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

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

a proportional hazards model for interval censored failure time data this paper develops a method for fitting the proportional hazards regression model when the data contain left right or interval censored observations results given for testing the hypothesis of a zero regression coefficient lead to a generalization of the log rank test for comparison of several survival curves the method is used to analyze data from an animal tumorigenicity study and also a clinical trial

differentiability of the metric projection in hilbert space a study is made of differentiability of the metric projection p onto a closed convex subset k of a hilbert space h when k has nonempty interior the gateaux or fr het smoothness of its boundary can be related with some precision to gateaux or frechet differentiability properties of p for instance combining results in 3 with earlier work of r d holmes shows that k has a c2 boundary if and only if p is cl in h k and its derivative p has a certain invertibility property at each point an example in 5 shows that if the c2 condition is relaxed even slightly then p can be nondifferentiable fr het in h k

spectrum and analytical indices of the c algebra of wiener hopf operators we study multivariate generalisations of the classical wiener hopf algebra which is the c algebra generated by the wiener hopf operators given by convolutions restricted to convex cones by the work of muhly and renault this c algebra is known to be isomorphic to the reduced c algebra of a certain restricted action groupoid given by the action of euclidean space on a certain compactification using groupoid methods we construct composition series for the wiener hopf c algebra by a detailed study of this compactification we compute the spectrum and express homomorphisms in k theory induced by the symbol maps which arise by the subquotients of the composition series in analytical terms namely these symbols maps turn out to be given by an analytical family index of a continuous family of fredholm operators in a subsequent paper we also obtain a topological expression of these indices

algebraic gauss manin systems and brieskorn modules we study the algebraic gauss manin system and the algebraic brieskorn module associated to a polynomial mapping with isolated singularities since the algebraic gauss manin system does not contain any information on the cohomology of singular fibers we first construct a non quasi coherent sheaf which gives the cohomology of every fiber then we study the algebraic brieskorn module and show that its position in the the algebraic gauss manin system is determined by a natural map to quotients of local analytic gauss manin systems and its pole part by the vanishing cycles at infinity comparing it with the deligne extension this implies for example a formula for the determinant of periods in the two dimensional case we can describe the global structure of the algebraic gauss manin system rather explicitly

perturbed optimization in banach spaces iii semi infinite optimization this paper is devoted to the study of perturbed semi infinite optimization problems i e minimization over er n with an infinite number of inequality constraints we obtain the second order expansion of the optimal value function and the first order expansion of approximate optimal solutions in two cases i when the number of binding constraints is finite and ii when the inequality constraints are parametrized by a real scalar these results are partly obtained by specializing the sensitivity theory for perturbed optimization developed in part i cf siam j control optim 34 1996 pp 1151 1171 and deriving specific sharp lower estimates for the optimal value function which take into account the curvature of the positive cone in the space c omega of continuous real valued functions

center conditions rigidity of logarithmic differential equations abstract in this paper we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again this is a generalization of ilyashenko s result on hamiltonian differential equations the main tools are picard lefschetz theory of a polynomial with complex coefficients in two variables specially the gusein zade a campo s theorem on calculating the dynkin diagram of the polynomial and the action of gauss manin connection on the so called brieskorn lattice petrov module of the polynomial we will also generalize j p francoise recursion formula and condition for a polynomial which is a product of lines in a general position some applications on the cyclicity of cycles and the bautin ideals will be given

inference based on imputed failure times for the proportional hazards model with interval censored data abstract we propose an approach to the proportional hazards model for interval censored data in which parameter estimates are obtained by solving estimating equations that are the partial likelihood score equations for the full data proportional hazards model averaged over all rankings of imputed failure times consistent with the observed censoring intervals imputed failure times are generated using a parametric estimate of the baseline distribution the parameters of the baseline distribution are estimated simultaneously with the proportional hazards regression parameters although a parametric form for the baseline distribution must be specified simulation studies show that the method performs well even when the baseline distribution is misspecified the estimating equations are solved using monte carlo techniques we present a recursive stochastic approximation scheme that converges to the zero of the estimating equations the solution has a random error that is asymptotically normally distributed w

optimal control problems with partially polyhedric constraints this paper discusses a class of state constrained optimal control problems for which it is possible to formulate second order necessary or sufficient conditions for local optimality or quadratic growth that do not involve all curvature terms for the constraints this kind of result is classical in the case of polyhedric control constraints our theory of optimization problems with partially polyhedric constraints allows to extend these results to the case when the control constraints are polyhedric in the presence of state constraints satisfying some specific hypotheses the analysis is based on the assumption that some strict semilinearized qualification condition is satisfied we apply the theory to some optimal control problems of elliptic equations with state and control constraints

perturbed optimization in banach spaces i a general theory based on a weak directional constraint qualification using a directional form of constraint qualification weaker than robinson s we derive an implicit function theorem for inclusions and use it for first and second order sensitivity analyses of the value function in perturbed constrained optimization we obtain holder and lipschitz properties and under a no gap condition first order expansions for exact and approximate solutions as an application differentiability properties of metric projections in hilbert spaces are obtained using a condition generalizing polyhedricity we also present in the appendix a short proof of a generalization of the convex duality theorem in banach spaces

differential stability of two stage stochastic programs two stage stochastic programs with random right hand side are considered optimal values and solution sets are regarded as mappings of the expected recourse functions and their perturbations respectively conditions are identified implying that these mappings are directionally differentiable and semidifferentiable on appropriate functional spaces explicit formulas for the derivatives are derived special attention is paid to the role of a lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function

efficient computation of subset selection probabilities with application to cox regression summary an efficient method is presented to compute the probability of selection of a specified subset from the set of all subsets of a fixed size where the subsets are taken from a population whose units have varying individual probabilities of selection the problem is motivated by the computation of the exact marginal likelihood for the cox proportional hazards model

an efficient alternative to the stratified cox model analysis consider a typical two treatment randomized clinical trial involving a time to event endpoint with randomization stratified by a categorical prognostic factor for example gender at the design stage it is often assumed that the treatment hazard ratio hr is constant across the strata and the data are commonly analyzed using the stratified cox proportional hazards model we caution that this ubiquitous approach is needlessly risky because departures from the assumption of the hr being the same for all the strata can result in a notably biased and or less powerful analysis an alternative approach is proposed in which first the log hr is estimated separately for each stratum using an unstratified cox model and then the stratum specific estimates are combined for overall inference using either sample size or minimum risk stratum weights the advantages of the proposed two step analysis versus the common one step stratified cox model analysis are illustrated using simulations that were conducted to support the design of a vaccine clinical trial

on the mixed hodge structure on the cohomology of the milnor fibre in 18 the second author introduced a mixed hodge structure on the cohomology of the milnor fibre of an isolated hypersurface singularity for the definition of a mixed hodge structure el 17 sect 3 4 the weight filtration is essentially the monodromy weight filtration 18 sect 4 and thus simplifies the jordan normal form of the unipotent part of the monodromy the significance of the hodge filtration however is not so dear and its description seems difficult to use this paper gives another description of the hodge filtration which we hope is easier to understand and apply its definition does not use resolution of singularities instead it relies on the theory of holonomic modules in one variable with regular singularities this paper arose from conversations at the 1980 arbeitstagung in bonn where varchenko s conjectured hodge filtration 22 had been discussed in brieskorn s seminar we have since learned that varchenko obtained similar results to those in this paper in the summer and autumn of 1980 23 25 in sect 1 the milnor fibration is embedded in a family of smooth projective hypersurfaces the hodge theory of a smooth projective hypersurface is explained in the fashion of brylinski 2 in sect 2 in sect 3 we recall the description of the gauss manin system from 8 our new formula for the hodge filtration is explained in sects 4 5 in sect 6 we prove that it gives the same result as 18 we prove a thom sebastiani formula for the hodge filtration in sect 7 which leads to a proof of conjecture 5 4 of 18 in sect 8 we show how to prove varchenko s result about the jordan normal forms of multiplication by f in the jacobian ring of f and the logarithm of the unipotent monodormy of f with our method in the last chapter the mixed hodge structure is calculated for two examples

second order optimality conditions based on parabolic second order tangent sets in this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints our analysis is based on various concepts of second order tangent sets and parametric duality we introduce a condition called second order regularity under which there is no gap between the corresponding second order necessary and second order sufficient conditions we show that the second order regularity condition always holds in the case of semidefinite programming

perturbed optimization in banach spaces ii a theory based on a strong directional constraint qualification we study the sensitivity of the optimal value and optimal solutions of perturbed optimization problems in two cases the first one is when multipliers exist but only the weak and not the strong second order sufficient optimality condition is satisfied the second case is when no lagrange multipliers exist to deal with these pathological cases we are led to introduce a directional constraint qualification stronger than in part i of this paper which reduces to the latter in the important case of equality inequality constrained problems we give sharp upper estimates of the cost based on paths varying as the square root of the perturbation parameter and under a no gap condition obtain the first term of the expansion for the cost when multipliers exist we study the expansion of approximate solutions as well we show in the appendix that the strong directional constraint qualification is satisfied for a large class of problems including regular problems in the sense of robinson

optimal control of a parabolic equation with time dependent state constraints in this paper we study the optimal control problem of the heat equation with a cubic nonlinearity by a distributed control over a subset of the domain in the presence of a state constraint the latter is integral over the space and has to be satisfied each time using for the first time the technique of alternative optimality systems in the context of optimal control of partial differential equations we show that both the control and multiplier are continuous in time under some natural geometric hypotheses we can prove that extended polyhedricity holds allowing us to obtain no gap second order optimality conditions that characterize quadratic growth an expansion of the value function and of approximate solutions can be computed for a directional perturbation of the right hand side of the state equation

on the choice of variance for the log rank test summary the log rank test is widely used for comparison of survival curves this paper examines various estimators for the variance of the log rank test statistic these include the mantel haenszel variance the permutation variance of peto peto and several newly proposed estimators the results generalize to a class of test statistics proposed by tarone ware the log rank test is widely used for comparison of survival curves there are two versions of the test which differ in the choice of variance for the same test statistic t these are the mantel haenszel variance an estimate of var t which is unbiased independently of differences in censoring and the permutation variance of peto peto 1972 for which the variance is computed under the assumption of equal censoring this paper examines the choice of variance it is observed that for equal sample sizes the mantel haenszel variance tends to underestimate the true variance when t is large in absolute value as this is the situation where rejection of the null hypothesis is to be seriously considered it appears that the true type i error will tend to be higher than that assumed and the reported p value will tend to be too low this phenomena is discussed also by morton 1978 in 4 it is shown that under equal sample sizes the permutation variance tends to overestimate the true variance when censoring is unequal it thus appears prudent to use the permutation variance rather than the mantel haenszel variance in assessing significance in 5 alternative variance estimators are proposed the results immediately generalize to a class of test statistics proposed by tarone ware 1977 this point is discussed in 6

sensitivity analysis of optimization problems under second order regular constraints we present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition this is a technical condition that is always satisfied in the case of semi definite optimization we derive lipschitz and holder expansions of approximate optimal solutions under a directional constraint qualification hypothesis and various second order sufficient conditions that take into account the curvature of the set defining the constraints of the problem we show how the theory applies to semi infinite programs in which the contact set is a smooth manifold and the quadratic growth condition in the constraint space holds and discuss the differentiability of metric projections as well as the moreau yosida regularization finally we show how the theory applies to semi definite optimization

relative cohomology with respect to a lefschetz pencil let m be a complex projective manifold of dimension n 1 and f a meromorphic function on m obtained by a generic pencil of hyperplane sections of m the n th cohomology vector bundle of f0 f m r where r is the set of indeterminacy points of f is defined on the set of regular values of f0 and we have the usual gauss manin connection on it following brieskorn s methods in br we extend the n th cohomology vector bundle of f0 and the associated gauss manin connection to p1 by means of differential forms the new connection turns out to be meromorphic on the critical values of f0 we prove that the meromorphic global sections of the vector bundle with poles of arbitrary order at p1 is isomorphic to the brieskorn module of f in a natural way and so the brieskorn module in this case is a free c t module of rank n where c t is the ring of polynomials in t and n is the dimension of n th cohomology group of a regular fiber of f0

relative risk estimation and inference using a generalized logrank statistic when comparing two survival distributions with proportional hazard functions the logrank test is optimal for testing the null hypothesis that the constant hazard ratio relative risk is one in this paper we focus on i testing for departures from a relative risk other than one and ii estimation of the relative risk the standard tool to address both i and ii is the cox proportional hazards model however the performance of the cox model can be less than optimal with small samples we show why this is the case and propose a simple alternative method of estimation and inference based on a generalized logrank glr statistic while the glr and cox model approaches are asymptotically similar empirical results reveal that the glr approach is notably more efficient than the cox model when the number of subjects is small 100 subjects per treatment group an example based on survival times of cervical cancer patients is used to illustrate the proposed methodology

the incidence of monotone likelihood in the cox model in estimating covariate coefficients for the cox proportional hazards model there is a nonzero probability for any finite sample that the maximum likelihood estimate will be infinite this implies that in monte carlo studies the variance is not a suitable estimator of precision this also may necessitate a stratified analysis when multiple covariates are involved a simple method is given for detecting and handling these infinite estimates

sensitivity analysis for relaxed optimal control problems with final state constraints in this article we compute a second order expansion of the value function of a family of relaxed optimal control problems with final state constraints parameterized by a perturbation variable the sensitivity analysis is performed for controls that we call r strong solutions they are optimal solutions with respect to the set of feasible controls with a uniform norm smaller than a given r and having an associated trajectory in a small neighborhood for the uniform norm in this framework relaxation enables us to consider a wide class of perturbations and therefore to derive sharp estimates of the value function

asymptotically efficient two step estimators of the hazards ratio for follow up studies and survival data summary asymptotically efficient estimators of a common hazard rate ratio for follow up studies and the proportional hazards ratio for survival studies are obtained by a single iteration of the mantelhaenszel estimator appropriate for each setting estimators of their variance are also developed the two step estimator for survival data and its variance estimator are shown by simulation to be minimally biased and the estimator is shown to be efficient relative to the cox partial likelihood estimator in small samples

directional differentiability of metric projections onto moving sets at boundary points soient x y des espaces normes de dimension finie et x y une multifonction assignant a chaque xex un ensemble ferme x cy a x sont associees la fonction distance d x y r et la projection metrique a valeur d ensemble x y y on etudie les proprietes differentielles de au voisinage d un point x 0 y 0 avec y 0 point limite de l ensemble x 0

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal