conditional scores and optimal scores for generalized linear measurement error models abstract this paper studies estimation of the parameters of generalized linear models in canonical form when the explanatory vector is measured with independent normal error for the functional case i e when the explanatory vectors are fixed constants unbiased score functions are obtained by conditioning on certain sufficient statistics this work generalizes results obtained for logistic regression in the case that the explanatory vectors are independent and identically distributed with unknown distribtuion efficient score functions are obtained using the theory developed in begun et al 1983 keywords conditional score function efficient score function functional model generalized linear model measurement error structural model

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

on errors in variables for binary regression models summary we consider binary regression models when some of the predictors are measured with error for normal measurement errors structural maximum likelihood estimates are considered we show that if the measurement error is large the usual estimate of the probability of the event in question can be substantially in error especially for high risk groups in the situation of large measurement error we investigate a conditional maximum likelihood estimator and its properties

elliptic regularization for the semi linear telegraph system abstract the solution of the semi linear telegraph system is compared with the solution of an elliptic regularization to which one associates two point boundary conditions an asymptotic approximation for the solution of the elliptic regularization is constructed the method employed here is the boundary function method due to vishik and lyusternik the problem is singularly perturbed of elliptic hyperbolic type to conduct this analysis high regularity with respect to t for the solutions of both problems is required finally the order of this approximation is found in different spaces of functions

semiparametric estimators of functional measurement error models with unknown error summary we consider functional measurement error models where the measurement error distribution is estimated non parametrically we derive a locally efficient semiparametric estimator but propose not to implement it owing to its numerical complexity instead a plug in estimator is proposed where the measurement error distribution is estimated through non parametric kernel methods based on multiple measurements the root n consistency and asymptotic normality of the plug in estimator are derived despite the theoretical inefficiency of the plug in estimator simulations demonstrate its near optimal performance computational advantages relative to the theoretically efficient estimator make the plug in estimator practically appealing application of the estimator is illustrated by using the framingham data example

a unified approach to regression analysis under double sampling designs we propose a unified approach to the estimation of regression parameters under double sampling designs in which a primary sample consisting of data on the rough or proxy measures for the response and or explanatory variables as well as a validation subsample consisting of data on the exact measurements are available we assume that the validation sample is a simple random subsample from the primary sample our proposal utilizes a specific parametric model to extract the partial information contained in the primary sample the resulting estimator is consistent even if such a model is misspecified and it achieves higher asymptotic efficiency than the estimator based only on the validation data specific cases are discussed to illustrate the application of the estimator proposed

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

a nonparametric method for dealing with mismeasured covariate data abstract mismeasurement of covariate data is a frequent problem in statistical data analysis however when true and mismeasured data are obtained for a subsample of the observations it is possible to estimate the parameters relating the outcome to the covariate of interest maximum likelihood methods that rely on parametric models for the mismeasurement have not met with much success realistic models for the mismeasurement process are difficult to construct the form of the likelihood is often intractable and more important such methods are not robust to model misspecification we propose an easily implemented method that is nonparametric with respect to the mismeasurement process and that is applicable when mismeasurement is due to the problem of missing data errors in variables or use of imperfect surrogate covariates specifically denote the outcome variable by y the covariate data subject to mismeasurement by x and the remaining covariates including perhaps surrogates or mismeasured values

regression analysis with missing covariate data using estimating equations in regression analysis missing covariate data has been among the most common problems frequently practitioners adopt the so called complete case analysis i e performing the analysis on only a complete dataset after excluding records with missing covariates performing a complete case analysis is convenient with existing statistical packages but it may be inefficient since the observed outcomes and covariates on those records with missing covariates are not used it can even give misleading statistical inference if missing is not completely at random this paper introduces a joint estimating equation jee for regression analysis in the presence of missing observations on one covariate which may be thought of as a method in a general framework for the missing covariate data problem proposed by robins rotnitzky and zhao 1994 journal of the american statistical association 89 846 866 a generalization of jee to more than one such covariate is discussed the jee is generally applicable to estimating regression coefficients from a regression model including linear and logistic regression provided that the missing covariate data is either missing completely at random or missing at random in addition to mild regularity conditions estimates of regression coefficients from the jee are consistent and have an asymptotic normal distribution simulation results show that the asymptotic distribution of estimated coefficients performs well in finite samples also shown through the simulation study is that the validity of jee estimates depends on the correct specification of the probability function that characterizes the missing mechanism suggesting a need for further research on how to robustify the estimation from making this nuisance assumption finally the jee is illustrated with an application from a case control study of diet and thyroid cancer

approximate quasi likelihood estimation in models with surrogate predictors abstract we consider quasi likelihood estimation with estimated parameters in the variance function when some of the predictors are measured with error we review and extend four approaches to estimation in this problem all of them based on small measurement error approximations a taxonomy of the data sets likely to be available in measurement error studies is developed an asymptotic theory based on this taxonomy is obtained and includes measurement error and berkson error models as special cases

some remarks on the asymptotic behaviour of the solutions of second order evolution equations the aim of this paper is to continue the study initiated in 9 concerning the asymptotic behaviour as t co of the solution of the boundary value problem where a is a maximal monotone operator on a real hilbert space h and u0 belongs to the domain of a as shown in lo the solution of 0 1 does not converge in general as t co even if a is the gradient of a c convex function however it was proved in 9 that in many important cases e g operators that satisfy the convergence condition of pazy the solution converges in the second section of the paper we introduce a general condition on the operator a which guarantees the convergence of the solution and which is independent of pazy s condition the main hypothesis on the operator a is a generalization of a previous one used by bruck 6 in the study of an averaging iteration scheme associated to monotone operators in view of the local character of the condition it is natural to expect that our result applies also to some cases of perturbation of odd operators but at the present we have no results in this direction section 3 contains a refining of theorem 2 1 of 9 it concern the existence of solutions which have compact support or more generally solutions which are constant after a finite time essentially we prove that if the operator a satisfies the uniform con vergence condition with gauge function p s cs see definition 3 1 where c o and 0 tl 2 then the solution of problem 0 1 is constant 211

an augmented probit model for missing predictable covariates in quantal bioassay with small sample size summary quantal bioassay experiments relate the amount or potency of some compound for example poison antibody or drug to a binary outcome such as death or infection in animals for infectious diseases probit regression is commonly used for inference and a key measure of potency is given by the idp the amount that results in p of the animals being infected in some experiments a validation set may be used where both direct and proxy measures of the dose are available on a subset of animals with the proxy being available on all the proxy variable can be viewed as a messy reflection of the direct variable leading to an errors in variables problem we develop a model for the validation set and use a constrained seemingly unrelated regression sur model to obtain the distribution of the direct measure conditional on the proxy we use the conditional distribution to derive a pseudo likelihood based on probit regression and use the parametric bootstrap for statistical inference we re evaluate an old experiment in 21 monkeys where neutralizing antibodies nabs to hiv were measured using an old proxy assay in all monkeys and with a new direct assay in a validation set of 11 who had sufficient stored plasma using our methods we obtain an estimate of the id1 for the new assay an important target for hiv vaccine candidates in simulations we compare the pseudo likelihood estimates with regression calibration and a full joint likelihood approach

convergence of solutions to a second order difference inclusion abstract we investigate the asymptotic behavior of solutions to the following second order difference inclusion u i 1 1 i u i i u i 1 c i a u i i 1 u 0 x sup i 0 u i where a is a maximal monotone operator in a real hilbert space h and c i and i are positive real sequences we show weak and strong convergence of solutions to an element of a 1 0 for general maximal monotone operator and when a where is a convex proper and lower semicontinuous function our results extend and improve previous results by morosanu 1979 18 mitidieri morosanu 1985 86 23 see also morosanu 1988 1 pp 156 168 and aperutesei 2003 21 22 as well as some recent works of djafari rouhani khatibzadeh 2010 2011 25 26 and khatibzadeh 2011 27 with more general assumptions on parameters c i and i in homogeneous case

asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations abstract by using previous results of djafari rouhani b djafari rouhani ergodic theorems for nonexpansive sequences in hilbert spaces and related problems ph d thesis yale university 1981 part i pp 1 76 b djafari rouhani asymptotic behaviour of quasi autonomous dissipative systems in hilbert spaces j math anal appl 147 1990 465 476 b djafari rouhani asymptotic behaviour of almost nonexpansive sequences in a hilbert space j math anal appl 151 1990 226 235 for dissipative systems we study the weak and strong convergence of solutions to the following class of second order nonhomogeneous evolution equations u t c u t a u t f t a e t 0 u 0 u 0 sup t 0 u t where a is a monotone operator in a real hilbert space h c 0 and f r h is a given function our results extend previous results by morosanu g morosanu nonlinear evolution equations and applications editura academiei romane and d reidel publishing company bucharest 1988 g morosanu asymptotic behaviour of solutions of differential equations associated to monotone operators nonlinear anal 3 1979 873 883 and mitidieri e mitidieri some remarks on the asymptotic behaviour of the solutions of second order evolution equations j math anal appl 107 1985 211 221 who studied the case c 0 and f 0 by assuming that a is maximal monotone and a 1 0 as well as previous results by veron l veron problemes d evolution du second ordre associes a des operateurs monotones c r acad sci paris ser a 278 1974 1099 1101 l veron equations d evolution du second ordre associees a des operateurs maximaux monotones proc roy soc edinburgh sect a 75 1975 76 131 147 and the authors b djafari rouhani h khatibzadeh asymptotic behavior of solutions to some homogeneous second order evolution equations of monotone type j c inequal appl 2007 art id 72931 8 pp b djafari rouhani h khatibzadeh asymptotic behavior of bounded solutions to some second order evolution systems rocky mountain j math in press who studied the case f 0 some applications are also presented

an estimated likelihood method for continuous outcome regression models with outcome dependent sampling many biomedical observational studies attempt to relate a continuous outcome to an environmental exposure and other important covariates if the outcome is easier or cheaper to measure relative to the exposure of interest then the outcome may be observed for every member of a finite study population whereas exposure measurements may be obtained only for a relatively small subsample of this population rather than selecting a simple random subsample of individuals for exposure measurement investigators may attempt to enhance study efficiency by allowing the selection probabilities to depend on the observed outcomes we refer to such sampling schemes as outcome dependent sampling ods standard estimation methods that ignore the ods design will yield biased and inconsistent parameter estimates furthermore it is generally desirable to use estimators that incorporate all available data as analyses restricted to subjects with complete information are inefficient to this end we extend an estimated likel

empirical likelihood based inference in linear errors in covariables models with validation data summary linear errors in covariables models are considered assuming the availability of independent validation data on the covariables in addition to primary data on the response variable and surrogate covariables we first develop an estimated empirical loglikelihood with the help of validation data and prove that its asymptotic distribution is that of a weighted sum of independent standard z2 random variables with unknown weights by estimating the unknown weights consistently we construct an estimated empirical likelihood confidence region for the regression parameter vector we also suggest an adjusted empirical loglikelihood and prove that its asymptotic distribution is a standard x2 to avoid estimating the unknown weights or the adjustment factor we propose a partially smoothed bootstrap empirical loglikelihood for constructing a confidence region which has asymptotically correct coverage probability a simulation study is conducted to compare the proposed methods with a method based on a normal approximation in terms of coverage accuracy and average length of the confidence interval

a note on the strong convergence of solutions to a second order evolution equation abstract in this note with suitable assumptions on p t and r t we establish the strong convergence of solutions to the following nonlinear second order evolution equation p t u t r t u t a u t a e on r u 0 u 0 sup t 0 u t to a zero of a without assuming a 1 0 0 and determine the rate of convergence where a is a general maximal monotone operator we present also some applications of our result

a bayesian approach to logistic regression models having measurement error following a mixture distribution to estimate the parameters in a logistic regression model when the predictors are subject to random or systematic measurement error we take a bayesian approach and average the true logistic probability over the conditional posterior distribution of the true value of the predictor given its observed value we allow this posterior distribution to consist of a mixture when the measurement error distribution changes form with observed exposure we apply the method to study the risk of alcohol consumption on breast cancer using the nurses health study data we estimate measurement error from a small subsample where we compare true with reported consumption some of the self reported non drinkers truly do not drink the resulting risk estimates differ sharply from those computed by standard logistic regression that ignores measurement error

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

generating iterated function systems of some planar self similar sets abstract suppose e e is a planar self similar set where e r is a self similar set satisfying the strong separation condition with 0 dim h e 1 by revealing a relationship between algebraic numbers cos 2 l and sin 2 l with l z we prove that for all generating iterated function systems ifss of e e the associated orthogonal matrices have at most 8 possibilities additionally we investigate all generating ifss of two typical planar self similar sets one has a minimal presentation every other generating ifs is an iteration of this one while the other has not

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

causal inference in outcome dependent two phase sampling designs summary we consider estimation of the causal effect of a treatment on an outcome from observational data collected in two phases in the first phase a simple random sample of individuals is drawn from a population on these individuals information is obtained on treatment outcome and a few low dimensional covariates these individuals are then stratified according to these factors in the second phase a random subsample of individuals is drawn from each stratum with known stratum specific selection probabilities on these individuals a rich set of covariates is collected in this setting we introduce five estimators simple inverse weighted simple doubly robust enriched inverse weighted enriched doubly robust locally efficient we evaluate the finite sample performance of these estimators in a simulation study we also use our methodology to estimate the causal effect of trauma care on in hospital mortality by using data from the national study of cost and outcomes of trauma

random effects logistic regression analysis with auxiliary covariates summary we study a semiparametric estimation method for the random effects logistic regression when there is auxiliary covariate information about the main exposure variable we extend the semiparametric estimator of pepe and fleming 1991 journal of the american statistical association86 108 113 to the random effects model using the best linear unbiased prediction approach of henderson 1975 biometrics31 423 448 the method can be used to handle the missing covariate or mismeasured covariate data problems in a variety of real applications simulation study results show that the proposed method outperforms the existing methods we analyzed a data set from the collaborative perinatal project using the proposed method and found that the use of ddt increases the risk of preterm births among us children

semiparametric methods for response selective and missing data problems in regression summary suppose that data are generated according to the model f ylx 0 g x where y is a response and x are covariates we derive and compare semiparametric likelihood and pseudolikelihood methods for estimating 0 for situations in which units generated are not fully observed and in which it is impossible or undesirable to model the covariate distribution the probability that a unit is fully observed may depend on y and there may be a subset of covariates which is observed only for a subsample of individuals our key assumptions are that the probability that a unit has missing data depends only on which of a finite number of strata that y x belongs to and that the stratum membership is observed for every unit applications include case control studies in epidemiology field reliability studies and broad classes of missing data and measurement error problems our results make fully efficient estimation of 0 feasible and they generalize and provide insight into a variety of methods that have been proposed for specific problems

conditional independence models for epidemiological studies with covariate measurement error we construct a unifying representation of the structure of measurement error problems with particular reference to situations commonly encountered in epidemiological studies and outline how estimation of the parameters of interest can be carried out in a bayesian framework using gibbs sampling we show how this approach can be implemented for designs involving continuous measurement errors assessed through a validation substudy and discuss our results on simulated data

self similar structure on intersections of triadic cantor sets abstract let c be the triadic cantor set we characterize the all real number such that the intersection c c is a self similar set and investigate the form and structure of the all iterated function systems which generate the self similar set

efficient regression calibration for logistic regression in main study internal validation study designs with an imperfect reference instrument an extension to the version of the regression calibration estimator proposed by rosner et al for logistic and other generalized linear regression models is given for main study internal validation study designs this estimator combines the information about the parameter of interest contained in the internal validation study with rosner et al s regression calibration estimate using a generalized inverse variance weighted average it is shown that the validation study selection model can be ignored as long as this model is jointly independent of the outcome and the incompletely observed covariates conditional at most upon the surrogates and other completely observed covariates in an extensive simulation study designed to follow a complex multivariate setting in nutritional epidemiology it is shown that with validation study sizes of 340 or more this estimator appears to be asymptotically optimal in the sense that it is nearly unbiased and nearly as efficient as a properly specified maximum likelihood estimator a modification to the regression calibration variance estimator which replaces the standard uncorrected logistic regression coefficient variance with the sandwich estimator to account for the possible misspecification of the logistic regression fit to the surrogate covariates in the main study was also studied in this same simulation experiment in this study the alternative variance formula yielded results virtually identical to the original formula a version of the proposed estimator is also derived for the case where the reference instrument available only in the validation study is imperfect but unbiased at the individual level and contains error that is uncorrelated with other covariates and with error in the surrogate instrument replicate measures are obtained in a subset of study participants in this case it is shown that the validation study selection model can be ignored when sampling into the validation study depends at most only upon perfectly measured covariates two data sets a study of fever in relation to occupational exposure to antineoplastics among hospital pharmacists and a study of breast cancer incidence in relation to dietary intakes of alcohol and vitamin a adjusted for total energy intake from the nurses health study were analysed using these new methods in these data because the validation studies contained less than 200 observations and the events of interest were relatively rare as is typical the potential improvements offered by this new estimator were not apparent

a semiparametric empirical likelihood method for biased sampling schemes with auxiliary covariates we consider a semiparametric inference procedure for data from epidemiologic studies conducted with a two component sampling scheme where both a simple random sample and multiple outcome or outcome auxiliary dependent samples are observed this sampling scheme allows the investigators to oversample certain subpopulations believed to have more information about the regression model while still gaining insights about the underlying population through the simple random sample we focus on settings where there is no additional information about the parent cohort and the sampling probability is nonidentifiable we motivate our problem with an ongoing study to assess the association between the mutation level of epidermal growth factor receptor egfr and the antitumor response to egfr targeted therapy among nonsmall cell lung cancer patients the proposed method applies to both binary and multicategorical outcome data and allows an arbitrary link function in the framework of generalized linear models simulation studies show that the proposed estimator has nice small sample properties the proposed method is illustrated with a data example

a strong convergence theorem for solutions to a nonhomogeneous second order evolution equation abstract in this paper we establish the strong convergence of possible solutions to the following nonhomogeneous second order evolution system u t c u t a u t f t a e t 0 u 0 u 0 sup t 0 u t to an element of a 1 0 with an exponential rate of convergence when f 0 where a is a general maximal monotone operator in a real hilbert space h c 0 is a real constant and f r h is a given function we show also that the curve u is almost nonexpansive and present some applications of our result

estimation of linear and nonlinear errors in variables models using validation data abstract consistent estimators for linear and nonlinear regression models with measurement errors in variables in the presence of validation data are proposed the estimation procedures are based on least squares methods with regression functions replaced by wide sense conditional expectation functions the methods do not depend on distributional assumptions and are robust against the misspecification of a measurement error model they are computationally and analytically simpler than semiparametric methods based on nonparametric regression or density functions

prospective analysis of logistic case control studies abstract in a classical case control study prentice and pyke proposed to ignore the study design and instead base estimation and inference on a random sampling i e prospective formulation we generalize this prospective formulation of case control studies to include multiplicative models stratification missing data measurement error robustness and other examples the resulting estimators which ignore the case control study aspect and instead are based on a random sampling formulation are typically consistent for nonintercept parameters and are asymptotically normally distributed we derive the resulting asymptotic covariance matrix of the parameter estimates the covariance matrix obtained by ignoring the case control sampling scheme and using prospective formulas instead is shown to be at worst asymptotically conservative and asymptotically correct in a variety of problems a simple sufficient condition guaranteeing the latter is obtained

estimation from truncated lifetime data with supplementary information on covariates and censoring times summary in epidemiology reliability and other areas one encounters situations where response times and covariates for individuals in a population are observed only if the response time does not exceed an associated censoring time methods for obtaining supplementary information about covariates and censoring times for nonresponding individuals are considered estimation procedures are developed for the resulting combination of truncated data and supplementary data theoretical and simulation results are given and the methodology illustrated on some automobile failure data collected under a warranty plan

approximations for regression with covariate measurement error abstract the effect of covariate measurement error on maximum likelihood and maximum quasilikelihood estimates mle s or mqle s of regression parameters is considered the naive estimates obtained by ignoring error are modified to account for it the modified estimates can be computed explicitly from the naive ones and have less bias for small error they are particularly useful for berkson error models examples illustrate their application to linear and nonlinear regression errors in continuous covariates and misclassification of discrete covariates the estimates are applied to data relating air pollution to respiratory disease in children analytic methods and simulations are used to compare the estimates to others that have been proposed for dealing with covariate measurement errors the basic idea is that the mle or mqle of the parameter is a function of the size of the error when 0 and there is no error is the usual mle or mqle for small an improvement over is in many cases of

a simple estimator for nonlinear error in variable models we propose a simple estimator for nonlinear method of moment models with measurement error of the classical type when no additional data such as validation data or double measurements are available we assume that the marginal distributions of the measurement errors are laplace double exponential with zero means and unknown variances and the measurement errors are independent of the latent variables and are independent of each other under these assumptions we derive simple revised moment conditions in terms of the observed variables they are used to make inference about the model parameters and the variance of the measurement error the results of this paper show that the distributional assumption on the measurement errors can be used to point identify the parameters of interest our estimator is a parametric method of moments estimator that uses the revised moment conditions and hence is simple to compute our estimation method is particularly useful in situations where no additional data are available which is the case in many economic data sets simulation study demonstrates good finite sample properties of our proposed estimator we also examine the performance of the estimator in the case where the error distribution is misspecified

semiparametric estimation in logistic measurement error models summary we describe semiparametric estimation and inference in a logistic regression model with measurement error in the predictors the particular measurement error model consists of a primary data set in which only the response y and a fallible surrogate w of the true predictor x are observed plus a smaller validation data set for which y x w are observed except for the underlying assumption of a logistic model in the true predictor no parametric distributional assumption is made about the true predictor or its surrogate we develop a semiparametric parameter estimate of the logistic regression parameter which is asymptotically normally distributed and computationally feasible the estimate relies on kernel regression techniques for scalar predictors by a detailed analysis of the mean squared error of the parameter estimate we obtain a representation for an optimal bandwidth

covariate measurement error in generalized linear models summary the em algorithm is used to obtain estimators of regression coefficients for generalized linear models with canonical link when normally distributed covariates are masked by normally distributed measurement errors by casting the true covariates as missing data the em procedure suggests an iterative scheme in which each cycle consists of an e step requiring the computation of approximate first and second conditional moments of the true covariates given the observed data followed by an m step in which regression parameters are updated by iteratively reweighted least squares based on these approximations the proposed procedure is compared numerically with the exact maximum likelihood solution obtained by using gaussian quadrature instead of the approximations in the e step of the em algorithm and with alternative estimators for simple logistic regression with measurement error the results for the proposed procedure are encouraging a procedure is proposed in this paper for estimating regression coefficients in generalized linear models when one or more of the covariates is measured with error this work is related to the paper by carroll et al 1984 in which maximum likelihood estimates are considered for the structural logistic regression model these authors provide estimates for the computationally simpler probit regression model and suggest that the structural logistic regression can be done in principle this paper follows up on that suggestion but differs in detail from similar work by armstrong 1985 other procedures with related aims have been suggested by stefanski carroll 1985 wolter fuller 1982 and prentice 1982 for a quick introduction to the proposed method it is convenient to display the naive estimator which ignores measurement error in its standard computational form if yi is an observed response variable and xi is the observed covariate then for the model which equates the canonical parameter of the distribution of yi to xtf3 the maximum likelihood estimator of f3 can be obtained by iteratively reweighted least squares the estimate of f3 after s 1 cycles is given by

modelling marginal hazards in multivariate failure time data we consider the problem of estimating relative risk parameters in the marginal distribution of failure times from a sample in which individuals are grouped into clusters and failure times for individuals within clusters may be dependent we assume that the marginal hazard of failure for individuals within a cluster is specified by a relative risk regression model and the structure of dependence between individuals within a cluster is left unspecified estimating equations for the regression parameters and an estimator for the cumulative base line marginal hazard are described

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

asymptotic behavior of bounded solutions to a nonhomogeneous second order evolution equation of monotone type abstract by using previous results of djafari rouhani b djafari rouhani ergodic theorems for nonexpansive sequences in hilbert spaces and related problems ph d thesis yale university 1981 part i pp 1 76 b djafari rouhani asymptotic behaviour of quasi autonomous dissipative systems in hilbert spaces j math anal appl 147 1990 465 476 b djafari rouhani asymptotic behaviour of almost nonexpansive sequences in a hilbert space j math anal appl 151 1990 226 235 for dissipative systems we study the asymptotic behavior of solutions to the following system of second order nonhomogeneous evolution equations u t a u t f t a e t 0 u 0 u 0 sup t 0 u t where a is a maximal monotone operator in a real hilbert space h and f r h is a suitable function we investigate weak and strong convergence theorems for solutions to this system our results extend previous results of morosanu g morosanu nonlinear evolution equations and applications editura academiei romane and d reidel publishing company bucharest 1988 g morosanu asymptotic behaviour of solutions of differential equations associated to monotone operators nonlinear anal 3 1979 873 883 and mitidieri e mitidieri some remarks on the asymptotic behaviour of the solutions of second order evolution equations j math anal appl 107 1985 211 221 who studied the case f 0 by assuming that a 1 0 as well as previous results of the authors b djafari rouhani h khatibzadeh asymptotic behavior of bounded solutions to some second order evolution systems rocky mountain j math in press who studied the homogeneous case without this additional assumption

a semiparametric correction for attenuation abstract a correction method is proposed for models including the generalized linear model when the covariate is measured with error the method requires a separate validation data set that consists of the surrogate w and the true covariate x or an unbiased estimate x of x we do not require the classical additive measurement error model in which the surrogate is unbiased for the true covariates we first obtain an estimate of e x w by using nonparametric kernel regression of x or x on w based on the validation data then we perform a standard analysis with the unknown x replaced by the estimate of e x w the asymptotic distribution of the resulting regression parameter estimator is obtained generalizations to include components of x measured without error are also discussed

case control studies with errors in covariates we devise methods for estimating the parameters of a prospective logistic model with dichotomous response d and arbitrary covariates x from case control data when these covariates are measured with error we suppose that some fraction of the cases and controls provide only the error prone covariate measurements w the incomplete or reduced data whereas some of the cases and controls provide measurements on x and w the complete data we assume a measurement error density with a finite set of parameters namely fw xd w x d and nondifferential error is treated as a special case of this model fw x w x our algorithm estimates both the logistic parameters and from a pseudolikelihood because empirical distribution functions are used in place of needed distributions in the pseudolikelihoods the required asymptotic theory is more elaborate than for pseudolikelihoods based on substitution for a finite number of nuisance parameters we also examine computationally simpler methods under t

asymptotic behavior of bounded solutions to a class of second order nonhomogeneous difference equations of monotone type abstract as a continuation of our previous work in djafari rouhani and khatibzadeh 2008 1 we investigate the asymptotic behavior of solutions to the following system of second order nonhomogeneous difference equations u n 1 1 n u n n u n 1 c n a u n f n n 1 u 0 a h sup n 0 u n where a is a maximal monotone operator in a real hilbert space h c n and n are positive real sequences and f n is a sequence in h with suitable conditions on a and the sequences c n n and f n we show the weak or strong convergence of u n or its weighted average to an element of a 1 0 which is also the asymptotic center of the sequence u n implying therefore in particular that the existence of a solution u n implies that a 1 0 0 our results extend some previous results by apreutesei 2007 2003 2003 13 23 24 morosanu 1988 1979 4 20 and mitidieri and morosanu 1985 86 31 whose proofs use the assumption a 1 0 0 as well as the authors djafari rouhani and khatibzadeh 2008 1 as mentioned there in the section on future directions to the nonhomogeneous case with n 1 we also present some applications of our results to optimization

on the structures of generating iterated function systems of cantor sets abstract a generating ifs of a cantor set f is an ifs whose attractor is f for a given cantor set such as the middle 3rd cantor set we consider the set of its generating ifss we examine the existence of a minimal generating ifs i e every other generating ifs of f is an iterating of that ifs we also study the structures of the semi group of homogeneous generating ifss of a cantor set f in r under the open set condition osc if dim h f 1 we prove that all generating ifss of the set must have logarithmically commensurable contraction factors from this logarithmic commensurability theorem we derive a structure theorem for the semi group of generating ifss of f under the osc we also examine the impact of geometry on the structures of the semi groups several examples will be given to illustrate the difficulty of the problem we study

existence results for second order monotone differential inclusions on the positive half line abstract consider in a real hilbert space h the differential equation inclusion e p t u t q t u t a u t f t a e in 0 with the condition b u 0 x d a where a d a h h is a possibly set valued maximal monotone operator whose range contains 0 p q l 0 with ess inf p 0 and q l 1 0 more than four decades ago v barbu established the existence of a unique bounded solution to e b in the particular case p 1 q 0 and f 0 subsequently the existence and uniqueness of bounded solutions in the homogeneous case f 0 have been further investigated by h brezis 1972 n pavel 1976 l veron 1974 1976 and by e i poffald and s reich 1986 when a is an m accretive operator in a banach space the non homogeneous case has received less attention from this point of view in this paper we prove existence and uniqueness of bounded solutions to e b in the general case of non constant functions p q satisfying the mild conditions above thus compensating for the lack of existence theory for such kind of second order problems note that our results open up the possibility to apply lions method of artificial viscosity towards approximating the solutions of some nonlinear parabolic and hyperbolic problems as shown in the last section of the paper

covariate measurement error in logistic regression abstract in a logistic regression model when covariates are subject to measurement error the naive estimator obtained by regressing on the observed covariates is asymptotically biased this document introduces a bias adjusted estimator and two estimators appropriate for normally distributed measurement errors a functional maximum likelihood estimator and an estimator which exploits the consequences of sufficiency the four proposals are studied asymptotically under conditions which are appropriate when the measurement error is small a small monte carlo study illustrates the superiority of the measurement error estimators in certain situations additional keywords mathematical models

semiparametric efficient estimation of a conditional density with missing or mismeasured covariates summary pepe and fleming and carroll and wand have recently proposed estimators in a parametric model for the density of a random variable y conditional on a vector of covariates x v when data on one of the regressors x is missing for some study subjects we propose a new class of estimators that remains consistent and asymptotically normal even when the probability that x is missing depends on the observed v and y includes an estimator whose asymptotic variance attains the semiparametric variance bound for the model and when the data are missing completely at random includes an estimator that is asymptotically equivalent to the inefficient estimators proposed by pepe and fleming and by carroll and wand the optimal estimator in our class depends on the unknown probability law generating the data when the vector v of non missing regressors has at most two continuous components we propose an adaptive semiparametric efficient estimator and compare the performance of the proposed semiparametric efficient estimator with the estimators proposed by pepe and fleming and carroll and wand in a small simulation study when v has many continuous components we propose an alternative class of adaptive estimators that should have high efficiency

auxiliary covariate data in failure time regression summary we consider the problem of missing covariate data in the context of censored failure time relative risk regression auxiliary covariate data which are considered informative about the missing data but which are not explicitly part of the relative risk regression model may be available full covariate information is available for a validation set an estimated partial likelihood method is proposed for estimating relative risk parameters this method is an extension of the estimated likelihood regression analysis method for uncensored data pepe 1992 pepe fleming 1991 a key feature of the method is that it is nonparametric with respect to the association between the missing and observed including auxiliary covariate components asymptotic distribution theory is derived for the proposed estimated partial likelihood estimator in the case where the auxiliary or mismeasured covariates are categorical asymptotic efficiencies are calculated for exponential failure times using an exponential relative risk model the estimated partial likelihood estimator compares favourably with a fully parametric maximum likelihood analysis comparisons are also made with a standard partial likelihood analysis which ignores the incomplete observations important efficiency gains can be made with the estimated partial likelihood method small sample properties are investigated through simulation studies

lipschitz equivalence of self similar sets and hyperbolic boundaries abstract kaimanovich 2003 9 introduced the concept of augmented tree on the symbolic space of a self similar set it is hyperbolic in the sense of gromov and it was shown by lau and wang 2009 12 that under the open set condition a self similar set can be identified with the hyperbolic boundary of the tree in the paper we investigate in detail a class of simple augmented trees and the lipschitz equivalence of such trees the main purpose is to use this to study the lipschitz equivalence problem of the totally disconnected self similar sets which has been undergoing some extensive development recently

measurement error regression with unknown link dimension reduction and data visualization abstract a general nonlinear regression problem is considered with measurement error in the predictors we assume that the response is related to an unknown linear combination of a multidimensional predictor through an unknown link function instead of observing the predictor we instead observe a surrogate with the property that its expectation is linearly related to the true predictor with constant variance we identify an important transformation of the surrogate variable using this transformed variable we show that if one proceeds with the usual analysis ignoring measurement error then both ordinary least squares and sliced inverse regression yield estimates which consistently estimate the true regression parameter up to a constant of proportionality we derive the asymptotic distribution of the estimates a simulation study is conducted applying sliced inverse regression in this context

locally ancillary quasi score models for errors in covariates we use the notion of locally ancillary estimating functions to develop a quasi score method for fitting regression models containing measurement error in the covariates suppose that interest is on the model e y u w for response y the observed data are y x w and x is a mismeasured surrogate for u we take a functional modeling approach treating the u as a fixed nuisance parameter beginning with quasi scores for the regression parameter and the unknown u we derive a bias corrected quasi score for the regression parameter that is second order locally ancillary for the nuisance u our method for this requires only the correct specification of the mean and variance functions for y and x in terms of u w and the regression parameter when an estimator for u is plugged into the corrected quasi score local approximations show that the bias is small we present simulations verifying this result and an example from child psychiatry both using log linear regression models

new results on the asymptotic behavior of solutions to a class of second order nonhomogeneous difference equations abstract we investigate the asymptotic behavior of solutions to the following system of second order nonhomogeneous difference equation u n 1 1 n u n n u n 1 c n a u n f n n 1 u 0 x sup n 0 u n where a is a maximal monotone operator in a real hilbert space h c n and n are positive real sequences and f n is a sequence in h we show the weak and strong convergence of solutions and their weighted averages to an element of a 1 0 which is the asymptotic center of the sequence u n under appropriate assumptions on the sequences c n n and f n our results continue our previous work in djafari rouhani and khatibzadeh 2008 2010 30 31 by presenting some new results on the asymptotic behavior of solutions including in particular a completely new strong convergence result and extend some previous results by apreutesei 2003 27 28 morosanu 1979 21 and mitidieri and morosanu 1985 86 22 to the nonhomogeneous case and without assuming a to have a nonempty zero set

logistic regression with incompletely observed categorical covariates investigating the sensitivity against violation of the missing at random assumption missing values in the covariates are a widespread complication in the statistical inference of regression models the maximum likelihood principle requires specification of the distribution of the covariates at least in part for categorical covariates log linear models can be used additionally the missing at random assumption is necessary which excludes a dependence of the occurrence of missing values on the unobserved covariate values this assumption is often highly questionable we present a framework to specify alternative missing value mechanisms such that maximum likelihood estimation of the regression parameters under a specified alternative is possible this allows investigation of the sensitivity of a single estimate against violations of the missing at random assumption the possible results of a sensitivity analysis are illustrated by artificial examples the practical application is demonstrated by the analysis of two case control studies

semiparametric estimation exploiting covariate independence in two phase randomized trials recent results for case control sampling suggest when the covariate distribution is constrained by gene environment independence semiparametric estimation exploiting such independence yields a great deal of efficiency gain we consider the efficient estimation of the treatment biomarker interaction in two phase sampling nested within randomized clinical trials incorporating the independence between a randomized treatment and the baseline markers we develop a newton raphson algorithm based on the profile likelihood to compute the semiparametric maximum likelihood estimate spmle our algorithm accommodates both continuous phase one outcomes and continuous phase two biomarkers the profile information matrix is computed explicitly via numerical differentiation in certain situations where computing the spmle is slow we propose a maximum estimated likelihood estimator mele which is also capable of incorporating the covariate independence this estimated likelihood approach uses a one step empirical covariate distribution thus is straightforward to maximize it offers a closed form variance estimate with limited increase in variance relative to the fully efficient spmle our results suggest exploiting the covariate independence in two phase sampling increases the efficiency substantially particularly for estimating treatment biomarker interactions

a mean score method for missing and auxiliary covariate data in regression models summary we consider regression analysis when incomplete or auxiliary covariate data are available for all study subjects and in addition for a subset called the validation sample true covariate data of interest have been ascertained the term auxiliary data refers to data not in the regression model but thought to be informative about the true missing covariate data of interest we discuss a method which is nonparametric with respect to the association between available and missing data allows missingness to depend on available response and covariate values and is applicable to both cohort and case control study designs the method previously proposed by flanders greenland 1991 and by zhao lipsitz 1992 is generalised and asymptotic theory is derived our expression for the asymptotic variance of the estimator provides intuition regarding performance of the method optimal sampling strategies for the validation set are also suggested by the asymptotic results