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

regression analysis for recurrent events data under dependent censoring summary recurrent events data are commonly seen in longitudinal follow up studies dependent censoring often occurs due to death or exclusion from the study related to the disease process in this article we assume flexible marginal regression models on the recurrence process and the dependent censoring time without specifying their dependence structure the proposed model generalizes the approach by ghosh and lin 2003 biometrics 59 877 885 the technique of artificial censoring provides a way to maintain the homogeneity of the hypothetical error variables under dependent censoring here we propose to apply this technique to two gehan type statistics one considers only order information for pairs whereas the other utilizes additional information of observed censoring times available for recurrence data a model checking procedure is also proposed to assess the adequacy of the fitted model the proposed estimators have good asymptotic properties their finite sample performances are examined via simulations finally the proposed methods are applied to analyze the aids linked to the intravenous experiences cohort data

information bounds for cox regression models with missing data we derive information bounds for the regression parameters in cox models when data are missing at random these calculations are of interest for understanding the behavior of efficient estimation in case cohort designs a type of two phase design often used in cohort studies the derivations make use of key lemmas appearing in robins rotnitzky and zhao j amer statist assoc 89 1994 846 866 and robins hsieh and newey j roy statist soc ser b 57 1995 409 424 but in a form suited for our purposes here we begin by summarizing the results of robins rotnitzky and zhao in a form that leads directly to the projection method which will be of use for our model of interest we then proceed to derive new information bounds for the regression parameters of the cox model with data missing at random mar in the final section we exemplify our calculations with several models of interest in cohort studies including an i i d version of the classical case cohort design of prentice biometrika 73 1986 1 11 and self and prentice ann statist 16 1988 64 81

regression methods for gap time hazard functions of sequentially ordered multivariate failure time data sequentially ordered multivariate failure time data are often observed in biomedical studies and inter event or gap times are often of interest generally standard hazard regression methods cannot be applied to the gap times because of identifiability issues and induced dependent censoring we propose estimating equations for fitting proportional hazards regression models to the gap times model parameters are shown to be consistent and asymptotically normal simulation studies reveal the appropriateness of the asymptotic approximations in finite samples the proposed methods are applied to renal failure data to assess the association between demographic covariates and both time until wait listing and time from wait listing to kidney transplantation

reweighting estimators for cox regression with missing covariates missingness in covariates is a common problem in survival data in this article we propose a reweighting method for estimating the regression parameters in the cox model with missing covariates we also consider the augmented reweighting method by subtracting the projection term onto the nuisance tangent space the proposed method provides consistent and asymptotically normally distributed estimators when the missing data mechanism depends on the outcome variables as well as on the observed covariates with either monotone or arbitrary nonmonotone missingness patterns simulation results indicate that in most situations the proposed reweighting estimators are more efficient than the inverse probability weighting estimators for the regression coefficients of the missing covariates and are as efficient as or more efficient than the inverse probability weighting estimators for the regression coefficients of the completely observed covariates the simple reweighting estimators can be easily implemented in st

the use of gaussian quadrature for estimation in frailty proportional hazards models in this paper we propose a novel gaussian quadrature estimation method in various frailty proportional hazards models we approximate the unspecified baseline hazard by a piecewise constant one resulting in a parametric model that can be fitted conveniently by gaussian quadrature tools in standard software such as sas proc nlmixed we first apply our method to simple frailty models for correlated survival data e g recurrent or clustered failure times then to joint frailty models for correlated failure times with informative dropout or a dependent terminal event such as death simulation studies show that our method compares favorably with the well received penalized partial likelihood method and the monte carlo em mcem method for both normal and gamma frailty models we apply our method to three real data examples 1 the time to blindness of both eyes in a diabetic retinopathy study 2 the joint analysis of recurrent opportunistic diseases in the presence of death for hiv infected patients and 3 the joint modeling of local distant tumor recurrences and patients survival in a soft tissue sarcoma study the proposed method greatly simplifies the implementation of the joint frailty models and makes them much more accessible to general statistical practitioners

numerical solution of a singularly perturbed two point boundary value problem using equidistribution analysis of convergence abstract adaptive grid methods are becoming established as valuable computational techniques for the numerical solution of differential equations with near singular solutions adaptive methods are equally effective in approximating solutions of problems with boundary layers or interior layers see for example mulholland et al siam j sci comput 19 4 1998 1261 1289 much is now being done in developing error analyes for methods that are based on adaptivity in this paper we present a rigorous error analysis for the solution of a singularly perturbed two point boundary value problem on a grid that is constructed adaptively from a knowledge of the exact solution the discrete solutions are generated by an upwind finite difference scheme and the grid is formed by equidistributing a monitor function based on arc length an error analysis shows that the discrete solutions are uniformly convergent with respect to the perturbation parameter epsilon the epsilon uniform convergence is confirmed by numerical computations

a semiparametric model for the analysis of recurrent event panel data summary in many longitudinal studies interest focuses on the occurrence rate of some phenomenon for the subjects in the study when the phenomenon is nonterminating and possibly recurring the result is a recurrent event data set examples include epileptic seizures and recurrent cancers when the recurring event is detectable only by an expensive or invasive examination only the number of events occurring between follow up times may be available this article presents a semiparametric model for such data based on a multiplicative intensity model paired with a fully flexible nonparametric baseline intensity function a random subject specific effect is included in the intensity model to account for the overdispersion frequently displayed in count data estimators are determined from quasi likelihood estimating functions because only first and second moment assumptions are required for quasi likelihood the method is more robust than those based on the specification of a full parametric likelihood consistency of the estimators depends only on the assumption of the proportional intensity model the semiparametric estimators are shown to be highly efficient compared with the usual parametric estimators as with semiparametric methods in survival analysis the method provides useful diagnostics for specific parametric models including a quasi score statistic for testing specific baseline intensity functions the techniques are used to analyze cancer recurrences and a pheromone based mating disruption experiment in moths a simulation study confirms that for many practical situations the estimators possess appropriate small sample characteristics

asymptotic theory for the frailty model the frailty model is a generalization of cox s proportional hazards model which includes a random effect nielsen gill andersen and sorensen 1992 proposed an em algorithm to estimate the cumulative baseline hazard and the variance of the random effect here the asymptotic distribution of the estimators is given along with a consistent estimator of the asymptotic variance

consistency in a proportional hazards model incorporating a random effect supported in part by the north atlantic treaty organization under a grant awarded in 1990 conducted at the mathematical institute of the university of utrecht the netherlands

simultaneous control of a rod equation and a simple schro dinger equation abstract we show that a corollary of a local approximate controllability result for the bilinear rod equation in 1 is that the controls which steer the modes of the rod equation also move the modes of a controlled schrodinger equation specifying a target point for the schrodinger system restricts but does not determine the outcome of the controlled rod equation the local result is a special case of a general local result for hyperbolic systems which is used in 1 to obtain a global approximate controllability result after modifying the latter we obtain a global result for the two systems

correcting for selective compliance in a re employment bonus experiment we propose a two stage instrumental variable estimator that is consistent if there is selective compliance in the treatment group of a randomized experiment and the outcome variable is a censored duration the estimator assumes full compliance in the control group we use the estimator to re analyze data from the illinois re employment bonus experiment

a robust adaptive method for a quasi linear one dimensional convection diffusion problem a quasi linear conservative convection diffusion two point boundary value problem is considered to solve it numerically an upwind finite difference scheme is applied the mesh used has a fixed number n 1 of nodes and is initially uniform but its nodes are moved adaptively using a simple algorithm of de boor based on equidistribution of the arc length of the current computed piecewise linear solution it is proved for the first time that a mesh exists that equidistributes the arc length along the polygonal solution curve and that the corresponding computed solution is first order accurate uniformly in varepsilon where varepsilon is the diffusion coefficient in the case when the boundary value problem is linear if n is sufficiently large independently of varepsilon it is shown that after o rm ln 1 varepsilon rm ln n iterations of the algorithm the piecewise linear interpolant of the computed solution achieves first order accuracy in the l infty 0 1 norm uniformly in varepsilon numerical experiments are presented that support our theoretical results

accelerated intensity frailty model for recurrent events data in this article we propose an accelerated intensity frailty aif model for recurrent events data and derive a test for the variance of frailty in addition we develop a kernel smoothing based em algorithm for estimating regression coefficients and the baseline intensity function the variance of the resulting estimator for regression parameters is obtained by a numerical differentiation method simulation studies are conducted to evaluate the finite sample performance of the proposed estimator under practical settings and demonstrate the efficiency gain over the gehan rank estimator based on the aft model for counting process lin et al 1998 our method is further illustrated with an application to a bladder tumor recurrence data

on a uniformly accurate finite difference approximation of a singulary peturbed reaction diffusion problem using grid equidistribution we examine the convergence properties of a finite difference approximation of a singularly perturbed reaction diffusion boundary value problem using a nonuniform grid the grid is based on the equidistribution of a positive monitor function that is a linear combination of a constant floor and a power of the second derivative of the solution analysis shows how the monitor function can be chosen to ensure that the accuracy of the numerical approximation is insensitive to the size of the singular perturbation parameter the use of equidistribution principles appears in many practical grid adaption schemes and our analysis provides insight into the convergence behaviour on such grids numerical results are given that confirm the uniform convergence rates

a booster method for singular perturbation problems arising in chemical reactor theory a numerical method for singularly perturbed two point boundary value problems bvps for second order ordinary differential equations odes arising in chemical reactor theory is proposed in this method an asymptotic approximation is incorporated into a suitable finite difference scheme to improve the numerical solution uniform error estimates are derived when implemented in known difference schemes numerical results are presented in support of this claim

modeling random effects for censored data by a multivariate normal regression model summary a normal distribution regression model with a frailty like factor to account for statistical dependence between the observed survival times is introduced this model as opposed to standard hazard based frailty models has survival times that conditional on the shared random effect have an accelerated failure time representation the dependence properties of this model are discussed and maximum likelihood estimation of the model s parameters is considered a number of examples are considered to illustrate the approach the estimated degree of dependence is comparable to other models but the present approach has the advantage that the interpretation of the random effect is simpler than in the frailty model

controllability for distributed bilinear systems this paper studies controllability of systems of the form dw dt mathcal a w p t mathcal b w where mathcal a is the infinitesimal generator of a c 0 semigroup of bounded linear operators e mathcal a t on a banach space x mathcal b x to x is a c 1 map and p in l 1 0 t mathbb r is a control the paper i gives conditions for elements of x to be accessible from a given initial state w 0 and ii shows that controllability to a full neighborhood in x of w 0 is impossible for dim x infty examples of hyperbolic partial differential equations are provided

recurrent events analysis in the presence of time dependent covariates and dependent censoring summary recurrent events models have had considerable attention recently the majority of approaches show the consistency of parameter estimates under the assumption that censoring is independent of the recurrent events process of interest conditional on the covariates that are included in the model we provide an overview of available recurrent events analysis methods and present an inverse probability of censoring weighted estimator for the regression parameters in the andersen gill model that is commonly used for recurrent event analysis this estimator remains consistent under informative censoring if the censoring mechanism is estimated consistently and it generally improves on the naive estimator for the andersen gill model in the case of independent censoring we illustrate the bias of ad hoc estimators in the presence of informative censoring with a simulation study and provide a data analysis of recurrent lung exacerbations in cystic fibrosis patients when some patients are lost to follow up

on fitting cox s regression model with time dependent coefficients summary this paper examines goodness of fit testing in the cox regression model with timevarying regression coefficients arjas 1988 approach is used to define test statistics of the kolmogorov smirnov and cramer von mises types their asymptotic limits are shown to be well known functionals of standard brownian motion leading to the construction of formal goodness of fit tests some numerical studies are also included to illustrate the performance of the tests for moderate sample sizes

analysis of survival data by the proportional odds model a model is presented for the analysis of lifetime data in which the rates of mortality for separate groups of patients converge with time a non parametric estimate is given for the survivor function the theoretical basis for the model assumes that prognostic factors have a multiplicative effect on the odds against survival beyond any given time the model is fitted to data using maximum likelihood estimation and an example of its use in the analysis of a lung cancer trial is given

joint modeling and analysis of longitudinal data with informative observation times in analysis of longitudinal data it is often assumed that observation times are predetermined and are the same across study subjects such an assumption however is often violated in practice as a result the observation times may be highly irregular it is well known that if the sampling scheme is correlated with the outcome values the usual statistical analysis may yield bias in this article we propose joint modeling and analysis of longitudinal data with possibly informative observation times via latent variables a two step estimation procedure is developed for parameter estimation we show that the resulting estimators are consistent and asymptotically normal and that the asymptotic variance can be consistently estimated using the bootstrap method simulation studies and a real data analysis demonstrate that our method performs well with realistic sample sizes and is appropriate for practical use

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

semiparametric modeling and estimation of the terminal behavior of recurrent marker processes before failure events abstractrecurrent event processes with marker measurements are mostly and largely studied with forward time models starting from an initial event interestingly the processes could exhibit important terminal behavior during a time period before occurrence of the failure event a natural and direct way to study recurrent events prior to a failure event is to align the processes using the failure event as the time origin and to examine the terminal behavior by a backward time model this article studies regression models for backward recurrent marker processes by counting time backward from the failure event a three level semiparametric regression model is proposed for jointly modeling the time to a failure event the backward recurrent event process and the marker observed at the time of each backward recurrent event the first level is a proportional hazards model for the failure time the second level is a proportional rate model for the recurrent events occurring before the failure event and the thi

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

regression analysis of multivariate incomplete failure time data by modeling marginal distributions abstract many survival studies record the times to two or more distinct failures on each subject the failures may be events of different natures or may be repetitions of the same kind of event in this article we consider the regression analysis of such multivariate failure time observations each marginal distribution of the failure times is formulated by a cox proportional hazards model no specific structure of dependence among the distinct failure times on each subject is imposed the regression parameters in the cox models are estimated by maximizing the failure specific partial likelihoods the resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated simultaneous inferential procedures are then proposed extensive monte carlo studies indicate that the normal approximation is adequate for practical use the new methods allow time dependent covariates missing observations and arbitrary patterns of censorship they are illustrat

the l 1 method for robust nonparametric regression abstract consider the problem of estimating the regression function underlying a set of data that is contaminated by a heavy tailed error distribution the two standard approaches to such a problem are each flawed robust parametric least squares is appropriate only if there is good reason to believe that the underlying function has some particular form whereas most nonparametric regression methods are asymptotically equivalent to kernel regression methods which are not resistant against outliers existing algorithms for robustifying nonparametric regression procedures use either nonlinear optimization of an influence function or iterative solution of local polynomial fitting using reweighted least squares neither of these two approaches combines computational ease with asymptotic theoretical results furthermore application of the robust procedure has been limited almost exclusively to the case of a single explanatory variable with the response variable in this article a new hybrid method is propose

survival analysis 1982 1991 the second decade of the proportional hazards regression model a short review of the cox regression model and its development from publication to 1982 is given i discuss how the model is used and presented in medical journals and how improvement may be achieved finally some extensions of the model are outlined

the analysis of multiple endpoints in clinical trials treatment comparisons in randomized clinical trials usually involve several endpoints such that conventional significance testing can seriously inflate the overall type i error rate one option is to select a single primary endpoint for formal statistical inference but this is not always feasible another approach is to apply bonferroni correction i e multiply each p value by the total number of endpoints its conservatism for correlated endpoints is examined for multivariate normal data a third approach is to derive an appropriate global test statistic and this paper explores one such test applicable to any set of asymptotically normal test statistics quantitative binary and survival endpoints are all considered within this general framework two examples are presented and the relative merits of the proposed strategies are discussed

score tests for homogeneity of regression effect in the proportional hazards model a simple model containing the proportional hazards regression model as a special case is presented the purpose of the model is to provide a framework in which specific alternatives to the proportional hazards assumption may be tested rank invariant score tests for linear quadratic or exponential trends can for instance all be undertaken within this framework in the case of the two sample problem the required calculations are shown to take a particularly simple form special consideration is given to the two sample case in which there is an inversion of the regression effect i e where the hazard functions cross at some given point both of the motivating examples are concerned with this problem computational aspects are relatively straightforward and some discussion on this is provided

covariance and survivor function estimation using censored multivariate failure time data summary the covariance between counting process martingales is used to characterize the dependence between two failure time variates a representation of the bivariate survivor function is obtained in terms of the marginal survivor functions and this covariance function a closely related representation expresses the bivariate survivor function in terms of marginal survivor functions and a conditional covariance function leading to a new nonparametric survivor function estimator generalizations to higher dimensional failure time variates are also given simulation evaluations of the survivor function estimator are presented and generalizations to regression problems are outlined

competing risks regression for stratified data for competing risks data the fine gray proportional hazards model for subdistribution has gained popularity for its convenience in directly assessing the effect of covariates on the cumulative incidence function however in many important applications proportional hazards may not be satisfied including multicenter clinical trials where the baseline subdistribution hazards may not be common due to varying patient populations in this article we consider a stratified competing risks regression to allow the baseline hazard to vary across levels of the stratification covariate according to the relative size of the number of strata and strata sizes two stratification regimes are considered using partial likelihood and weighting techniques we obtain consistent estimators of regression parameters the corresponding asymptotic properties and resulting inferences are provided for the two regimes separately data from a breast cancer clinical trial and from a bone marrow transplantation registry illustrate the potential utility of the stratified fine gray model

quasi maximum likelihood estimation of volatility with high frequency data this paper investigates the properties of the well known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise by extending the classic asymptotic results of quasi maximum likelihood estimation when trying to estimate the integrated volatility and the variance of noise this parametric approach remains consistent efficient and robust as a quasi estimator under misspecified assumptions moreover it shares the model free feature with nonparametric alternatives for instance realized kernels while being advantageous over them in terms of finite sample performance in light of quadratic representation this estimator behaves like an iterative exponential realized kernel asymptotically comparisons with a variety of implementations of the tukey hanning2 kernel are provided using monte carlo simulations and an empirical study with the euro us dollar future illustrates its application in practice

some results on controllability of a nonlinear degenerate parabolic system by bilinear control in this paper we discuss the controllability of a nonlinear degenerate parabolic system with bilinear control based on the shrinking property of the solutions we prove that the system is not globally approximately controllable furthermore we give an approximate null controllability result we also prove that the system is not globally exactly null controllable by a comparison principle

nested frailty models using maximum penalized likelihood estimation the frailty model is a random effect survival model which allows for unobserved heterogeneity or for statistical dependence between observed survival data the nested frailty model accounts for the hierarchical clustering of the data by including two nested random effects nested frailty models are particularly appropriate when data are clustered at several hierarchical levels naturally or by design in such cases it is important to estimate the parameters of interest as accurately as possible by taking into account the hierarchical structure of the data we present a maximum penalized likelihood estimation mpnle to estimate non parametrically a continuous hazard function in a nested gamma frailty model with right censored and left truncated data the estimators for the regression coefficients and the variance components of the random effects are obtained simultaneously the simulation study demonstrates that this semi parametric approach yields satisfactory results in this complex setting in order to illustrate the mpnle method and the nested frailty model we present two applications one is for modelling the effect of particulate air pollution on mortality in different areas with two levels of geographical regrouping the other application is based on recurrent infection times of patients from different hospitals we illustrate that using a shared frailty model instead of a nested frailty model with two levels of regrouping leads to inaccurate estimates with an overestimation of the variance of the random effects we show that even when the frailty effects are fairly small in magnitude they are important since they alter the results in a systematic pattern

on the regression analysis of multivariate failure time data summary the paper is concerned with the analysis of regression effects when individual study subjects may experience multiple failures the proposed methods are most likely to be useful when there are a fairly large number of study subjects two general classes of regression models are considered in order to relate the hazard or intensity function to covariates and to preceding failure time history both models are of a stratified proportional hazards type one model includes baseline intensity functions that are arbitrary as a function of time from the beginning of study while the other includes baseline intensity functions that are arbitrary as a function of time from the study subject s immediately preceding failure partial likelihood functions are derived for the regression coefficients generalizations and illustrations are given

accelerated failure time models for counting processes summary we present a natural extension of the conventional accelerated failure time model for survival data to formulate the effects of covariates on the mean function of the counting process for recurrent events a class of consistent and asymptotically normal rank estimators is developed for estimating the regression parameters of the proposed model in addition a nelson aalen type estimator for the mean function of the counting process is constructed which is consistent and properly normalised converges weakly to a zeromean gaussian process we assess the finite sample properties of the proposed estimators and the associated inference procedures through monte carlo simulation and provide an application to a well known bladder cancer study

maximum likelihood estimation in semiparametric regression models with censored data summary semiparametric regression models play a central role in formulating the effects of covariates on potentially censored failure times and in the joint modelling of incomplete repeated measures and failure times in longitudinal studies the presence of infinite dimensional parameters poses considerable theoretical and computational challenges in the statistical analysis of such models we present several classes of semiparametric regression models which extend the existing models in important directions we construct appropriate likelihood functions involving both finite dimensional and infinite dimensional parameters the maximum likelihood estimators are consistent and asymptotically normal with efficient variances we develop simple and stable numerical techniques to implement the corresponding inference procedures extensive simulation experiments demonstrate that the inferential and computational methods proposed perform well in practical settings applications to three medical studies yield important new insights we conclude that there is no reason theoretical or numerical not to use maximum likelihood estimation for semiparametric regression models we discuss several areas that need further research

asymptotic behavior and positive solutions of a chemical reaction diffusion system abstract this paper is concerned with some qualitative analysis for a coupled system of three reaction diffusion equations which arises from certain chemical reactions first discovered by belousov and zhabotinskii the analysis includes the existence of a bounded global time dependent solutions the stability and instability of the zero solution and the existence and nonexistence of a positive steady state solution including a global attractor of the system the global existence and stability problem is determined by the method of upper and lower solutions and the existence of a positive steady state solution is based on the fixed point index and bifurcation theory this analysis leads to a necessary and sufficient condition for the existence and nonexistence of a positive steady state solution in relation to the various physical parameters of the system

a large sample study of cox s regression model strong consistency and asymptotic normality are established for the maximum partial likelihood estimate of the regression parameter in cox s regression model estimates are also derived for the underlying cumulative hazard function and survival distribution we establish the asymptotic normality of these estimates and calculate the limiting variances

assessment of stratum covariate interactions in cox s proportional hazards regression model we assess stratum e g treatment interactions with covariates and with the baseline hazard function in the proportional hazards ph regression model for lifetime data we consider models incorporating stratum interactions both with and without stratification of the risk sets in the likelihood function and describe likelihood ratio statistics for tests of the presence of these interactions we also present step down methods for building reduced models which include stratum specific parameters corresponding to covariates which interact with treatment we apply ph models with such interactions to a clinical trial of des in the treatment of prostate cancer to determine optimal treatment conditional on each patient s covariates

convergence rates of moving mesh methods for moving boundary partial integro differential equations from regime switching jump diffusion asian option pricing abstract this paper studies the convergence rates of moving mesh methods for a system of moving boundary partial integro differential equations pides which arise in the asian option pricing under the state dependent regime switching jump diffusion models the value function of the asian option under the model is governed by a system of two dimensional pides in this paper the two dimensional pides are recast into a one dimensional moving boundary problem of the pides a moving finite difference method fdm is proposed to solve the one dimensional moving boundary problem and the convergence rates are proved numerical examples are provided to confirm the theoretical results

point process modelling for directed interaction networks abstract network data often take the form of repeated interactions between senders and receivers tabulated over time a primary question to ask of such data is which traits and behaviors are predictive of interaction to answer this question a model is introduced for treating directed interactions as a multivariate point process a cox multiplicative intensity model using covariates that depend on the history of the process consistency and asymptotic normality are proved for the resulting partial likelihood based estimators under suitable regularity conditions and an efficient fitting procedure is described multicast interactions those involving a single sender but multiple receivers are treated explicitly the resulting inferential framework is then employed to model message sending behavior in a corporate e mail network the analysis gives a precise quantification of which static shared traits and dynamic network effects are predictive of message recipient selection

efficient estimation for the accelerated failure time model the accelerated failure time model provides a natural formulation of the effects of covariates on potentially censored response variable the existing semiparametric estimators are computationally intractable and statistically inefficient in this article we propose an approximate nonparametric maximum likelihood method for the accelerated failure time model with possibly time dependent covariates we estimate the regression parameters by maximizing a kernel smoothed profile likelihood function the maximization can be achieved through conventional gradient based search algorithms the resulting estimators are consistent and asymptotically normal the limiting covariance matrix attains the semiparametric efficiency bound and can be consistently estimated we also provide a consistent estimator for the error distribution extensive simulation studies demonstrate that the asymptotic approximations are accurate in practical situations and the new estimators are considerably more efficient than the existing o

stratified partial likelihood estimation when multiple durations are generated by a single unit they may be related in a way that is not fully captured by the regressors the omitted unit specific variables might vary over the durations they might also be correlated with the variables in the regression component we propose an estimator that responds to these concerns and develop a specification test for detecting unobserved unit specific effects data from malaysia reveal that concentration of child mortality in some families is imperfectly explained by observed explanatory variables and that failure to control for unobserved heterogeneity seriously biases the parameter estimates

kernel smoothed profile likelihood estimation in the accelerated failure time frailty model for clustered survival data clustered survival data frequently arise in biomedical applications where event times of interest are clustered into groups such as families in this article we consider an accelerated failure time frailty model for clustered survival data and develop nonparametric maximum likelihood estimation for it via a kernel smoother aided em algorithm we show that the proposed estimator for the regression coefficients is consistent asymptotically normal and semiparametric efficient when the kernel bandwidth is properly chosen an em aided numerical differentiation method is derived for estimating its variance simulation studies evaluate the finite sample performance of the estimator and it is applied to the diabetic retinopathy data set

flood frequency analysis using the cox regression model procedures for incorporating time varying exogenous information into flood frequency analyses are developed using the cox regression model for counting processes in this statistical model the probability of occurrence of a flood peak in a short interval t t dt depends in an explicit manner on the values at t of k covariate processes z1 zk specifically letting dn t be 1 if a flood peak occurs in t t dt and 0 otherwise dn t a t exp j 1kbjzj t dm t where a the baseline intensity is an unknown function b is a vector of unknown regression parameters and the error dm t is conditionally orthogonal to the past history two applications assessment of relative importance of physical processes such as snow melt or soil moisture storage on flood frequency at a site and derivation of time varying flood frequency estimates are considered

some graphical displays and marginal regression analyses for recurrent failure times and time dependent covariates abstract recurrent event time data are common in medical research examples include infections in aids patients and seizures in epilepsy patients in this context as well as in the more usual context of a single failure time variable time dependent covariates are frequently of interest we suggest some rate functions that might be displayed when analyzing recurrent failure time data or when the effect of a categorical time dependent covariate is of interest estimators of these functions are provided along with two sample test statistics a new approach to regression modeling of these data is suggested and contrasted with existing methods our methods do not require that an explicit model be formulated for the probabilistic association between failure times within an individual this is in line with the currently popular generalized estimating equation approach to longitudinal data if the nature of such associations is known or is of particular interest then alternative methods may be appropriate

numerical solution of 2d singularly perturbed reaction diffusion system with multiple scales abstract in this article a robust numerical method is studied to approximate singularly perturbed system of reaction diffusion problems with multiple scales the analytical properties of the exact solution have been studied the numerical method consists of the classical central difference scheme on a shishkin mesh for spatial semidiscretization processes and the implicit euler scheme on a uniform time stepping for temporal derivative the error estimate is deduced which exhibits that the numerical approximation is uniformly convergent of almost second order in spatial variable and first order in temporal variable numerical experiments are given which reveals the effectiveness of the proposed scheme

on profile likelihood abstract we show that semiparametric profile likelihoods where the nuisance parameter has been profiled out behave like ordinary likelihoods in that they have a quadratic expansion in this expansion the score function and the fisher information are replaced by the efficient score function and efficient fisher information the expansion may be used among others to prove the asymptotic normality of the maximum likelihood estimator to derive the asymptotic chi squared distribution of the log likelihood ratio statistic and to prove the consistency of the observed information as an estimator of the inverse of the asymptotic variance

semiparametric transformation models with time varying coefficients for recurrent and terminal events in this article we propose a family of semiparametric transformation models with time varying coefficients for recurrent event data in the presence of a terminal event such as death the new model offers great flexibility in formulating the effects of covariates on the mean functions of the recurrent events among survivors at a given time for the inference on the proposed models a class of estimating equations is developed and asymptotic properties of the resulting estimators are established in addition a lack of fit test is provided for assessing the adequacy of the model and some tests are presented for investigating whether or not covariate effects vary with time the finite sample behavior of the proposed methods is examined through monte carlo simulation studies and an application to a bladder cancer study is also illustrated

methods for the analysis of sampled cohort data in the cox proportional hazards model methods are provided for regression parameter and cumulative baseline hazard estimation in the cox proportional hazards model when the cohort is sampled according to a predictable sampling probability law the key to the methodology is to define counting processes which count joint failure and sampled risk sets occurrences and to derive the appropriate intensities for these counting processes leading to estimation methods parallel to those for full cohort data these techniques are illustrated for a number of sampling designs including three novel designs counter matching with additional randomly sampled controls quota sampling of controls and nested case control sampling with number of controls dependent on the failure s exposure status general asymptotic theory is developed for the maximum partial likelihood estimator and cumulative baseline hazard estimator and is used to derive the asymptotic distributions for estimators from a large class of designs

robust inference for univariate proportional hazards frailty regression models we consider a class of semiparametric regression models which are one parameter extensions of the cox j roy statist soc ser b 34 1972 187 220 model for right censored univariate failure times these models assume that the hazard given the covariates and a random frailty unique to each individual has the proportional hazards form multiplied by the frailty the frailty is assumed to have mean 1 within a known one parameter family of distributions inference is based on a nonparametric likelihood the behavior of the likelihood maximizer is studied under general conditions where the fitted model may be misspecified the joint estimator of the regression and frailty parameters as well as the baseline hazard is shown to be uniformly consistent for the pseudo value maximizing the asymptotic limit of the likelihood appropriately standardized the estimator converges weakly to a gaussian process when the model is correctly specified the procedure is semiparametric efficient achieving the semiparametric information bound for all parameter components it is also proved that the bootstrap gives valid inferences for all parameters even under misspecification we demonstrate analytically the importance of the robust inference in several examples in a randomized clinical trial a valid test of the treatment effect is possible when other prognostic factors and the frailty distribution are both misspecified under certain conditions on the covariates the ratios of the regression parameters are still identifiable the practical utility of the procedure is illustrated on a non hodgkin s lymphoma dataset

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

semiparametric transformation models for point processes in this article we propose a family of semiparametric transformation models for point processes with positive jumps of arbitrary sizes these models offer great flexibilities in formulating the effects of covariates on the mean function of the point process while leaving the stochastic structure completely unspecified we develop a class of estimating equations for the baseline mean function and the vector valued regression parameter based on censored point processes and covariate data these equations can be solved easily by the standard newton raphson algorithm the resultant estimator of the regression parameter is consistent and asymptotically normal with a covariance matrix that can be estimated consistently furthermore the estimator of the baseline mean function is uniformly consistent and upon proper normalization converges weakly to a zero mean gaussian process with an easily estimated covariance function we demonstrate through extensive simulation studies that the proposed inference procedu

the asymptotic joint distribution of the efficient scores test for the proportional hazards model calculated over time the log rank test derived by mantel 1966 and peto 1972 is a special case of the efficient scores test for the proportional hazards model it is used frequently in the analysis of survival duration in clinical trials where patients entering the study are assigned at random to different treatments and are followed until they either fail or the study is terminated most survival studies are designed so that the log rank test is calculated at the termination of the study however it is not unusual for the investigator to monitor the experiment periodically it would be very useful to know the joint distribution of the test statistic at different points in time in order to adjust for the effect of repeated significance testing although the joint distribution has not been derived previously sequential procedures for the analysis of survival data in clinical trials have been given by several authors including armitage 1975 pp 140 6 jones whitehead 1979 and nagelkerke hart 1980 in this paper we establish the asymptotic joint distribution of the more general efficient scores test for the proportional hazards model calculated at different points in time and indicate how this can be used for repeated significance testing

optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction diffusion boundary value problems in this article we study the problem of determining an appropriate grading of meshes for a system of coupled singularly perturbed reaction diffusion problems having diffusion parameters with different magnitudes the central difference scheme is used to discretize the problem on adaptively generated mesh where the mesh equation is derived using an equidistribution principle an a priori monitor function is obtained from the error estimate a suitable a posteriori analogue of this monitor function is also derived for the mesh construction which will lead to an optimal second order parameter uniform convergence we present the results of numerical experiments for linear and semilinear reaction diffusion systems to support the effectiveness of our preferred monitor function obtained from theoretical analysis

robust rank estimation for transformation models with random effects summarysemiparametric transformation models with random effects are useful in analysing recurrent and clustered data with specified error and random effect distributions zeng lin 2007a proved that nonparametric maximum likelihood estimators are semiparametric efficient in this paper we consider a more general class of transformation models with random effects under which an unknown monotonic transformation of the response is linearly related to the covariates and the random effects with unspecified error and random effect distributions this includes many popular models we propose an estimator based on the maximum rank correlation which relies on symmetry of the random effect distribution and establish its consistency and asymptotic normality a random weighting resampling scheme is employed for inference the proposed method can be extended to censored and clustered data numerical studies demonstrate that the proposed method performs well in practical situations application of the method is illustrated with the framingham cholesterol data

a numerical method for a system of singularly perturbed reaction diffusion equations a dirichlet problem for a system of two coupled singularly perturbed reaction diffusion ordinary differential equations is examined a numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed numerical results are presented which illustrate the theoretical results

asymptotic theory for the correlated gamma frailty model the frailty model is a generalization of cox s proportional hazard model where a shared unobserved quantity in the intensity induces a positive correlation among the survival times murphy showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator npmle for the shared gamma frailty model without covariates in this paper we extend this result to the correlated gamma frailty model and we allow for covariates we discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator which goes back to wald our proof of the consistency for the npmle is essentially the same as the classical proof for the maximum likelihood estimator a new central limit theorem for processes of bounded variation is given furthermore we prove that a consistent estimator for the asymptotic variance of the npmle is given by the inverse of a discrete observed information matrix

a robust computational method for singularly perturbed coupled system of reaction diffusion boundary value problems abstract in this paper we devise a robust computational method for singularly perturbed system of reaction diffusion boundary value problems bvps we use cubic spline on nonuniform mesh to obtain the difference scheme and apply it to the system of bvps on a piecewise uniform shishkin mesh on the whole domain we observed that the cubic spline scheme is not stable in the regular outer region where the mesh is coarse to overcome this difficulty we use the classical central difference scheme only for that portion and the cubic spline scheme elsewhere this newly proposed scheme is uniformly stable throughout the domain and provides second order uniform convergence results we derive the uniform error estimate for this scheme and apply it to a test problem to verify the efficiency and accuracy of the method

a dual approach to inference for partially identified econometric models abstract this paper considers inference for the set i of parameter values that minimize a criterion function chernozhukov et al 2007 cht develop a general theory of estimation and inference using the level set of a criterion function we establish a dual relationship between the level set estimator and its support function and show that the properly normalized support function provides alternative wald type inference methods these methods can be used to obtain confidence sets for i and points inside it for models with finitely many moment inequalities we show that our wald type statistic is asymptotically equivalent to cht s statistic under regularity conditions

shooting method for the solution of singularly perturbed two point boundary value problems having less severe boundary layer singularly perturbed two point boundary value problems bvps having less severe boundary layers are considered a method is presented in this paper to solve these problems numerically the domain of definition of the differential equation is divided into two subintervals in one subinterval inner region an initial value problem ivp obtained from the given bvp is solved by an exponentially fitted difference efd scheme whereas a classical upwind scheme is used to solve the bvp in the other subinterval outer region in order to obtain initial and boundary conditions an asymptotic approximation is used error estimates are derived for the numerical solution the implementation of the method on parallel architectures is discussed numerical results are also presented in support of the proposed method

tests for detecting overdispersion in poisson regression models abstract poisson regression models are widely used in analyzing count data this article develops tests for detecting extra poisson variation in such situations the tests can be obtained as score tests against arbitrary mixed poisson alternatives and are generalizations of tests of fisher 1950 and collings and margolin 1985 accurate approximations for computing significance levels are given and the power of the tests against negative binomial alternatives is compared with those of the pearson and deviance statistics one way to test for extra poisson variation is to fit models that parametrically incorporate and then test for the absence of such variation within the models for example negative binomial models can be used in this way cameron and trivedi 1986 lawless 1987a the tests in this article require only the poisson model to be fitted two test statistics are developed that are motivated partly by a desire to have good distributional approximations for computing significance levels simu

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

a modification of the redistribution to the right algorithm using disease markers summary disease markers are time dependent covariates which describe progression towards development of disease traditional methods in survival analysis do not make use of available data on these markers to recover additional information from censored individuals using a heuristic modification of the redistribution to the right algorithm efron 1967 a new approach for recovering information for censored individuals using disease markers is proposed additionally the statistical properties of the proposed method are examined there are two possible advantages to this modification i bias reduction when censoring is informative and ii an increase in efficiency in the case of truly noninformative censoring

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

singular perturbation methods for a class of initial and boundary value problems in discrete systems a class of singularly perturbed initial and boundary value problems occurring in discrete systems is formulated singular perturbation methods are developed separately for initial value problems and boundary value problems the methods essentially consist of seeking an approximate solution in terms of an outer series based on the degenerate problem and a correction series to recover the auxiliary conditions sacrificed in the process of degeneration some interesting features are explored and the asymptotic correctness of the formal series expansions is clearly established illustrative examples are provided in support of the proposed methods

a parameter robust higher order numerical method for singularly perturbed two parameter problems with non smooth data a singularly perturbed second order ordinary differential equation having two parameters with a discontinuous source term is presented for numerical analysis theoretical bounds on the derivatives regular and singular components of the solution are derived a hybrid monotone difference scheme with the method of averaging at the discontinuous point is constructed on shishkin mesh parameter uniform error bounds for the numerical approximation are established numerical results are presented which support the theoretical results

comparison of a higher order method and the simple upwind and non monotone methods for singularly perturbed boundary value problems in this paper we will present a new scheme for discretization of singularly perturbed boundary value problems based on finite difference methods this method is a combination of simple upwind scheme and central difference method on a special non uniform mesh shishkin mesh for the space discretization numerical results show that the convergence of method is uniform with respect to singular perturbation parameter and has a higher order of convergence

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

semiparametric transformation models with random effects for recurrent events in this article we study a class of semiparametric transformation models with random effects for the intensity function of the counting process these models provide considerable flexibility in formulating the effects of possibly time dependent covariates on the developments of recurrent events while accounting for the dependence of the recurrent event times within the same subject we show that the nonparametric maximum likelihood estimators npmles for the parameters of these models are consistent and asymptotically normal the limiting covariance matrices for the estimators of the regression parameters achieve the semiparametric efficiency bounds and can be consistently estimated the limiting covariance function for the estimator of any smooth functional of the cumulative intensity function also can be consistently estimated we develop a simple and stable em algorithm to compute the npmles as well as the variance and covariance estimators simulation studies demonstrate that the proposed methods per

an almost third order finite difference scheme for singularly perturbed reaction diffusion systems this paper addresses the numerical approximation of solutions to coupled systems of singularly perturbed reaction diffusion problems in particular a hybrid finite difference scheme of hodie type is constructed on a piecewise uniform shishkin mesh it is proved that the numerical scheme satisfies a discrete maximum principle and also that it is third order except for a logarithmic factor uniformly convergent even for the case in which the diffusion parameter associated with each equation of the system has a different order of magnitude numerical examples supporting the theory are given

some simple robust methods for the analysis of recurrent events nelson discussed a method of estimating the cumulative mean function for identically distributed processes of recurrent events we show that a similar approach can be used with more general models including regression the key idea is to use point estimates based on poisson models and to develop robust variance estimates that are valid more generally the methods are illustrated on reliability and warranty data

generalized survival models for correlated time to event data our aim is to develop a rich and coherent framework for modeling correlated time to event data including 1 survival regression models with different links and 2 flexible modeling for time dependent and nonlinear effects with rich postestimation we extend the class of generalized survival models which expresses a transformed survival in terms of a linear predictor by incorporating a shared frailty or random effects for correlated survival data the proposed approach can include parametric or penalized smooth functions for time time dependent effects nonlinear effects and their interactions the maximum penalized marginal likelihood method is used to estimate the regression coefficients and the variance for the frailty or random effects the optimal smoothing parameters for the penalized marginal likelihood estimation can be automatically selected by a likelihood based cross validation criterion for models with normal random effects gauss hermite quadrature can be used to obtain the cluster level marginal likelihoods the akaike information criterion can be used to compare models and select the link function we have implemented these methods in the r package rstpm2 simulating for both small and larger clusters we find that this approach performs well through 2 applications we demonstrate 1 a comparison of proportional hazards and proportional odds models with random effects for clustered survival data and 2 the estimation of time varying effects on the log time scale age varying effects for a specific treatment and two dimensional splines for time and age

estimating time varying effects for overdispersed recurrent events data with treatment switching in the analysis of multivariate event times frailty models assuming time independent regression coefficients are often considered mainly due to their mathematical convenience in practice regression coefficients are often time dependent and the temporal effects are of clinical interest motivated by a phase iii clinical trial in multiple sclerosis we develop a semiparametric frailty modelling approach to estimate time varying effects for overdispersed recurrent events data with treatment switching the proposed model incorporates the treatment switching time in the time varying coefficients theoretical properties of the proposed model are established and an efficient expectation maximization algorithm is derived to obtain the maximum likelihood estimates simulation studies evaluate the numerical performance of the proposed model under various temporal treatment effect curves the ideas in this paper can also be used for time varying coefficient frailty models without treatment switching as well as for alternative models when the proportional hazard assumption is violated a multiple sclerosis dataset is analysed to illustrate our methodology

semiparametric regression for the mean and rate functions of recurrent events the counting process with the cox type intensity function has been commonly used to analyse recurrent event data this model essentially assumes that the underlying counting process is a time transformed poisson process and that the covariates have multiplicative effects on the mean and rate function of the counting process recently pepe and cai and lawless and co workers have proposed semiparametric procedures for making inferences about the mean and rate function of the counting process without the poisson type assumption in this paper we provide a rigorous justification of such robust procedures through modern empirical process theory furthermore we present an approach to constructing simultaneous confidence bands for the mean function and describe a class of graphical and numerical techniques for checking the adequacy of the fitted mean rate model the advantages of the robust procedures are demonstrated through simulation studies an illustration with multiple infection data taken from a clinical study on chronic granulomatous disease is also provided