cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

misspecified proportional hazard models summary the properties of an estimator based on a proportional hazards model are investigated when the model is incorrect the estimator from the partial likelihood is shown to be consistent for a parameter that is defined implicitly the results are used to investigate the effects on estimation if the true model is accelerated failure time or if covariates are omitted from the proportional hazards model

semiparametric regression calibration for general hazard models in survival analysis with covariate measurement error surprising performance under linear hazard observational epidemiological studies often confront the problem of estimating exposure disease relationships when the exposure is not measured exactly regression calibration rc is a common approach to correct for bias in regression analysis with covariate measurement error in survival analysis with covariate measurement error it is well known that the rc estimator may be biased when the hazard is an exponential function of the covariates in the paper we investigate the rc estimator with general hazard functions including exponential and linear functions of the covariates when the hazard is a linear function of the covariates we show that a risk set regression calibration rrc is consistent and robust to a working model for the calibration function under exponential hazard models there is a trade off between bias and efficiency when comparing rc and rrc however one surprising finding is that the trade off between bias and efficiency in measurement error research is not seen under linear hazard when the unobserved covariate is from a uniform or normal distribution under this situation the rrc estimator is in general slightly better than the rc estimator in terms of both bias and efficiency the methods are applied to the nutritional biomarkers study of the women s health initiative

negative order mkdv hierarchy and a new integrable neumann like system abstract the purpose of this paper is to develop the negative order mkdv hierarchy and to present a new related integrable neumann like hamiltonian flow from the view point of inverse recursion operator and constraint method the whole mkdv hierarchy both positive and negative is generated by the kernel elements of lenard s operators pair and recursion operator through solving a key operator equation the whole mkdv hierarchy is shown to have the lax representation in particular some new integrable equation together with the liouville equations the sine gordon equation and the sinh gordon equation are derived from the negative order mkdv hierarchy it is very interesting that the restricted flow corresponding to the negative order mkdv hierarchy is just a new kind of neumann like system this new neumann like system is obtained through restricting the mkdv spectral problem onto a symplectic submanifold and is proven to be completely integrable under the dirac poisson bracket which we define on the symplectic submanifold finally with the help of the constraint between the neumann like system and the negative order mkdv hierarchy all equations in the hierarchy are proven to have the parametric representations of solutions in particular we obtain the parametric solutions of the sine gordon equation and the sinh gordon equation

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

proportional hazards model with covariates subject to measurement error when covariates of a proportional hazards model are subject to measurement error the maximum likelihood estimates of regression coefficients based on the partial likelihood are asymptotically biased prentice 1982 biometrika 69 331 342 presents an example of such bias and suggests a modified partial likelihood this paper applies the corrected score function method nakamura 1990 biometrika 77 127 137 to the proportional hazards model when measurement errors are additive and normally distributed the result allows a simple correction to the ordinary partial likelihood that yields asymptotically unbiased estimates the validity of the correction is confirmed via a limited simulation study

failure time regression with continuous covariates measured with error summary we consider failure time regression analysis with an auxiliary variable in the presence of a validation sample we extend the nonparametric inference procedure of zhou and pepe to handle a continuous auxiliary or proxy covariate we estimate the induced relative risk function with a kernel smoother and allow the selection probability of the validation set to depend on the observed covariates we present some asymptotic properties for the kernel estimator and provide some simulation results the method proposed is illustrated with a data set from an on going epidemiologic study

new set of symmetries of the integrable equations lie algebra and non isospectral evolution equations ii akns system the authors define directly a new set of symmetries for the akns system and prove that they constitute an infinite dimensional lie algebra with the old symmetries they also point out the relation between the new symmetry and the non isospectral problem they use the reduction technique and point out that the nls mkdv sg and sinh g hierarchies have two sets of symmetries which constitute an infinite dimensional lie algebra these results are extensions of those of chen et al 1982

propagation of ultra short optical pulses in cubic nonlinear media abstract we derive a partial differential equation that approximates solutions of maxwell s equations describing the propagation of ultra short optical pulses in nonlinear media and which extends the prior analysis of alterman and rauch phys lett a 264 2000 390 diffractive nonlinear geometric optics for short pulses preprint 2002 we discuss non rigorously conditions under which this approximation should be valid but the main contributions of this paper are 1 an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration 2 a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation 3 a local well posedness result for the model equation and 4 a proof that in contrast to the nonlinear schrodinger equation which models slowly varying wavetrains this equation has no smooth pulse solutions which propagate with fixed shape and speed

estimation of dependence between paired correlated failure times in the presence of covariate measurement error summary in many biomedical studies covariates are subject to measurement error although it is well known that the regression coefficients estimators can be substantially biased if the measurement error is not accommodated there has been little study of the effect of covariate measurement error on the estimation of the dependence between bivariate failure times we show that the dependence parameter estimator in the clayton oakes model can be considerably biased if the measurement error in the covariate is not accommodated in contrast with the typical bias towards the null for marginal regression coefficients the dependence parameter can be biased in either direction we introduce a bias reduction technique for the bivariate survival function in copula models while assuming an additive measurement error model and replicated measurement for the covariates and we study the large and small sample properties of the dependence parameter estimator proposed

survival analysis with heterogeneous covariate measurement error this article is motivated by a time to event analysis where the covariate of interest was measured at the wrong time we show that the problem can be formulated as a special case of survival analysis with heterogeneous covariate measurement error and develop a general analytic framework we study the asymptotic behavior of the naive partial likelihood estimates and analytically demonstrate that under the heterogeneous measurement error structure and the assumption that all components of the covariate vector and the measurement error vector combined are mutually independent these naive estimates will shrink toward 0 and that the degree of attenuation increases as the measurement error increases we also give counterexamples for reverse attenuation when the independence conditions are violated we use our analytical results to derive a simple bias correcting estimator that performs well in simulations for small and moderate amounts of measurement error our framework can be used to provide insight into th

proportional hazards regression with missing covariates abstract nonparametric maximum likelihood npml is used to estimate regression parameters in a proportional hazards regression model with missing covariates the npml estimator is shown to be consistent and asymptotically normally distributed under some conditions em type algorithms are applied to solve the maximization problem variance estimates of the regression parameters are obtained by a profile likelihood approach that uses em aided numerical differentiation simulation results indicate that the npml estimates of the regression parameters are more efficient than the approximate partial likelihood estimates and estimates from complete case analysis when missing covariates are missing completely at random and that the proposed method corrects for bias when the missing covariates are missing at random

hazard ratio estimation for biomarker calibrated dietary exposures uncertainty concerning the measurement error properties of self reported diet has important implications for the reliability of nutritional epidemiology reports biomarkers based on the urinary recovery of expended nutrients can provide an objective measure of short term nutrient consumption for certain nutrients and when applied to a subset of a study cohort can be used to calibrate corresponding self report nutrient consumption assessments a non standard measurement error model that makes provision for systematic error and subject specific error along with the usual independent random error is needed for the self report data three estimation procedures for hazard ratio cox model parameters are extended for application to this more complex measurement error structure these procedures are risk set regression calibration conditional score and nonparametric corrected score an estimator for the cumulative baseline hazard function is also provided the performance of each method is assessed in a simulation study the methods are then applied to an example from the women s health initiative dietary modification trial

on a class of physically important integrable equations abstract a methodology introduced by fuchssteiner and the author is used to derive a class of physically important integrable evolution equations among these equations are integrable generalizations of the korteweg devries kdv of the modified kdv of the nonlinear schrodinger nls and of the sine gordon equations the modeling of water waves as well as general asymptotic considerations are used to illustrate the occurrence of the generalized modified kdv and nls equations respectively

the sylvester equation and the elliptic korteweg de vries system the elliptic potential korteweg de vries lattice system is a multi component extension of the lattice potential korteweg de vries equation whose soliton solutions are associated with an elliptic cauchy kernel i e a cauchy kernel on the torus in this paper we generalize the class of solutions by allowing the spectral parameter to be a full matrix obeying a matrix version of the equation of the elliptic curve and for the cauchy matrix to be a solution of a sylvester type matrix equation subject to this matrix elliptic curve equation the construction involves solving the matrix elliptic curve equation by using toeplitz matrix techniques and analysing the solution of the sylvester equation in terms of jordan normal forms furthermore we consider the continuum limit system associated with the elliptic potential korteweg de vries system and analyse the dynamics of the soliton solutions which reveals some new features of the elliptic system in comparison to the non elliptic case

soliton scattering with amplitude changes of a negative order akns equation the paper investigates a nonlinear model which provides soliton scattering with amplitude changes this is a nonlinear equation in negative order akns hierarchy we derive its n soliton solutions through hirota method for dynamics 1 soliton is characterized by an asymmetric shape and time dependent amplitude then by means of asymptotic analysis it is found that two soliton scattering provides not only phase shifts but also amplitude changes for each soliton which differs from classical soliton interactions where only phase shifts appear appendix a gives a tentative discussion on lax integrability and formal hamiltonian structures of the negative order hierarchy and appendix b lists reductions of equations and solutions

covariate measurement errors and parameter estimation in a failure time regression model summary estimation in cox s failure time regression model is considered when the regression vector is subject to measurement error a hazard function model is induced for the failure rate given the measured covariate and a partial likelihood function is derived for the relative risk parameters this partial likelihood function may involve the baseline hazard function as well as the regression parameter but useful inference techniques arise for testing whether the regression parameter equals zero and for more general inferences in important special cases explicit consideration is given to testing equality of survival curves when group membership is subject to misclassification and to relative risk estimation with normally distributed covariates approaches to regression estimation using the overall likelihood function and a marginal likelihood function based on failure time ranks are also indicated illustration of the possible effect of covariate errors on relative risk estimation is provided

the generalized wronskian solutions of a inverse kdv hierarchy abstract a hierarchy of the inverse kdv equation is discussed through the bilinear form of lax pairs we prove a generalized darboux crum theorem of the hierarchy the backlund transformation and the generalized wronskian solutions are presented the soliton solutions explicit rational solutions are obtained then

the robust inference for the cox proportional hazards model abstract we derive the asymptotic distribution of the maximum partial likelihood estimator for the vector of regression coefficients under a possibly misspecified cox proportional hazards model as in the parametric setting this estimator converges to a well defined constant vector in addition the random vector n 1 2 is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated the newly proposed robust covariance matrix estimator is similar to the so called sandwich variance estimators that have been extensively studied for parametric cases for many misspecified cox models the asymptotic limit or part of it can be interpreted meaningfully in those circumstances valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of and the related results for the score statistics extensive studies demonstrate that the proposed robust tests and interval estimation procedures

negative powers of olver recursion operators olver s concept of a recursion operator for symmetries of an evolution equation is extended to include negative powers of the operator some negative order kdv equations are derived in this way in particular the inverse image of the trivial zero symmetry generator has an elegant formulation in terms of an eigenfunction of the associated schrodinger equation used in the inverse scattering solution of the kdv equation and this formulation is used to show that this new equation is the miura transform of a sinh gordon equation

survival analysis with error prone time varying covariates a risk set calibration approach occupational environmental and nutritional epidemiologists are often interested in estimating the prospective effect of time varying exposure variables such as cumulative exposure or cumulative updated average exposure in relation to chronic disease endpoints such as cancer incidence and mortality from exposure validation studies it is apparent that many of the variables of interest are measured with moderate to substantial error although the ordinary regression calibration approach is approximately valid and efficient for measurement error correction of relative risk estimates from the cox model with time independent point exposures when the disease is rare it is not adaptable for use with time varying exposures by re calibrating the measurement error model within each risk set a risk set regression calibration method is proposed for this setting an algorithm for a bias corrected point estimate of the relative risk using an rrc approach is presented followed by the derivation of an estimate of its variance resulting in a sandwich estimator emphasis is on methods applicable to the main study external validation study design which arises in important applications simulation studies under several assumptions about the error model were carried out which demonstrated the validity and efficiency of the method in finite samples the method was applied to a study of diet and cancer from harvard s health professionals follow up study hpfs

on using the cox proportional hazards model with missing covariates summary we propose two methods for handling missing covariates in using the cox proportional hazards model the maximum partial likelihood estimator based only on study subjects having complete covariates does not utilise all available information also it is biased when the probability of missingness depends on the failure or censoring time our suggestion is to impute the conditional expectation of the statistic involving missing covariates given the available information the proposed method provides a consistent regression parameter estimator when the probability of missingness depends on the failure or censoring time as well as on the observed covariates also the proposed estimator is more efficient than the estimator suggested previously by lin ying 1993 when data are missing completely at random

regression calibration in failure time regression in this paper we study a regression calibration method for failure time regression analysis when data on some covariates are missing or mismeasured the method estimates the missing data based on the data structure estimated from a validation data set a random subsample of the study cohort in which covariates are always observed ordinary cox 1972 journal of the royal statistical society series b 34 187 220 regression is then applied to estimate the regression coefficients using the observed covariates in the validation data set and the estimated covariates in the nonvalidation data set the method can be easily implemented we present the asymptotic theory of the proposed estimator finite sample performance is examined and compared with an estimated partial likelihood estimator and other related methods via simulation studies where the proposed method performs well even though it is technically inconsistent finally we illustrate the method with a mouse leukemia data set

the associated camassa holm equation and the kdv equation recently the associated camassa holm ach equation related to the fuchssteiner fokas camassa holm equation by a reciprocal transformation was introduced by schiff who derived backlund transformations by a loop group technique and used these to obtain some simple soliton and rational solutions we show how the ach equation is related to schrodinger operators and kdv and describe how to construct solutions of ach from tau functions of the kdv hierarchy rational n soliton and elliptic solutions are considered as well as exact solutions given by a particular case of the third painleve transcendent

computational complexity of spl mu calculation the structured singular value spl mu measures the robustness of uncertain systems numerous researchers over the last decade have worked on developing efficient methods for computing spl mu this paper considers the complexity of calculating spl mu with general mixed real complex uncertainty in the framework of combinatorial complexity theory in particular it is proved that the spl mu recognition problem with either pure real or mixed real complex uncertainty is np hard this strongly suggests that it is futile to pursue exact methods for calculating spl mu of general systems with pure real or mixed uncertainty for other than small problems

the rank one mixed m problem and kharitonov type analysis the general mixed problem has been shown to be np hard so that the exact solution of the general problem is computationally intractable except for small problems in this paper we consider not the general problem but a particular special case of this problem the rank one mixed problem we show that for this case the mixed problem is equivalent to its upper bound which is convex and it can in fact be computed easily and exactly this special case is shown to be equivalent to the so called affine parameter variation problem for a polynomial with perturbed coefficients which has been examined in detail in the literature and for which several celebrated kharitonov type results have been proven

estimating the parameters in the cox model when covariate variables are measured with error the cox proportional hazards model is commonly used to model survival data as a function of covariates because of the measuring mechanism or the nature of the environment covariates are often measured with error and are not directly observable a naive approach is to use the observed values of the covariates in the cox model which usually produces biased estimates of the true association of interest an alternative strategy is to take into account the error in measurement which may be carried out for the cox model in a number of ways we examine several such approaches and compare and contrast them through several simulation studies we introduce a likelihood based approach which we refer to as the semiparametric method and show that this method is an appealing alternative the methods are applied to analyze the relationship between survival and cd4 count in patients with aids

a risk set calibration method for failure time regression by using a covariate reliability sample regression parameter estimation in the cox failure time model is considered when regression variables are subject to measurement error assuming that repeat regression vector measurements adhere to a classical measurement model we can consider an ordinary regression calibration approach in which the unobserved covariates are replaced by an estimate of their conditional expectation given available covariate measurements however since the rate of withdrawal from the risk set across the time axis due to failure or censoring will typically depend on covariates we may improve the regression parameter estimator by recalibrating within each risk set the asymptotic and small sample properties of such a risk set regression calibration estimator are studied a simple estimator based on a least squares calibration in each risk set appears able to eliminate much of the bias that attends the ordinary regression calibration estimator under extreme measurement error circumstances corresponding asymptotic distribution theory is developed small sample properties are studied using computer simulations and an illustration is provided

k symplectic structures the basic properties of k symplectic structures are introduced and developed in analogy with the well known symplectic differential geometry examples of such structures related to k symplectic lie algebras are given

a modified kdv equation with self consistent sources in non uniform media and soliton dynamics two non isospectral modified kdv equations with self consistent sources are derived which correspond to the time dependent spectral parameter satisfying t and t 3 respectively gauge transformation between the first non isospectral equation corresponding to t and its isospectral counterpart is given from which exact solutions and conservation laws for the non isospectral one are easily listed besides solutions to the two nonisospectral modified kdv equations with self consistent sources are derived by means of the hirota method and the wronskian technique respectively nonisospectral dynamics and source effects including one soliton characteristics in non uniform media two solitons scattering and special behaviours related to sources for example the ghost solitons in the degenerate two soliton case are investigated analytically

a pseudo partial likelihood method for semiparametric survival regression with covariate errors this article presents an estimator for the regression coefficient vector in the cox proportional hazards model with covariate error the estimator is obtained by maximizing a likelihood type function similar to the cox partial likelihood the likelihood function involves the cumulative baseline hazard function for which a simple estimator is substituted the method is capable of handling general covariate error structures it is not restricted to the independent additive error model it can be applied to studies with either an external or internal validation sample and also to studies with replicate measurements of the surrogate covariate the estimator is shown to be consistent and asymptotically normal and an estimate of the asymptotic covariance matrix is derived some extensions to general transformation survival models are indicated simulation studies are presented for a setup with a single error prone binary covariate and a setup with a single error prone normally distributed covariate these si

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

semiparametric modeling of longitudinal measurements and time to event data a two stage regression calibration approach summary in this article we investigate regression calibration methods to jointly model longitudinal and survival data using a semiparametric longitudinal model and a proportional hazards model in the longi tudinal model a biomarker is assumed to followa semiparametric mixed model where covariate effects are modeled parametrically and subject specific time profiles are modeled nonparametrially using a population smoothing spline and subject specific random stochastic processes the cox model is assumed for survival data by including both the current measure and the rate of change of the underlying longitudinal trajectories as covariates as motivated by a prostate cancer study application we develop a two stage semiparametric regression calibration rc method two variations of the rc method are considered risk set regression calibration and a computationally simpler ordinary regression calibration simulation results showthat the two stage rc approach performs well in practice and effectively corrects the bias from the naive method we apply the proposed methods to the analysis of a dataset for evaluating the effects of the longitudinal biomarker psa on the recurrence of prostate cancer

cox regression with accurate covariates unascertainable a nonparametric correction approach abstract many survival studies involve covariates that are not accurately ascertainable cd4 lymphocyte count in hiv aids research is a typical example for which the gold standard of the measurement is not available this article proposes a consistent estimation procedure for cox regression under the additive measurement error model distinct from existing methods the proposed estimation for regression coefficients does not require any additional assumptions we establish the normalized partial score function as a functional of empirical processes which facilitates the construction of an estimating function with the same limit using replicated mismeasured covariates the resulting regression coefficient estimators are shown to be consistent and asymptotically normal a consistent sandwich variance estimate is presented we also suggest estimators for the baseline cumulative hazard function numerical studies demonstrate that the procedure performs well under practical sample sizes application to an aid

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

a convex parameterization of robustly stabilizing controllers this paper treats synthesis of robust controllers for linear time invariant systems uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial the main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations with the new parameterization certain robust performance problems can be stated in terms of quasi convex optimization

hamiltonian structure of discrete soliton systems we describe an approach for investigating the hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem by using the so called implicit representations of the isospectral flows we demonstrate the existence of the recursion operator l which is a strong and hereditary symmetry of the flows it is then proven that every equation in the isospectral hierarchy possesses the hamiltonian structure if l has a skew symmetric factorization and the first equation ut k 0 in the hierarchy satisfies some simple condition we obtain related properties such as the implectic symplectic factorization of l the liouville complete integrability and the multi hamiltonian structures of the isospectral hierarchy four examples are given

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries

properties of the mixed spl mu problem and its bounds upper and lower bounds for the mixed spl mu problem have recently been developed and here we examine the relationship of these bounds to each other and to spl mu a number of interesting properties are developed and the implications of these properties for the robustness analysis of linear systems and the development of practical computation schemes are discussed in particular we find that current techniques can only guarantee easy computation for large problems when spl mu equals its upper bound and computational complexity results prohibit this possibility for general problems in this context we present some special cases where computation is easy and make some direct comparisons between mixed spl mu and kharitonov type analysis methods

cox regression with dependent error in covariates many survival studies have error contaminated covariates due to the lack of a gold standard of measurement furthermore the error distribution can depend on the true covariates but the structure may be difficult to characterize heteroscedasticity is a common manifestation we suggest a novel dependent measurement error model with minimal assumptions on the dependence structure and propose a new functional modeling method for cox regression when an instrumental variable is available this proposal accommodates much more general error contamination than existing approaches including nonparametric correction methods of huang and wang 2000 journal of the american statistical association 95 1209 1219 2006 statistica sinica 16 861 881 the estimated regression coefficients are consistent and asymptotically normal and a consistent variance estimate is provided for inference simulations demonstrate that the procedure performs well even under substantial error contamination illustration with a clinical study is provided