temporal process regression summary we consider regression for response and covariates which are temporal processes observed over intervals a functional generalised linear model is proposed which includes extensions of standard models in multi state survival analysis simple nonparametric esti mators of time indexed parameters are developed using working independence estimating equations and are shown to be uniformly consistent and to converge weakly to gaussian processes the procedure does not require smoothing or a markov assumption unlike approaches based on transition intensities the usual definition of optimal estimating equations for parametric models is then generalised to the functional model and the optimum is identified in a class of functional generalised estimating equations simulations demonstrate large efficiency gains relative to working independence at times where censoring is heavy the estimators are the basis for new tests of the covariate effects and for the estimation of models in which greater structure is imposed on the parameters providing novel goodness of fit tests the methodology s practical utility is illustrated in a data analysis

on polynomial factorial diophantine equations we study equations of the form p x n and show that for some classes of polynomials p the equation has only finitely many solutions this is the case say if p is irreducible of degree greater than 1 or has an irreducible factor of relatively large degree this is also the case if the factorization of p contains some large power s of irreducible s for example we can show that the equation x r x 1 n has only finitely many solutions for r 4 but not that this is the case for 1 r 3 although it undoubtedly should be we also study the equation p x h n where h n is one of several other highly divisible sequences proving again that for various classes of polynomials these equations have only finitely many solutions

nonparametric association analysis of bivariate competing risks data although nonparametric analyses of bivariate failure times under independent censoring have been widely studied nonparametric analyses of bivariate competing risks data have not yet been investigated such analyses are important in familial association studies where multiple interacting failure types may violate the independent censoring assumption we develop nonparametric estimators for the bivariate cause specific hazards function and the bivariate cumulative incidence function that are natural analogs of their univariate counterparts and make no assumptions about the dependence of the risks the estimators are shown to be uniformly consistent and to converge weakly to gaussian processes they provide the basis for novel time dependent association measures with the associated inferences yielding tests of cause specific independence in pairs the methodology performs well in simulations with realistic sample sizes its practical utility is illustrated in an analysis of dementia in the cache county st

on the turan number for the hexagon abstract a long standing conjecture of erd s and simonovits is that ex n c 2 k the maximum number of edges in an n vertex graph without a 2 k gon is asymptotically 1 2 n 1 1 k as n tends to infinity this was known almost 40 years ago in the case of quadrilaterals in this paper we construct a counterexample to the conjecture in the case of hexagons for infinitely many n we prove that ex n c 6 3 5 2 5 1 4 3 n 4 3 o n 0 5338 n 4 3 we also show that ex n c 6 n 4 3 o n 0 6272 n 4 3 if n is sufficiently large where is the real root of 16 3 4 2 3 0 this yields the best known upper bound for the number of edges in a hexagon free graph the same methods are applied to find a tight bound for the maximum size of a hexagon free 2 n n bipartite graph

convergence of conditional expectations and strong laws of large numbers for multivalued random variables on etablit des lemmes de fatou et des theoremes de convergence de lebesgue pour des esperances conditionnelles multivaluees de variables aleatoires a valeurs dans des sous ensembles fermes d un espace de banach separable

bivariate survival models induced by frailties abstract when two observed survival times depend via a proportional hazards model on the same unobserved variable called a frailty this common dependence induces an association between the observed times this article considers the class of bivariate survival distributions that can arise in this way extends it to allow negative association and shows that the observable bivariate distribution determines the unobserved frailty distribution up to a scale parameter a cross ratio function easily estimated even from censored data is the key to both the characterization results and inferential procedures and diagnostic plots that are introduced and illustrated by real examples the models considered are the natural counterpart for survival data of the latent variable models that have long been used in factor analysis for continuous data or in latent structure analysis for binary data the main distinguishing feature of survival data is the possibility of censoring which makes the standard methods difficu

the statistical analysis of failure time data preface 1 introduction 1 1 failure time data 1 2 failure time distributions 1 3 time origins censoring and truncation 1 4 estimation of the survivor function 1 5 comparison of survival curves 1 6 generalizations to accommodate delayed entry 1 7 counting process notation bibliographic notes exercises and complements 2 failure time models 2 1 introduction 2 2 some continuous parametric failure time models 2 3 regression models 2 4 discrete failure time models bibliographic notes exercises and complements 3 inference in parametric models and related topics 3 1 introduction 3 2 censoring mechanisms 3 3 censored samples from an exponential distribution 3 4 large sample likelihood theory 3 5 exponential regression 3 6 estimation in log linear regression models 3 7 illustrations in more complex data sets 3 8 discrimination among parametric models 3 9 inference with interval censoring 3 10 discussion bibliographic notes exercises and complements 4 relative risk cox regression models 4 1 introduction 4 2 estimation of beta 4 3 estimation of the baseline hazard or survivor function 4 4 inclusion of strata 4 5 illustrations 4 6 counting process formulas 4 7 related topics on the cox model 4 8 sampling from discrete models bibliographic notes exercises and complements 5 counting processes and asymptotic theory 5 1 introduction 5 2 counting processes and intensity functions 5 3 martingales 5 4 vector valued martingales 5 5 martingale central limit theorem 5 6 asymptotics associated with chapter 1 5 7 asymptotic results for the cox model 5 8 asymptotic results for parametric models 5 9 efficiency of the cox model estimator 5 10 partial likelihood filtration bibliographic notes exercises and complements 6 likelihood construction and further results 6 1 introduction 6 2 likelihood construction in parametric models 6 3 time dependent covariates and further remarks on likelihood construction 6 4 time dependence in the relative risk model 6 5 nonnested conditioning events 6 6 residuals and model checking for the cox model bibliographic notes exercises and complements 7 rank regression and the accelerated failure time model 7 1 introduction 7 2 linear rank tests 7 3 development and properties of linear rank tests 7 4 estimation in the accelerated failure time model 7 5 some related regression models bibliographic notes exercises and complements 8 competing risks and multistate models 8 1 introduction 8 2 competing risks 8 3 life history processes bibliographic notes exercises and complements 9 modeling and analysis of recurrent event data 9 1 introduction 9 2 intensity processes for recurrent events 9 3 overall intensity process modeling and estimation 9 4 mean process modeling and estimation 9 5 conditioning on aspects of the counting process history bibliographic notes exercises and complements 10 analysis of correlated failure time data 10 1 introduction 10 2 regression models for correlated failure time data 10 3 representation and estimation of the bivariate survivor function 10 4 pairwise dependency estimation 10 5 illustration australian twin data 10 6 approaches to nonparametric estimation of the bivariate survivor function 10 7 survivor function estimation in higher dimensions bibliographic notes exercises and complements 11 additional failure time data topics 11 1 introduction 11 2 stratified bivariate failure time analysis 11 3 fixed study period survival studies 11 4 cohort sampling and case control studies 11 5 missing covariate data 11 6 mismeasured covariate data 11 7 sequential testing with failure time endpoints 11 8 bayesian analysis of the proportional hazards model 11 9 some analyses of a particular data set bibliographic notes exercises and complements glossary of notation appendix a some sets of data appendix b supporting technical material bibliography author index subject index

smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications inspired by a recent work by alexander et al j bank finance 30 583 605 2006 which proposes a smoothing method to deal with nonsmoothness in a conditional value at risk problem we consider a smoothing scheme for a general class of nonsmooth stochastic problems assuming that a smoothed problem is solved by a sample average approximation method we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w p 1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem moreover under some metric regularity conditions we obtain an error bound on approximate stationary points the convergence result is applied to a conditional value at risk problem and an inventory control problem

a cutting surface method for uncertain linear programs with polyhedral stochastic dominance constraints in this paper we study linear optimization problems with a newly introduced concept of multidimensional polyhedral linear second order stochastic dominance constraints by using the polyhedral properties of this dominance condition we present a cutting surface algorithm and show its finite convergence the cut generation problem is a difference of convex functions dc optimization problem we exploit the polyhedral structure of this problem to present a novel branch and cut algorithm that incorporates concepts from concave minimization and binary integer programming a linear programming problem is formulated for generating concavity cuts in our case where the polyhedra are unbounded we also present duality results for this problem relating the dual multipliers to utility functions without the need to impose constraint qualifications which again is possible because of the polyhedral nature of the problem numerical examples are presented showing the nature of solutions of our model a corrected pdf has been appended to the original

stability and sensitivity of optimization problems with first order stochastic dominance constraints we analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order we consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance we show that the graph of the feasible set mapping is closed under rather general assumptions we obtain conditions for the continuity of the optimal value and upper semicontinuity of the optimal solutions as well as quantitative stability estimates of lipschitz type furthermore we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional derivatives of the optimal value the estimates are formulated in terms of the dual utility functions associated with the dominance constraints

the difference between consecutive primes ii with enough effort the value of x0 could be determined effectively the paper has much in common with 1 in particular we use the sieve method of harman 4 5 we no longer use zero density estimates however but rather mean value results on dirichlet polynomials similar to those that give rise to such estimates compare for example work of iwaniec and pintz 9 and baker harman and pintz 2 much of the improvement over 1 arises from the use of watt s theorem 11 on a particular kind of mean value more accurate estimates for six dimensional integrals are also used to good effect there is in addition a device which uses a two dimensional sieve to get an asymptotic formula for a one dimensionally sieved set see lemmas 16 17 unfortunately these lemmas which would be of great signi cance for v 0 53 are not very numerically signi cant when v drops to 0 525 the same applies to the roale reversals discussed below let us introduce enough notation to permit an outline of the proof when e is a nite sequence of positive integers counted with multiplicity we write jej for the number of terms of e and ed fm dm 2 eg let p z y

uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming abstract sample average approximation saa is one of the most popular methods for solving stochastic optimization and equilibrium problems research on saa has been mostly focused on the case when sampling is independent and identically distributed iid with exceptions dai et al 2000 9 homem de mello 2008 16 in this paper we study saa with general sampling including iid sampling and non iid sampling for solving nonsmooth stochastic optimization problems stochastic nash equilibrium problems and stochastic generalized equations to this end we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem exponential convergence of estimators of both optimal solutions and m stationary points characterized by mordukhovich limiting subgradients mordukhovich 2006 23 rockafellar and wets 1998 32 are established under mild conditions we also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem

analysis of sample path optimization sample path optimization is a method for optimizing limit functions occurring in stochastic modeling problems such as steady state functions in discrete event dynamic systems it is closely related to retrospective optimization techniques and to m estimation the method has been computationally tested elsewhere on problems arising in production and in project planning with apparent success in this paper we provide a mathematical justification for sample path optimization by showing that under certain assumptions which hold for the problems just mentioned the method will almost surely find a point that is in a specified sense sufficiently close to the set of optimizers of the limit function

long gaps between primes let pn denote the n th prime we prove that max pn 1 x pn 1 pn log x log log x log log log log x log log log x for sufficiently large x improving upon recent bounds of the first three and fifth authors and of the fourth author our main new ingredient is a generalization of a hypergraph covering theorem of pippenger and spencer proven using the rodl nibble method

quantitative stability analysis for minimax distributionally robust risk optimization this paper considers distributionally robust formulations of a two stage stochastic programming problem with the objective of minimizing a distortion risk of the minimal cost incurred at the second stage we carry out a stability analysis by looking into variations of the ambiguity set under the wasserstein metric decision spaces at both stages and the support set of the random variables in the case when the risk measure is risk neutral the stability result is presented with the variation of the ambiguity set being measured by generic metrics of zeta structure which provides a unified framework for quantitative stability analysis under various metrics including total variation metric and kantorovich metric when the ambiguity set is structured by a zeta ball we find that the hausdorff distance between two zeta balls is bounded by the distance of their centers and difference of their radii the findings allow us to strengthen some recent convergence results on distributionally robust optimization where the center of the wasserstein ball is constructed by the empirical probability distribution

quantitative stability in stochastic programming in this paper we study stability of optimal solutions of stochastic programming problems with fixed recourse an upper bound for the rate of convergence is given in terms of the objective functions of the associated deterministic problems as an example it is shown how it can be applied to derivation of the law of iterated logarithm for the optimal solutions it is also shown that in the case of simple recourse this upper bound implies upper lipschitz continuity of the optimal solutions with respect to the kolmogorov smirnov distance between the corresponding cumulative probability distribution functions

a globally convergent sqp method for semi infinite nonlinear optimization abstract a new approach for semi infinite programming problems is presented which belongs to the class of successive quadratic programming sqp methods with trust region technique the proposed algorithm employs the exact l penalty function as a criterion function and incorporates an appropriate scheme for estimating active constraints it is proved that the algorithm is globally convergent under some assumptions numerical experiments show that the algorithm is very promising in practice

the distribution of integers with a divisor in a given interval we determine the order of magnitude of h x y z the number of integers n x having a divisor in y z for all x y and z we also study hr x y z the number of integers n x having exactly r divisors in y z whenr 1 we establish the order of magnitude ofh1 x y z for allx y z satisfying z x 1 2 for every r 2 c 1 and 0 we determine the order of magnitude of hr x y z uniformly for y large and y y logy log 4 1 z min y c x 1 2 as a consequence of these bounds we settle a 1960 conjecture of erd os and some conjectures of tenenbaum one key element of the proofs is a new result on the distribution of uniform order statistics

numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization inspired by the successful applications of the stochastic optimization with second order stochastic dominance ssd model in portfolio optimization we study new numerical methods for a general ssd model where the underlying functions are not necessarily linear specifically we penalize the ssd constraints to the objective under slater s constraint qualification and then apply the well known stochastic approximation sa method and the level function method to solve the penalized problem both methods are iterative the former requires to calculate an approximate subgradient of the objective function of the penalized problem at each iterate while the latter requires to calculate a subgradient under some moderate conditions we show that w p 1 the sequence of approximated solutions generated by the sa method converges to an optimal solution of the true problem as for the level function method the convergence is deterministic and in some cases we are able to estimate the number of iterations for a given precision both methods are applied to portfolio optimization problem where the return functions are not necessarily linear and some numerical test results are reported

error bounds for infinite systems of convex inequalities without slater s condition the feasible set of a convex semi infinite program is described by a possibly infinite system of convex inequality constraints we want to obtain an upper bound for the distance of a given point from this set in terms of a constant multiplied by the value of the maximally violated constraint function in this point apart from this lipschitz case we also consider error bounds of holder type where the value of the residual of the constraints is raised to a certain power we give sufficient conditions for the validity of such bounds our conditions do not require that the slater condition is valid for the definition of our conditions we consider the projections on enlarged sets corresponding to relaxed constraints we present a condition in terms of projection multipliers a condition in terms of slater points and a condition in terms of descent directions for the lipschitz case we give five equivalent characterizations of the validity of a global error bound we extend previous results in two directions first we consider infinite systems of inequalities instead of finite systems the second point is that we do not assume that the slater condition holds which has been required in almost all earlier papers

fast algorithms for multiple evaluations of the riemann zeta function the best previously know algorithm for evaluating the riemann zeta function it with bounded and t large to moderate accuracy was based on the rienmann siegel formula and required on the order of t 1 2 operations for each value that was computed new algorithms are presented in this paper which enable one to compute any single value of it with fixed and t t t t 1 2 in o te operations on numbers of o log t bits for any e 0 for example these algorithms lead to methods for numerically verifying the riemann hypothesis for the first n zeros in what is expected to be o n 1 e operations

sample average approximation of stochastic dominance constrained programs in this paper we study optimization problems with second order stochastic dominance constraints this class of problems allows for the modeling of optimization problems where a risk averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks here we deal with the case of multi variate stochastic dominance under general distributions and nonlinear functions we introduce the concept of mathcal c dominance which generalizes some notions of multi variate dominance found in the literature we apply the sample average approximation saa method to this problem which results in a semi infinite program and study asymptotic convergence of optimal values and optimal solutions as well as the rate of convergence of the feasibility set of the resulting semi infinite program as the sample size goes to infinity we develop a finitely convergent method to find an epsilon optimal solution of the saa problem an important aspect of our contribution is the construction of practical statistical lower and upper bounds for the true optimal objective value we also show that the bounds are asymptotically tight as the sample size goes to infinity

nonparametric bivariate estimation with randomly censored data summary the estimation of a bivariate distribution function with randomly censored data is considered it is assumed that the censoring occurs independently of the lifetimes and that deaths and losses which occur simultaneously can be separated two estimators are developed a reduced sample estimator and a self consistent one it is shown that the latter estimator satisfies a nonparametric likelihood function and is unique up to the final uncensored values in any dimension it jumps at the points of double deaths in both dimensions some key word8 censored data kaplan meier estimator life table product limit estimator survival estimation

non parametric inference for cumulative incidence functions in competing risks studies in the competing risks problem a useful quantity is the cumulative incidence function which is the probability of occurrence by time t for a particular type of failure in the presence of other risks the estimator of this function as given by kalbfleisch and prentice is consistent and properly normalized converges weakly to a zero mean gaussian process with a covariance function for which a consistent estimator is provided a resampling technique is developed to approximate the distribution of this process which enables one to construct confidence bands for the cumulative incidence curve over the entire time span of interest and to perform kolmogorov smirnov type tests for comparing two such curves an aids example is provided

nonparametric estimation of bivariate failure time associations in the presence of a competing risk most research on the study of associations among paired failure times has either assumed time invariance or been based on complex measures or estimators little has accommodated competing risks this paper targets the conditional cause specific hazard ratio henceforth called the cause specific cross ratio a recent modification of the conditional hazard ratio designed to accommodate competing risks data estimation is accomplished by an intuitive non parametric method that localizes kendall s tau time variance is accommodated through a partitioning of space into bins between which the strength of association may differ inferential procedures are developed small sample performance is evaluated and the methods are applied to the investigation of familial association in dementia onset

counting sums of two squares the meissel lehmer method on developpe une variante de la methode de meissel lehmer pour compter les nombres ne depassant pas x qui sont les sommes de deux carres

identifiability of bivariate survival curves from censored data abstract we show that the survival curve is identifiable in bivariate censored data problems under weaker independence assumptions than have commonly been made the common assumption has been mutual independence of t 1 t 2 and z 1 z 2 where t 1 t 2 is the true survival vector z 1 z 2 is a nuisance censoring vector and bivariate right censored data is observed we show that the distribution of t 1 t 2 is identifiable under weaker conditional independence assumptions for distributions with full support bivariate survival analysis is a more powerful analysis tool than univariate analysis if multiple possibly related times are of interest the mutual independence model has become popular as a nonparametric way of analyzing such data analysis of the bivariate problem and analogy with univariate models are used to show that the conditional independence model is more widely applicable as a general nonparametric model for bivariate survival data

lipschitzian properties and stability of a class of first order stochastic dominance constraints considering first order stochastic dominance constraints for random variables arising as optimal values of stochastic programs with linear recourse verifiable sufficient conditions for metric regularity are presented a growth condition developed in r henrion and w romisch math program 84 1999 pp 55 88 has a crucial role in the analysis of the present paper implications regarding stability and sensitivity of optimal values and optimal solutions of stochastic optimization problems involving the dominance constraints considered conclude the paper

counting primes in residue classes we explain how the meissel lehmer lagarias miller odlyzko me thod for computing x can be used to compute ecien tly x k l the number of primes congruent to l modulo k up to x as an application we computed the number of prime numbers of the form 4n 1 less than x for several values of x up to 10 20 and found a new region where x 4 3 is less than x 4 1 near x 10 18 in the 1870 s the german astronomer meissel designed a method to compute the value of x the number of prime numbers up to x the method has been improved by many authors since then the most important improvement is due to lagarias miller odlyzko lmo85 which obtained a method requiring o x 2 3 log x time and computed the value of 4 10 16 further improvements were obtained by the rst author and rivat dr96 with o x 2 3 log 2 x time and who computed 10 18 finally gourdon using ideas originating from lagarias miller odlyzko implemented a parallel version of the algorithm and computed to date values of x up to 4 10 22 for l and k two relatively prime positive integers one denes x k l as the number of prime numbers up to x that are congruent to l modulo k asymptotically the numbers x k l are all of same size k 1 x log x however it has been known for quite some time that there are more primes in the congruence classes that are non quadratic residues modulo k than in those that are heuristically this bias can be explained from the fact that these classes contain more composite numbers than the latter since they contain all the squares see also rs94 for k 4 there are two classes the numbers congruent to 1 modulo 4 the quadratic residues and the numbers congruent to 3 modulo 4 the non quadratic residues in this setting littlewood proved that see ing90 for the notation

stability analysis of two stage stochastic mathematical programs with complementarity constraints via nlp regularization this paper presents numerical approximation schemes for a two stage stochastic programming problem where the second stage problem has a general nonlinear complementarity constraint first the complementarity constraint is approximated by a parameterized system of inequalities with a well known regularization approach by scholtes siam j optim 11 2001 pp 918 936 in deterministic mathematical programs with equilibrium constraints the distribution of the random variables of the regularized two stage stochastic program is then approximated by a sequence of probability measures by treating the approximation problems as a perturbation of the original true problem we carry out a detailed stability analysis of the approximated problems including continuity and local lipschitz continuity of optimal value functions and outer semicontinuity and continuity of the set of optimal solutions and stationary points a particular focus is given to the case where the probability distribution is approximated by

optimization with stochastic dominance constraints we introduce stochastic optimization problems involving stochastic dominance constraints we develop necessary and sufficient conditions of optimality and duality theory for these models and show that the lagrange multipliers corresponding to dominance constraints are concave nondecreasing utility functions the models and results are illustrated on a portfolio optimization problem

statistical robustness in utility preference robust optimization models utility preference robust optimization pro concerns decision making problems where information on decision maker s utility preference is incomplete and has to be elicited through partial information and the optimal decision is based on the worst case utility function elicited a key assumption in the pro models is that the true probability distribution is either known or can be recovered by real data generated by the true distribution in data driven optimization this assumption may not be satisfied when perceived data differ from real data and consequently it raises a question as to whether statistical estimators of the pro models based on perceived data are reliable in this paper we investigate the issue which is also known as qualitative robustness in the literature of statistics huber in robust statistics 3rd edn wiley new york 1981 and risk management kratschmer et al in finance stoch 18 271 295 2014 by utilizing the framework proposed by kratschmer et al 2014 we derive moderate sufficient conditions under which the optimal value and optimal solution of the pro models are robust against perturbation of the exogenous uncertainty data and examine how the tail behaviour of utility functions affects the robustness moreover under some additional conditions on the lipschitz continuity of the underlying functions with respect to random data we establish quantitative robustness of the statistical estimators under the kantorovich metric finally we investigate uniform consistency of the optimal value and optimal solution of the pro models the results cover utility selection problems and stochastic optimization problems as special cases

computing p x the meissel lehmer lagarias miller odlyzko method let x denote the number of primes x our aim in this paper is to present some refinements of a combinatorial method for computing single values of x initiated by the german astronomer meissel in 1870 extended and simplified by lehmer in 1959 and improved in 1985 by lagarias miller and odlyzko we show that it is possible to compute x in o x 2 3 log 2 x time and o x 1 3l log 3 x log log x space the algorithm has been implemented and used to compute 10 18

convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints sample average approximation saa method has recently been applied to solve stochastic programs with second order stochastic dominance ssd constraints in particular hu et al math program 133 171 201 2012 presented a detailed convergence analysis of epsilon optimal values and epsilon optimal solutions of sample average approximated stochastic programs with polyhedral ssd constraints in this paper we complement the existing research by presenting convergence analysis of stationary points when saa is applied to a class of stochastic minimization problems with ssd constraints specifically under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts moreover by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations we derive exponential rate of convergence of stationary points

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

kaplan meier estimate on the plane estimation of the bivariate survival function from censored data is considered the product integral representation of univariate survival functions is generalized to the bivariate case and used to determine identifiability of the survival function of the partially observed data a bivariate analogue of the kaplan meier estimate is introduced and its almost sure consistency is studied extensions to the general multivariate case are sketched

on the mathematical foundations of nondifferentiable optimization in engineering design it is shown by example that a large class of engineering design problems can be transcribed into the form of a canonical optimization problem with inequality constraints involving mar functions such problems are commonly referred to as semi infinite optimization problems the bulk of this paper is devoted to the development of a mathematical theory for the construction of first order nondifferentiable optimization algorithms related to phase i phase ii methods of feasible directions which solve these semi infinite optimization problems the applicability of the theory is illustrated with examples that are relevant to engineering design

on rates of convergence for stochastic optimization problems under non independent and identically distributed sampling in this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size this is a well studied problem in case the samples are independent and identically distributed i e when standard monte carlo simulation is used here we study the case where that assumption is dropped broadly speaking our results show that under appropriate assumptions the rates of convergence for pointwise estimators under a sampling scheme carry over to the optimization case in the sense that convergence of approximating optimal solutions and optimal values to their true counterparts has the same rates as in pointwise estimation we apply our results to two well established sampling schemes namely latin hypercube sampling and randomized quasi monte carlo qmc the novelty of our work arises from the fact that while there has been some work on the use of variance reduction techniques and qmc methods in stochastic optimization none of the existing work to the best of our knowledge has provided a theoretical study on the effect of these techniques on rates of convergence for the optimization problem we present numerical results for some two stage stochastic programs from the literature to illustrate the discussed ideas

cumulative incidence association models for bivariate competing risks data association models like frailty and copula models are frequently used to analyze clustered survival data and evaluate within cluster associations the assumption of noninformative censoring is commonly applied to these models though it may not be true in many situations in this paper we consider bivariate competing risk data and focus on association models specified for the bivariate cumulative incidence function cif a nonparametrically identifiable quantity copula models are proposed which relate the bivariate cif to its corresponding univariate cifs similarly to independently right censored data and accommodate frailty models for the bivariate cif two estimating equations are developed to estimate the association parameter permitting the univariate cifs to be estimated either parametrically or nonparametrically goodness of fit tests are presented for formally evaluating the parametric models both estimators perform well with moderate sample sizes in simulation studies the practical use of the methodology is illustrated in an analysis of dementia associations

quantitative stability analysis of stochastic generalized equations we consider the solution of a system of stochastic generalized equations sge where the underlying functions are mathematical expectation of random set valued mappings sge has many applications such as characterizing optimality conditions of a nonsmooth stochastic optimization problem or equilibrium conditions of a stochastic equilibrium problem we derive quantitative continuity of expected value of the set valued mapping with respect to the variation of the underlying probability measure in a metric space this leads to the subsequent qualitative and quantitative stability analysis of solution set mappings of the sge under some metric regularity conditions we derive aubin s property of the solution set mapping with respect to the change of probability measure the established results are applied to stability analysis of stochastic variational inequality stationary points of classical one stage and two stage stochastic minimization problems two stage stochastic mathematical programs with equilibriu

diophantine approximation with arithmetic functions i we prove a strong simultaneous diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes

on the sign of the difference a l aide d une methode due a sherman lehman on montre qu entre 6 62 10 370 et 6 69 10 370 il y a plus de 10 180 entiers successifs x pour lesquels x li x 0 ce resultat est base sur l hypothese de riemann

nonparametric association analysis of exchangeable clustered competing risks data summary the work is motivated by the cache county study of aging a population based study in utah in which sibship associations in dementia onset are of interest complications arise because only a fraction of the population ever develops dementia with the majority dying without dementia the application of standard dependence analyses for independently right censored data may not be appropriate with such multivariate competing risks data where death may violate the independent censoring assumption nonparametric estimators of the bivariate cumulative hazard function and the bivariate cumulative incidence function are adapted from the simple nonexchangeable bivariate setup to exchangeable clustered data as needed with the large sibships in the cache county study time dependent association measures are evaluated using these estimators large sample inferences are studied rigorously using empirical process techniques the practical utility of the methodology is demonstrated with realistic samples both via simulations and via an application to the cache county study where dementia onset clustering among siblings varies strongly by age

subdifferential convergence in stochastic programs in this paper we discuss convergence behavior of subdifferentials in approximation schemes for stochastic programs this information is useful for solving stochastic programs by nonlinear programming techniques wets variational inequalities and complementarity problems john wiley new york 1980 pp 375 404 showed that epiconvergence of closed convex functions implies the set convergence of the graph of the subdifferentials of these functions this conclusion is not true in general by a counterexample of higle and sen math oper res 17 1992 pp 112 311 we show that epiconvergence of closed convex functions implies set convergence of subdifferentials of these functions at points where the limit function is differentiable and apply this result to convex stochastic programs we also show that similar results can be achieved in three other cases of expectational functionals piecewise smooth integrands continuous probability distributions and loss functions in the case of loss functions we

a semiparametric random effects model for multivariate competing risks data we propose a semiparametric random effects model for multivariate competing risks data when the failures of a particular type are of interest under this model the marginal cumulative incidence functions follow a generalized semiparametric additive model the associations between the cause specific failure times can be studied through dependence parameters of copula functions that are allowed to depend on cluster level covariates a cross odds ratio type measure is proposed to describe the associations between cause specific failure times and its relationship to the dependence parameters is explored we develop a two stage estimation procedure where the marginal models are estimated in the first stage and the dependence parameters are estimated in the second stage the large sample properties of the proposed estimators are derived the proposed procedures are applied to danish twin data to model the cumulative incidence for the age of natural menopause and to investigate the association in the onset of natural menopause between monozygotic and dizygotic twins

efficient estimation in the bivariate censoring model and repairing npmle the npmle in the bivariate censoring model is not consistent the problem is caused by the singly censored observations in this paper we prove that the npmle based on interval censoring the singly censored observations is ecient for this reduced data and moreover if we let the width of the interval converge to zero slowly enough then the npmle is also ecient for the original data we are able to determine a lower bound for the rate at which the bandwidth should converge to zero the eciency proof uses the general identity which holds for npmle of a linear parameter in convex models as proved in van der laan 1993a

modelling multivariate failure time associations in the presence of a competing risk there has been much research on analysing multivariate failure times but little that has accommodated failures that arise in the presence of a competing failure process this paper studies the problem of describing associations among times to such failures it proposes a modified conditional hazard ratio measure of association that is tailored to competing risks data develops frailty models and a nonparametric method for describing the proposed measure and contrasts estimation by proposed methods with the standard of treating competing risks as independently censoring failure times due to targeted causes the methods are investigated on simulated and real data

optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints we consider a new class of optimization problems involving stochastic dominance constraints of second order we develop a new splitting approach to these models optimality conditions and duality theory these results are used to construct special decomposition methods

inference for events with dependent risks in multiple endpoint studies abstract kaplan meier and cumulative incidence functions are not sufficient descriptive devices for studies that have multiple time to event endpoints for example in cancer treatment research the probability of tumor recurrence conditional on not having died from treatment related toxicities and the prevalence of graft versus host disease among leukemia free patients surviving a bone marrow transplant are of interest these quantities can be estimated nonparametrically using simple functions of several kaplan meier and cumulative incidence estimates for events with possibly dependent risks we derive asymptotic distribution theory for such functions by representing kaplan meier cumulative incidence and cumulative hazard estimators as sums of iid random variables variance estimation also follows directly from this representation two sample test statistics with asymptotic null distribution theory are presented several examples illustrate the utility of these results

holder and lipschitz stability of solution sets in programs with probabilistic constraints we study perturbations of a stochastic program with a probabilistic constraint and r concave original probability distribution first we improve our earlier results substantially and provide conditions implying holder continuity properties of the solution sets w r t the kolmogorov distance of probability distributions secondly we derive an upper lipschitz continuity property for solution sets under more restrictive conditions on the original program and on the perturbed probability measures the latter analysis applies to linear quadratic models and is based on work by bonnans and shapiro the stability results are illustrated by numerical tests showing the different asymptotic behaviour of parametric and nonparametric estimates in a program with a normal probabilistic constraint

a new bound for the smallest x with p x li x let x denote the number of primes li x this improves earlier bounds of skewes lehman and te riele we also plot more than 10000 values of x li x in four different regions including the regions discovered by lehman te riele and the authors of this paper and a more distant region in the vicinity of 1 617 x 10 9608 where x appears to exceed li x by more than 18x 1 2 log x the plots strongly suggest although upper bounds derived to date for li x x are not sufficient for a proof that x exceeds li x for at least 10 311 integers in the vicinity of 1 398 x 10 316 if it is possible to improve our bound for x li x by finding a sign change before 10 316 our first plot clearly delineates the potential candidates finally we compute the logarithmic density of li x x and find that as x departs from the region in the vicinity of 1 62 x 10 9608 the density is 1 2 7 x 10 7 99999973 and that it varies from this by no more than 9 x 10 8 over the next 10 30000 integers this should be compared to rubinstein and sarnak

computing prime harmonic sums 1801241484456448000 andthenusedaprecomputedtableofsumsintheschoenfeldinterval therebyavoiding interpolation or binary search and multiple evaluations of s x todiscovers 1801241230056600467 3 99999 99999 99999 99966 ands 1801241230056600523 4 00000 00000 00000 00021 there are no primes between these two xvalues this computation took roughly one week on two workstations franz mertensjosephine mitchell lowell schoenfeld

stability analysis of stochastic programs with second order dominance constraints in this paper we present a stability analysis of a stochastic optimization problem with stochastic second order dominance constraints we consider a perturbation of the underlying probability measure in the space of regular measures equipped with pseudometric discrepancy distance romisch in stochastic programming elsevier amsterdam pp 483 554 2003 by exploiting a result on error bounds in semi infinite programming due to gugat math program ser b 88 255 275 2000 we show under the slater constraint qualification that the optimal value function is lipschitz continuous and the optimal solution set mapping is upper semicontinuous with respect to the perturbation of the probability measure in particular we consider the case when the probability measure is approximated by an empirical probability measure and show an exponential rate of convergence of the sequence of optimal solutions obtained from solving the approximation problem the analysis is extended to the stationary points

uniform laws of large numbers for set valued mappings and subdifferentials of random functions we derive a uniform strong law of large numbers lln for random set valued mappings the result can be viewed as an extension of both a uniform lln for random functions and lln for random sets we apply the established results to a consistency analysis of stationary points of sample average approximations of nonsmooth stochastic programs

stability in two stage stochastic programming we analyze the effect of changes in problem functions and or distributions in certain two stage stochastic programming problems with recourse under reasonable assumptions the locally optimal value of the perturbed problem will be continuous and the corresponding set of local optimizers will be upper semicontinuous with respect to the parameters including the probability distribution in the second stage

power spectrum of the difference between the prime number counting function and riemann s function 1 f2 abstract gamba et al phys lett a 145 1990 106 numerically estimated the power spectrum of the difference between the prime number counting function x and its approximate given by riemann s function r x but did not determine the functional dependence of the spectrum on the frequency our numerical estimates based on samples of the difference function r x x at integer x up to x 109 strongly suggest that the difference function has a 1 f2 power spectrum

new maximal prime gaps and first occurrences the search for first occurrences of prime gaps and maximal prime gaps is extended to 10 15 new maximal prime gaps of 806 and 906 are found and sixty two previously unpublished first occurrences are found for gaps varying from 676 to 906

estimating twin concordance for bivariate competing risks twin data for twin time to event data we consider different concordance probabilities such as the casewise concordance that are routinely computed as a measure of the lifetime dependence correlation for specific diseases the concordance probability here is the probability that both twins have experienced the event of interest under the assumption that both twins are censored at the same time we show how to estimate this probability in the presence of right censoring and as a consequence we can then estimate the casewise twin concordance in addition we can model the magnitude of within pair dependence over time and covariates may be further influential on the marginal risk and dependence structure we establish the estimators large sample properties and suggest various tests for example for inferring familial influence the method is demonstrated and motivated by specific twin data on cancer events with the competing risk death we thus aim to quantify the degree of dependence through the casewise concordance function and show a significant genetic component

convergence analysis for distributionally robust optimization and equilibrium problems in this paper we study distributionally robust optimization approaches for a one stage stochastic minimization problem where the true distribution of the underlying random variables is unknown but it is possible to construct a set of probability distributions which contains the true distribution and optimal decision is taken on the basis of the worst possible distribution from that set we consider the case when the distributional set which is also known as the ambiguity set varies and its impact on the optimal value and the optimal solutions a typical example is when the ambiguity set is constructed through samples and we need to look into the impact of increasing the sample size the analysis provides a unified framework for convergence of some problems where the ambiguity set is approximated in a process with increasing information on uncertainty and extends the classical convergence analysis in stochastic programming the discussion is extended briefly to a stochastic nash equilibrium problem wh