirical likelihood approach to inference for the mean of the response variable is developed a nonparametric version of wilks theorem is proved for the adjusted empirical log likelihood ratio by showing that it has an asymptotic standard chi squared distribution and the corresponding empirical likelihood confidence interval for the mean is constructed with auxiliary information an empirical likelihood based estimator is defined and an adjusted empirical log likelihood ratio is derived asymptotic normality of the estimator is proved also it is shown that the adjusted empirical log likelihood ratio obeys wilks theorem a simulation study is conducted to compare the adjusted empirical likelihood and the normal approximation methods in terms of coverage accuracies and average lengths of confidence intervals based on biases and standard errors a comparision is also made by simulation between the empirical likelihood based estimator and related estimators our simulation indicates that the adjusted empirical likelihood method performs competitively and that the use of auxiliary information provides improved inferences

the spectrum of a family of circulant preconditioned toeplitz systems the solutions of symmetric positive definite toeplitz systems ax b are studied by the preconditioned conjugate gradient method the preconditioner is the circulant matrix c that minimizes the frobenius norm c a f t chan an optimal circulant preconditioner for toeplitz systems ucla department of mathematics cam report 87 06 june 1987 the convergence rate of these iterative methods is known to depend on the distribution of the eigenvalues of c 1 a for toeplitz matrix a with entries which are fourier coefficients of a positive function in the wiener class this paper establishes the invertibility of c finds the asymptotic behaviour of the eigenvalues of the preconditioned matrix c 1 a as the dimension increases and proves that they are clustered around 1

fast superoptimal preconditioning of multiindex toeplitz matrices abstract in this article three well known methods of circulant preconditioning of finite multiindex toeplitz linear systems that is linear systems indexed by integers i 1 i d with 0 i s n s s 1 d are studied in detail a general algorithm for the construction of the so called superoptimal preconditioner is also given and it is shown that this procedure requires o n 1 n d log n 1 n d floating point operations

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

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

propriety of the posterior distribution and existence of the mle for regression models with covariates missing at random characterizing model identifiability in the presence of missing covariate data is a very important issue in missing data problems in this article we characterize the propriety of the posterior distribution of the regression coefficients for some general classes of regression models including the class of generalized linear models glm s and parametric survival models with right censored data toward this goal we derive some very general and easy to check conditions for the matrix of covariates we also derive sufficient conditions for the existence of the maximum likelihood estimates and establish novel results for checking propriety of the posterior when the sample size is large several theorems are given to establish propriety of the posterior and the existence of the maximum likelihood estimator the conditions reduce to solving a system of linear equations which can be carried out using software such as maple imsl or sas we assume that the missing covariates are missing at random and assume

a decomposition of toeplitz matrices and optimal circulant preconditioning abstract a factorization of the given toeplitz matrix t obtained in this work is used to derive an optimal circulant preconditioner c in the sense of minimizing the frobenius norm of i ct by use of the fast fourier transform the computation of the new preconditioner can be performed in o n log n operations