the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

on continuous solutions of a class of systems of nonlinear functional difference equations with deviating argument abstract bounded continuous solutions on r of a class of systems of nonlinear functional difference equations with a complicated nonlinear deviation of the argument that depends on unknown functions are studied here

wazewski s method for systems of dynamic equations on time scales abstract in the paper the wazewski s method which is well known for ordinary differential equations is developed for a system of dynamic equations on an arbitrary time scale sufficient conditions guaranteeing the existence of at least one solution with a graph staying in a previously defined open set are derived this result generalizing some previous results concerning the asymptotic behavior of solutions of discrete equations is suitable for investigating of asymptotic behavior of solutions of dynamical systems a relevant example is considered

bounded solutions of nonlinear discrete equations abstract the goal of this paper is to give sufficient conditions for the existence of bounded solutions of a system of discrete equations u k 1 f k u k k a a 1 a 2 with a n n 0 1 and f n a r n r n a principle giving sufficient conditions for the existence of at least one initial problem generating solution with its graph staying in a previously prescribed domain for every k a a 1 a 2 is formulated illustrative examples are considered as well

error estimates for the numerical approximation of time dependent flow of bingham fluid in cylindrical pipes by the regularization method the flow of a bingham fluid in a cylindrical pipe can give rise to free boundary problems the fluid behaves like a viscous fluid if the shear stress expressed as a linear function of the shear rate exceeds a yield value and like a rigid body otherwise the surfaces dividing fluid and rigid zones are the free boundaries therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods considerable progress has been made in the development of numerical algorithms for bingham fluids 2 20 27 29 30 32 however very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution that is the error estimate of these numerical methods error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters this paper concerns the error estimates of a unsteady bingham fluid modeled as a variational inequality due to duvaut lions 16 and glowinski 30 the difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term a common technique called the regularization method is to replace the non differentiable term by a perturbed differentiable term which depends on a small regularization parameter the regularization method effectively reduces the variational inequality to an equation a regularized problem which is much easier to cope with this paper has achieved the following 1 error estimates are derived for a continuous time galerkin method in suitable norms 2 we give an estimate of the difference between the true solution and the regularized solution in terms of 3 some regularity properties for both regularized solution and the true solution are proved 4 the error estimates for full discretization of the regularized problem using piecewise linear finite elements in space and backward differencing in time are established for the first time by coupling the regularization parameter and the discretization parameters h and t 5 we are able to improve our estimates in the one dimensional case or under stronger regularity assumptions on the true solution the estimates for the one dimensional case are optimal and confirmed by numerical experiment the estimates from 4 and 5 provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h t and in order to control the error see corollary 4 4 the above estimates extend the error bounds derived in glowinski lions and tremolieres 32 chapter 5 pp 348 404 for the stationary bingham fluid to the time dependent one which is the main contribution of this paper

statistical inference procedures for bivariate archimedean copulas abstract a bivariate distribution function h x y with marginals f x and g y is said to be generated by an archimedean copula if it can be expressed in the form h x y 1 f x g y for some convex decreasing function defined on 0 1 in such a way that 1 0 many well known systems of bivariate distributions belong to this class including those of gumbel ali mikhail haq thelot clayton frank and hougaard frailty models also fall under that general prescription this article examines the problem of selecting an archimedean copula providing a suitable representation of the dependence structure between two variates x and y in the light of a random sample x 1 y 1 x n y n the key to the estimation procedure is a one dimensional empirical distribution function that can be constructed whether the uniform representation of x and y is archimedean or not and independently of their marginals this semiparametric estimator based on a decomposition of kendall s tau statistic

nonparametric estimation of successive duration times under dependent censoring summary this paper considers nonparametric inference for duration times of two successive events since the second duration process becomes observable only if the first event has occurred the length of the first duration affects the probability of the second duration being censored dependent censoring arises if the two duration times are correlated which is often the case standard approaches to this problem fail because of dependent censoring mechanism a new product limit estimator for the second duration variable and a pathdependent joint survival function estimator are proposed both modified for the dependent censoring properties of the estimators are discussed an example from lawless 1982 is studied for illustrative purposes as well as a simulation study

anl2 error estimate for an approximation of the solution of a parabolic variational inequality we estimate the order of the difference between the numerical approximation and the solution of a parabolic variational inequality the numerical approximation is obtained using a finite element discretization in space and a finite difference discretization in time which is more general than is used in the literature we obtain better error estimates than those given in the literature the error estimates are compared with numerical experiments

inhomogeneous dirichlet boundary condition in the a posteriori error control of the obstacle problem abstract we study a posteriori error control of finite element approximation of the elliptic obstacle problem with nonhomogeneous dirichlet boundary condition the results in the article are two fold firstly we address the influence of the inhomogeneous dirichlet boundary condition in residual based a posteriori error control of the elliptic obstacle problem secondly by rewriting the obstacle problem in an equivalent form we derive a posteriori error bounds which are in simpler form and efficient to accomplish this we construct and use a post processed solution u h of the discrete solution u h which satisfies the exact boundary conditions sharply although the discrete solution u h may not satisfy we propose two post processing methods and analyze them namely the harmonic extension and a linear extension the theoretical results are illustrated by the numerical results

estimating the quality of life adjusted gap time distribution of successive events subject to censoring summary when treatment effects are studied in the context of successive or recurrent life events separate analyses of the quality of life scores and of the inter event gap times might lead to possibly contradictory conclusions in an attempt to reconcile this we propose a unitary and more comprehensive nonparametric analysis that combines the two separate analyses by introducing the quality of life adjusted gap time concept inverse probability of censoring estimators of the quality of life adjusted gap time joint and con ditional distributions are proposed and are shown to be consistent and asymptotically normal simulations performed in a variety of scenarios indicate that the joint and conditional quality of life adjusted gap time distribution estimators are virtually unbiased with properly estimated standard errors and asymptotic normality features an example from the international breast cancer study group trial v illustrates the use of the proposed estimators

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

on convergence of laplace inversion for the american put option under the cev model in this paper we study the convergence of the inverse laplace transform for valuing american put options when the dynamics of the risky asset is governed by the constant elasticity of variance cev model the cev model is one popular alternative of the black scholes model to describe well the real financial market we calculate various coefficients explicitly and prove that the inverse laplace transform converges absolutely using the properties of whittaker functions

finite element approximation of the parabolic fractional obstacle problem we study a discretization technique for the parabolic fractional obstacle problem in bounded domains the fractional laplacian is realized as the dirichlet to neumann map for a nonuniformly elliptic equation posed on a semi infinite cylinder which recasts our problem as a quasi stationary elliptic variational inequality with a dynamic boundary condition the rapid decay of the solution suggests a truncation that is suitable for numerical approximation we discretize the truncation with a backward euler scheme in time and for space we use first degree tensor product finite elements we present an error analysis based on different smoothness assumptions

estimating survival and association in a semicompeting risks model in many follow up studies patients are subject to concurrent events in this article we consider semicompeting risks data as defined by fine jiang and chappell 2001 biometrika 88 907 919 where one event is censored by the other but not vice versa the proposed model involves marginal survival functions for the two events and a parametric family of copulas for their dependency this article suggests a general method for estimating the dependence parameter when the dependency is modeled with an archimedean copula it uses the copula graphic estimator of zheng and klein 1995 biometrika 82 127 138 for estimating the survival function of the nonterminal event subject to dependent censoring asymptotic properties of these estimators are derived simulations show that the new methods work well with finite samples the copula graphic estimator is shown to be more accurate than the estimator proposed by fine et al 2001 its performances are similar to those of the self consistent estimator of jiang fine kosorok and chappell 2005 scandinavian journal of statistics 33 1 20 the analysis of a data set emphasizing the estimation of characteristics of the observable region is presented as an illustration

copula inference under censoring this paper discusses copula model selection procedures and goodness of fit tests under censoring the proposed methodology is based on a comparison of nonparametric and model based estimators of the probability integral transformation k new weighted estimators for k are introduced the resulting tests are compared to an existing approach by simulation and illustrated with an example involving bleeding changes in a woman s reproductive history

unique existence of bounded continuous solutions on the real line of a class of nonlinear functional equations with complicated deviations abstract the existence and uniqueness of bounded continuous solutions on the real line of a general class of nonlinear functional equations with quite complicated deviations dependent of unknown functions are established in this paper

inferences on the association parameter in copula models for bivariate survival data we investigate two stage parametric and two stage semi parametric estimation procedures for the association parameter in copula models for bivariate survival data where censoring in either or both components is allowed we derive asymptotic properties of the estimators and compare their performance by simulations both parametric and semi parametric estimators of the association parameter are efficient at independence and the parameter estimates in the margins have high efficiency and are robust to misspecification of dependency structures in addition we propose a consistent variance estimator for the semi parametric estimator of the association parameter we apply the proposed methods to an aids data set for illustration

on semi competing risks data summary we consider a variation of the competing risks problem in which a terminal event censors a non terminal event but not vice versa the joint distribution of the events is formulated via a gamma frailty model in the upper wedge where data are observable day et al 1997 with the marginal distributions unspecified an estimator for the association parameter is obtained from a concordance estimating function a novel plug in estimator for the marginal distribution of the non terminal event is shown to be uniformly consistent and to converge weakly to a gaussian process the assumptions on the joint distribution outside the upper wedge are weaker than those usually made in competing risks analyses simulations demonstrate that the methods work well with practical sample sizes the proposals are illustrated with data on morbidity and mortality in leukaemia patients

finite element error estimates for a nonlocal problem in american option valuation based on new exact formulations of american option problems on bounded domains error estimates are established for finite element approximations of american option prices under admissible regularity some numerical results are also presented

a concordance test for independence in the presence of censoring an extension of the definition of kendall s coefficient of concordance is proposed for testing independence in bivariate censored survival times the test statistic counts the number of known concordances minus the number of known discordances the variance of the statistic is calculated for an random censorship model which however does not require independence of the components of the censoring distribution two examples are given

nonparametric estimation of the gap time distributions for serial events with censored data summary in many follow up studies each subject can potentially experience a series of events which may be repetitions of essentially the same event or may be events of entirely diverent natures this paper provides a simple nonparametric estimator for the multivariate distribution function of the gap times between successive events when the follow up time is subject to right censoring the estimator is consistent and upon proper normalisation converges weakly to a zero mean gaussian process with an easily estimated covariance function numerical studies demonstrate that both the distribution function estimator and its covariance function estimator perform well for practical sample sizes an application to a colon cancer study is presented

discrete retract principle for systems of discrete equations abstract a power tool for investigation of various problems of ordinary differential equations as well as delayed differential equations is a retraction method the development of this method is discussed here in the case of system discrete equations corresponding definition of a point of a type of strict egress for a given set with respect to the system of discrete equations u k f k u k is given in an example it is shown that this definition cannot exactly copy the analogous one known from the theory of ordinary differential equations conditions for existence of at least one solution with graph remaining in a given set are formulated the proof is based on idea of a retract principle in construction of a retract mapping the property of continuous dependence on their initial data is used illustrative applications are considered too

non parametric estimation and model checking procedures for marginal gap time distributions for recurrent events for recurrent events there is evidence that misspecification of the frailty distribution can cause severe bias in estimated regression coefficients am j epidemiol 1998 149 404 411 statist med 2006 25 1672 1684 in this paper we adapt a procedure originally suggested in biometrika 1999 86 381 393 for parallel data for checking the gamma frailty to recurrent events to apply the model checking procedure a consistent non parametric estimator for the marginal gap time distributions is needed this is in general not possible due to induced dependent censoring in the recurrent events setting however in biometrika 1999 86 59 70 a non parametric estimator for the joint gap time distributions based on the principle of inverse probability of censoring weights is suggested here we attempt to apply this estimator in the model checking procedure and the performance of the method is investigated with simulations and applied to danish registry data the method is further investigated using the usual kaplan meier estimator and a marginalized estimator for the marginal gap time distributions we conclude that the procedure only works when the recurrent event is common and when the intra individual association between gap times is weak

nonparametric analysis of bivariate gap time with competing risks this article considers nonparametric methods for studying recurrent disease and death with competing risks we first point out that comparisons based on the well known cumulative incidence function can be confounded by different prevalence rates of the competing events and that comparisons of the conditional distribution of the survival time given the failure event type are more relevant for investigating the prognosis of different patterns of recurrence disease we then propose nonparametric estimators for the conditional cumulative incidence function as well as the conditional bivariate cumulative incidence function for the bivariate gap times that is the time to disease recurrence and the residual lifetime after recurrence to quantify the association between the two gap times in the competing risks setting a modified kendall s tau statistic is proposed the proposed estimators for the conditional bivariate cumulative incidence distribution and the association measure account for the induced dependent censoring for the second gap time uniform consistency and weak convergence of the proposed estimators are established hypothesis testing procedures for two sample comparisons are discussed numerical simulation studies with practical sample sizes are conducted to evaluate the performance of the proposed nonparametric estimators and tests an application to data from a pancreatic cancer study is presented to illustrate the methods developed in this article

anti lyapunov method for systems of discrete equations abstract in the present paper problems concerning the qualitative behaviour of solutions of discrete equations are considered a general principle which gives a guarantee that the graph of at least one solution stays in a prescribed domain is formulated as a consequence the auxiliary tool namely a method of regular polyfacial sets is developed and an application of the general principle is given this application is based with exceptions on the fact that points of the boundary of a domain considered are strict egress points such a situation is often in the theory of ordinary differential equations termed the anti lyapunov case illustrative applications and examples are given and open problems are formulated

l error estimate for an approximation of a parabolic variational inequality almost optimall convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase stefan problem the discretization employs piecewise linear finite elements in space and the backward euler scheme in time by means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation the latter bound is obtained by using the smoothing property of the galerkin method

stability of finite delay difference systems the stability of finite delay difference systems has recently been discussed in l in this paper we consider the delay difference systems with infinite delay of the following form x n 1 g n x s s 1 1 1 n g n x 1 where 1 is an integer or w we assume g n 0 0 forn l l l so that 1 has the zero solution x n 0 clearly for any given integer n l 1 and a given bounded initial function 4 l 1 1 no rk there is a unique solution of l denoted by x n n 4 such that it satisfies 1 for all integers n 2 n and x n no 4 444 forn f l l no for any integer m 2 co let 2 m m 1 i and 2 z the integer set if m a for any n e z define ii411 tlnl sup l s 15 s 5 n and for any h 0 let c n 4 z rk i nl h in the sequel we always assume that the variables n r and s take integer values and the corresponding intervals are discrete intervals

estimating survival under a dependent truncation summary the product limit estimator calculated from data subject to random left truncation relies on the testable assumption of quasi independence between the failure time and the truncation time in this paper we propose a model for a truncated sample of pairs xi y satisfying 1 xi a possible dependency between the truncation time and the variable of interest is modelled with a parametric family of copulas the model also features a distribution function fx and a survival distribution sy associated with the marginal behaviours of x and y in the observable region y x semiparametric estimators for these two functions are proposed they do not make any parametric assumption about either fx or sy we derive an estimator for the copula parameter cx based on the conditional kendall s tau we generalise the copula graphic estimators of zheng klein 1995 to truncated variables the asymptotic distributions of all these estimators are then investigated the methods are illustrated with a real dataset on hiv infection by transfusion and by simulations

inverse probability of censoring weighted estimates of kendall s τ for gap time analyses summary in life history studies interest often lies in the analysis of the interevent or gap times and the association between event times gap time analyses are challenging however even when the length of follow up is determined independently of the event process because associations between gap times induce dependent censoring for second and subsequent gap times this article discusses nonparametric estimation of the association between consecutive gap times based on kendall s in the presence of this type of dependent censoring a nonparametric estimator that uses inverse probability of censoring weights is provided estimates of conditional gap time distributions can be obtained following specification of a particular copula function simulation studies show the estimator performs well and compares favorably with an alternative estimator generalizations to a piecewise constant clayton copula are given several simulation studies and illustrations with real data sets are also provided

nonparametric inference on bivariate survival data with interval sampling association estimation and testing in many biomedical applications interest focuses on the occurrence of two or more consecutive failure events and the relationship between event times such as age of disease onset and residual lifetime bivariate survival data with interval sampling arise frequently when disease registries or surveillance systems collect data based on disease incidence occurring within a specific calendar time interval the initial event is then retrospectively confirmed and the subsequent failure event may be observed during follow up in life history studies the initial and two consecutive failure events could correspond to birth disease onset and death the statistical features and bias of observed data in relation to interval sampling were discussed by zhu wang 2012 here we propose nonparametric estimation of the association between bivariate failure times based on kendall s tau for data collected with interval sampling a nonparametric estimator is given where the contribution of each comparable and orderable pair is weighted by the inverse of the associated selection probability analysis methods for bivariate survival data with interval sampling rely on the assumption of quasi independence i e that bivariate failure times and the time of the initial event are independent in the observable region this paper develops a nonparametric test of quasi independence based on a bivariate conditional kendall s tau for such data simulation studies demonstrate that the association estimator and testing procedure perform well with moderate sample sizes illustrations with two real datasets are provided

residual type a posteriori error estimates for elliptic obstacle problems a posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems an instrumental ingredient is a new interpolation operator which requires minimal regularity exhibits optimal approximation properties and preserves positivity both upper and lower bounds are proved and their optimality is explored with several examples sharp a priori bounds for the a posteriori estimators are given and extensions of the results to double obstacle problems are briefly discussed

finite difference methods for pricing american put option with rationality parameter numerical analysis and computing abstract in this paper finite difference methods for pricing american option with rationality parameter are proposed the irrational exercise policy arising in american options is characterized in terms of a rationality parameter the model is formulated in terms of a new nonlinear black scholes equation that requires specific numerical methods although the solution converges to the solution of the classical american option price when the parameter tends to infinity for finite values of the parameter the classical boundary conditions cannot apply and we propose specific ones a logarithmic transformation is used to improve properties of the numerical solution that is constructed by explicit finite difference method numerical analysis provides stability conditions for the methods and its positivity properties of intensity function are studied from the point of view of numerical solution concerning the numerical methods for the original problem we propose the method for time discretization thus including explicit fully implicit and crank nicolson schemes as particular cases the nonlinear term is treated by a newton method the convergence rate is illustrated by numerical examples

an extension of kendall s coefficient of concordance to bivariate interval censored data non parametric tests of independence as well as accompanying measures of association are essential tools for the analysis of bivariate data such tests and measures have been developed for uncensored and right censored failure time data but have not been developed for interval censored failure time data bivariate interval censored data arise in aids studies in which screening tests for early signs of viral and bacterial infection are done at clinic visits because of missed clinic visits the actual times of first positive screening tests are interval censored to handle such data we propose an extension of kendall s coefficient of concordance we apply it to data from an aids study that recorded times of shedding of cytomegalovirus cmv and times of colonization of mycobacterium avium complex mac we examine the performance of our proposed measure through a simulation study

asymptotic formulae for the solutions of a linear delay difference equation consider the linear delay difference equation xn 1 xn anxnk where an s are real numbers and k is a positive integer in this paper we establish asymptotic formulae for the solutions of the above equation assuming n 0 an p for some p 1 the asymptotic formulae are given by an appropriate recursive sequence for p 1 2 3 4 we present explicit formulae we also construct an example which shows that some comparison results in a recent paper of yan and qian are incorrect

flexible maximum likelihood methods for bivariate proportional hazards models this article presents methodology for multivariate proportional hazards ph regression models the methods employ flexible piecewise constant or spline specifications for baseline hazard functions in either marginal or conditional ph models along with assumptions about the association among lifetimes because the models are parametric ordinary maximum likelihood can be applied it is able to deal easily with such data features as interval censoring or sequentially observed lifetimes unlike existing semiparametric methods a bivariate clayton model 1978 biometrika 65 141 151 is used to illustrate the approach taken because a parametric assumption about association is made efficiency and robustness comparisons are made between estimation based on the bivariate clayton model and working independence methods that specify only marginal distributions for each lifetime variable

a posteriori error estimates for variable time step discretizations of nonlinear evolution equations we study the backward euler method with variable time steps for abstract evolution equations in hilbert spaces exploiting convexity of the underlying potential or the angle bounded condition thereby assuming no further regularity we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual these estimators depend solely on the discrete solution and data and impose no constraints between consecutive time steps we also prove that they converge to zero with an optimal rate with respect to the regularity of the solution we apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character 2000 john wiley sons inc

a posteriori error estimations of residual type for signorini s problem this paper presents an a posteriori error analysis for the linear finite element approximation of the signorini problem in two space dimensions a posteriori estimations of residual type are defined and upper and lower bounds of the discretization error are obtained we perform several numerical experiments in order to compare the convergence of the terms in the error estimator with the discretization error

convergence analysis of finite element method for a parabolic obstacle problem abstract a conforming finite element method is proposed and analyzed for numerical approximation of the solution of a parabolic variational inequality of the obstacle problem the model problem which is originally proposed using a general obstacle is reframed as a model problem with zero obstacle but with non homogeneous dirichlet boundary conditions subsequently the discrete problem is reframed and the error analysis proving the convergence of the method is performed the analysis requires a positive preserving interpolation with non homogeneous dirichlet boundary condition and a post processed solution that satisfies the boundary conditions sharply the results in the article extend the results of johnson sinum 1976 for a zero obstacle function to a more general obstacle function

two sample multistate accelerated sojourn times model abstract many medical studies involve observations of patients experiencing multiple consecutive states during their follow up since treatments may have differential effects on these states comparison is of interest with respect to individual sojourn times in this article i generalize the univariate accelerated failure time model for multistate processes and model the treatment effects as state specific time scale changes i propose a two sample inference procedure that accommodates incomplete follow up data this semiparametric procedure based on estimating equations yields a class of estimators that are consistent and asymptotically normal sample based consistent variance estimates are derived numerical studies demonstrate that the procedure performs well for practical sample sizes application to a cancer clinical trial is presented for illustration

boundedness of infinite delay difference systems x n 1 g n x s s i 1 1 n g n x 1 where i is an integer or co g is a volterra functional determined by the integer n and the values of x s for s i 1 1 n and taking values in rk the stability results of 1 have been recently established in 11 however in order to explore the existence of periodic solutions of 1 we need to know not only the stability but also the boundedness of solutions for the linear case of 1 for the relevant results we refer to 2 while for the corresponding results of infinite delay differential systems see 3 41 in this paper we are going to establish theorems on boundedness for 1 in terms of both liapunov functionals and functions with the so called razumikhin techniques before doing this we give the following notations and definitions for any integer m 2 00 let z m m 1 and z z the integer set if m a trivially for any ie e z and a given bounded initial function 4 i i 1 no rk there is a unique solution of l denoted by x n n d such that it satisfies 1 for all integers 17 2 n and x n n 4 d n forn l l l n