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

the efficiency and consistency of approximations to the jackknife variance estimators abstract the problem considered is the computation reduction for general delete d jackknife variance estimators the delete d jackknife estimator was proved consistent shao and wu 1986 and in this article its mean squared error is shown to have order o n 2 where n is the sample size these properties are not shared by the traditional delete 1 jackknife in some situations use of the delete d jackknife however requires n d recomputations of a point estimate which increases rapidly as n and d increase using techniques from survey sampling a shortcut can be taken with evaluated only m times m n d the efficiency and consistency of the resulting jackknife sampling hybrid variance estimators jsve s are studied if m is chosen so that n m 0 the increase in mean squared error by using the jsve is relatively negligible for the consistency of jsve m is sufficient hence the jsve with m n can also be used to alleviate the computational burden for the delete 1 jackknife in th

lorelogram a regression approach to exploring dependence in longitudinal categorical responses abstract we propose flexible regression estimators of the marginal pairwise log odds ratio measure of association for longitudinal categorical responses the function that we estimate is the log odds ratio analog of the correlogram hence we name the function the lorelogram measuring the association of categorical responses on the log odds scale allows ease of interpretation and allows pairwise association to remain unconstrained by the marginal means a feature not shared by correlations with binary or multinomial responses estimation of the function is achieved through the use of standard parametric estimating equations or through an extension of generalized additive models that allows nonparametric estimation of dependence functions for fixed smoothing parameters we apply the methodology to binary longitudinal data where scientific interest focuses on the dependence structure

analysis of longitudinal data random coefficient regression modelling we review random coefficient regression rcr models and methods for fitting these models from an applications perspective methods for data with exponential family distributions are presented with the gaussian distribution as a special case attention is given to interpretation of fixed effects and the correlation structures implied by rcr models estimation methods are presented wtih computational approaches problems associated with testing fixed effects include accurate variance estimation and robustness to misspecification of the covariance structure methods for model selection and assessment are presented an example is used to demonstrate recommended approaches

jackknifing l statistics with smooth weight functions abstract the behavior of linear combinations of order statistics l statistics under jackknifing is discussed the asymptotic behavior of a jackknifed l statistic is described and the pseudovalue based variance estimate is shown to be consistent under moderate smoothness and a trimming condition on the weight function the results extend easily to functions of l statistics and thus answer a question posed in miller 1974 monte carlo results support small sample applicability of the large sample results for the construction of approximate confidence intervals

on asymptotic normality and variance estimation for nondifferentiable survey estimators survey estimators of population quantities such as distribution functions and quantiles contain nondifferentiable functions of estimated quantities the theoretical properties of such estimators are substantially more complicated to derive than those of differentiable estimators in this article we provide a unified framework for obtaining the asymptotic design based properties of two common types of nondifferentiable estimators estimators of the first type have an explicit expression while those of the second are defined only as the solution to estimating equations we propose both analytical and replication based design consistent variance estimators for both cases based on kernel regression the practical behaviour of the variance estimators is demonstrated in a simulation experiment

asymptotic results in jackknifing nonsmooth functions of the sample mean vector the asymptotic behavior of jackknife estimators and jackknife variance estimators is investigated for nonsmooth functions of the sample mean vector an application of jackknifing a suitable estimator of the intrinsic diversity profile is also presented

incorrect asymptotic size of subsampling procedures based on post consistent model selection estimators abstract subsampling and the m out of n bootstrap have been suggested in the literature as methods for carrying out inference based on post model selection estimators and shrinkage estimators in this paper we consider a subsampling confidence interval ci that is based on an estimator that can be viewed either as a post model selection estimator that employs a consistent model selection procedure or as a super efficient estimator we show that the subsampling ci of nominal level 1 for any 0 1 has asymptotic confidence size defined to be the limit of finite sample size equal to zero in a very simple regular model the same result holds for the m out of n bootstrap provided m 2 n 0 and the observations are i i d similar zero asymptotic confidence size results hold in more complicated models that are covered by the general results given in the paper and for super efficient and shrinkage estimators that are not post model selection estimators based on these results subsampling and the m out of n bootstrap are not recommended for obtaining inference based on post consistent model selection or shrinkage estimators

three circle theorem and dimension estimate for holomorphic functions on kahler manifolds the classical hadamard three circle theorem is generalized to complete kahler manifolds more precisely we show that the nonnegativity of the holomorphic sectional curvature is a necessary and sufficient condition for the three circle theorem two sharp monotonicity formulae are derived as corollaries among applications we obtain sharp dimension estimates with rigidity of holomorphic functions with polynomial growth when the holomorphic sectional curvature is nonnegative when the bisectional curvature is nonnegative the sharp dimension estimate was due to ni

poisson equation poincare lelong equation and curvature decay on complete kahler manifolds in the first part of this work the poisson equation on complete noncompact manifolds with nonnegative ricci curvature is studied sufficient and necessary conditions for the existence of solutions with certain growth rates are obtained sharp estimates on the solutions are also derived in the second part these results are applied to the study of curvature decay on complete kahler manifolds in particular the poincare lelong equation on complete noncompact kahler manifolds with nonnegative holomorphic bisectional curvature is studied several applications are then derived which include the steinness of the complete kahler manifolds with nonnegative curvature and the flatness of a class of complete kahler manifolds satisfying a curvature pinching condition liouville type results for plurisubharmonic functions are also obtained 0 introduction in this paper we will discuss the poisson equation on complete noncompact manifolds and derive some applications on kahler manifolds let mm be a complete noncompact kahler manifold where m 2 is the complex dimension assume m has nonnegative holomorphic bisectional curvature and has maximal volume growth such that the scalar curvature decays like r 2 where r is the distance from a fixed point then it was proved in 20 by mok siu yau that one can solve the following poincare lelong equation 0 1 1 u received november 25 2000 and in revised form june 28 2001 the first author was partially supported by nsf grant dms9970284 usa the second author was partially supported by nsf of china project 10001001 and the third author was partially supported by earmarked grant of hong kong cuhk4217 99p

model checking techniques based on cumulative residuals depuis longtemps on utilise les residus pour apprecier graphiquement et muneriquement l adequation de modeles de regression l analyse conventionnelle des residus fondee sur les graphes des residus bruts ou des courbes lissees est eminemment subjective tandis que la plupart des tests numeriques d ajustement informent peu sur la nature de l inadequation nous developpons des techniques d examen d un modele objectives et informatives qui reposent soit sur les sommes des residus cumulees selon certaines coordonnees comme une covariable on le predicteur lineaire soit sur des regroupements partiels sommes ou moyennes mobiles selon ces coordonnees pour une grande variete de modeles statistiques et de structures des observations dont les modeles lineaires generalises sur observations independantes ou correlees les distributions de ces processus stochastiques sous le modele postule peuvent etre approchees par des processus gaussiens centres faciles a simuler on pent done comparer graphiquement et numeriquement chaque processus observe a une serie de realisations du ment chaque processus observe a une serie de realisations du processus gaussien ces comparaisons permettent d evaluer objectivement dans quelle mesure une tendance observee sur un graphique serait attribuable a une mauvaise specification du modele plutot qu aux fluctuations d echantillonnage ces techniques sont particulierement utiles pour l examen de la fonction de lien ou de la forme fonctionnelle d une covariable plusieurs exemples medicaux illustrent cette approche

nonsmooth analysis and frechet differentiability of m functionals abstract a necessary requirement for existence of the frechet derivative is that the defining psi function is uniformly bounded and this naturally excludes those nonrobust estimators such as the maximum likelihood estimator in normal parametric models in this paper the methods of nonsmooth analysis described in the book by f h clarke 1983 are introduced to the theory of statistical expansions and are used here in the proofs of weak continuity and frechet differentiability of m functionals subsequently the conditions for frechet differentiability given in clarke 1983 can be relaxed to include most popular m functionals additional keywords distribution functions m estimators robustness gross error sensitivity weak continuity asymptotic expansions asymptotic normality selection functional local uniqueness author

semiparametric modeling of repeated measurements under outcome dependent follow up in regression analysis of repeated measurements that are taken at subject specific times the availability of the outcome data may be related to the past outcome and to other measured variables that are not in the intended regression model in this paper we propose a natural extension of the semiparametric regression procedure of lin and ying j am stat assoc 2001 96 103 126 by building a class of inverse intensity rate ratio weighted estimators that accommodate such outcome dependent follow up the estimators have a closed form are radicaln consistent asymptotically normal and do not require estimation of any infinite dimensional parameters we give several simulations to demonstrate the estimator s performance and show a sensitivity study under follow up with various degrees of dependence on outcome related variables we illustrate our approach using data from a randomized health services research study with noncompliance to scheduled visits

a note on bootstrapping the sample median efron 1979 1982 in his treatment of the bootstrap discusses its use for estimation of the asymptotic variance of the sample median in the sampling situation of independent and identically distributed random variables with common distribution function f having a positive derivative continuous in a neighborhood of the true median mu the natural conjecture that the bootstrap variance estimator converges almost surely to the asymptotic variance is shown by an example to be false unless a tail condition is imposed on f we prove that such strong convergence does hold under the fairly nonrestrictive condition that e lbrack mid x alpha rbrack 0

differentiability of statistical functionals and consistency of the jackknife in statistical applications the unknown parameter of interest can frequently be defined as a functional 0 t f where f is an unknown population statistical inferences about 0 are usually made based on the statistic t fn where fn is the empirical distribution assessing t fn as an estimator of 0 or making large sample inferences usually requires a consistent estimator of the asymptotic variance of t fn asymptotic behaviour of the jackknife variance estimator is closely related to the smoothness of the functional t this paper studies the smoothness of t through the differentiability of t and establishes some general results for the consistency of the jackknife variance estimators the results are applied to some examples in which the statistics t fn are l m estimators and some test statistics

classification of manifolds with weakly 1 4 pinched curvatures we show that a compact riemannian manifold with weakly pointwise 1 4 pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form more generally we classify all compact locally irreducible riemannian manifolds m with the property that m r 2 has non negative isotropic curvature

asymptotic properties of the balanced repeated replication method for sample quantiles inference including variance estimation can be made from stratified samples by selecting a balanced set of subsamples this balanced subsampling method is generically called the balanced repeated replication method in survey data analysis which includes mccarthy s balanced half samples method and its extensions for more general stratified designs we establish the asymptotic consistency of the balanced repeated replication variance estimators when the parameter of interest is the population quantile the consistency results also hold when balanced subsampling is replaced by random subsampling as a key technical prerequisite we prove a bahadurtype representation for sample quantiles in stratified random sampling

consistency of the jackknife after bootstrap variance estimator for the bootstrap quantiles of a studentized statistic efron j roy statist soc ser b 54 1992 83 1111 proposed a computationally efficient method called the jackknife after bootstrap for estimating the variance of a bootstrap estimator for independent data for dependent data a version of the jackknife after bootstrap method has been recently proposed by lahiri econometric theory 18 2002 79 98 in this paper it is shown that the jackknife after bootstrap estimators of the variance of a bootstrap quantile are consistent for both dependent and independent data results from a simulation study are also presented

on the optimality of prediction based selection criteria and the convergence rates of estimators several estimators of squared prediction error have been suggested for use in model and bandwidth selection problems among these are cross validation generalized cross validation and a number of related techniques based on the residual sum of squares for many situations with squared error loss e g nonparametric smoothing these estimators have been shown to be asymptotically optimal in the sense that in large samples the estimator minimizing the selection criterion also minimizes squared error loss however cross validation is known not to be asymptotically optimal for some easy location problems we consider selection criteria based on estimators of squared prediction risk for choosing between location estimators we show that criteria based on adjusted residual sum of squares are not asymptotically optimal for choosing between asymptotically normal location estimators that converge at rate n 1 2 but are when the rate of convergence is slower we also show that leave one out cross validation is not asymptotically optimal for choosing between n differentiable statistics but leave d out cross validation is optimal when d oo at the appropriate rate

quantifying the totality of treatment effect with multiple event time observations in the presence of a terminal event from a comparative clinical study to evaluate the totality of one treatment s benefit risk profile relative to an alternative treatment via a longitudinal comparative clinical study the timing and occurrence of multiple clinical events are typically collected during the patient s follow up these multiple observations reflect the patient s disease progression burden over time the standard practice is to create a composite endpoint from the multiple outcomes the timing of the occurrence of the first clinical event to evaluate the treatment via the standard survival analysis techniques by ignoring all events after the composite outcome this type of assessment may not be ideal various parametric or semiparametric procedures have been extensively discussed in the literature for the purposes of analyzing multiple event time data many existing methods were developed based on extensive model assumptions when the model assumptions are not plausible the resulting inferences for the treatment effect may be misleading in this article we propose a simple nonparametric inference procedure to quantify the treatment effect which has an intuitive clinically meaningful interpretation we use the data from a cardiovascular clinical trial for heart failure to illustrate the procedure a simulation study is also conducted to evaluate the performance of the new proposal

modular exponentiation via the explicit chinese remainder theorem fix pairwise coprime positive integers p 1 p 2 p a we propose representing integers u modulo m where m is any positive integer up to roughly p 1 p 2 p s as vectors u mod p 1 u mod p 2 u mod p s we use this representation to obtain a new result on the parallel complexity of modular exponentiation there is an algorithm for the common crcw pram that given positive integers x e and m in binary of total bit length n computes x e mod m in time o n lglgn using n o 1 processors for comparison a parallelization of the standard binary algorithm takes superlinear time adleman and kompella gave an o lg n 3 expected time algorithm using exp o n lg n processors von zur gathen gave an nc algorithm for the highly special case that m is polynomially smooth

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

predicting longitudinal trajectories of health probabilities with random effects multinomial logit regression researchers often encounter longitudinal health data characterized with three or more ordinal or nominal categories random effects multinomial logit models are generally applied to account for potential lack of independence inherent in such clustered data when parameter estimates are used to describe longitudinal processes however random effects both between and within individuals need to be retransformed for correctly predicting outcome probabilities this study attempts to go beyond existing work by developing a retransformation method that derives longitudinal growth trajectories of unbiased health probabilities variances of the predicted probabilities are estimated by using the delta method additionally the covariates regression coefficients on the multinomial logit function not substantively meaningful are transformed to the conditional effects on the predicted probabilities the empirical illustration uses the longitudinal data from the asset and health dynamics among the oldest old ahead our analysis compares three sets of the predicted probabilities of three health states at six time points obtained from respectively the retransformation method the blup and the fixed effects approach the results demonstrate that neglect of retransforming random errors in the random effects multinomial logit model results in severely biased longitudinal trajectories of health probabilities as well as overestimated effects of covariates on the probabilities

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

a copula based mixed poisson model for bivariate recurrent events under event dependent censoring in many chronic disease processes subjects are at risk of two or more types of events we describe a bivariate mixed poisson model in which a copula function is used to model the association between two gamma distributed random effects the resulting model is a bivariate negative binomial process in which each type of event arises from a negative binomial process methods for parameter estimation are described for parametric and semiparametric models based on an em algorithm we also consider the issue of event dependent censoring based on one type of event which arises when one event is sufficiently serious that its occurence may influence the decision of whether to withdraw a patient from a study the asymptotic biases of estimators of rate and mean functions from naive marginal analyses are discussed as well as associated treatment effects because the joint model is fit based on a likelihood consistent estimates are obtained simulation studies are carried out to evaluate the empirical performance of the proposed estimators with independent and event dependent censoring and applications to a trial of breast cancer patients with skeletal metastases and a study of patients with chronic obstructive pulmonary disease illustrate the approach

on a geometric notion of quantiles for multivariate data abstract an extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered the approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles both of which have been extensively studied in the literature these geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other l estimates of multivariate location and they lead to a directional notion of central and extreme points in a multidimensional setup such quantiles can be defined as meaningful and natural objects even in infinite dimensional hilbert and banach spaces and they yield an effective generalization of quantile regression in multiresponse linear model problems desirable equivariance properties are shown to hold for these multivariate quantiles and issues related to their computation for data in finite dimensional spaces are discussed n 1 2 consistenc

a general theory for jackknife variance estimation the delete 1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles this well known deficiency can be rectified by using a more general jackknife with d the number of observations deleted depending on a smoothness measure of the point estimator our general theory explains why jackknife works or fails it also shows that i for sufficiently smooth estimators the jackknife variance estimators with bounded d are consistent and asymptotically unbiased and ii for nonsmooth estimators d has to go to infinity at a rate explicitly determined by a smoothness measure to ensure consistency and asymptotic unbiasedness improved results are obtained for several classes of estimators in particular for the sample p quantiles the jackknife variance estimators with d satisfying n 1 2 d rightarrow 0 and n d rightarrow infty are consistent and asymptotically unbiased

checking the cox model with cumulative sums of martingale based residuals summary this paper presents a new class of graphical and numerical methods for checking the adequacy of the cox regression model the procedures are derived from cumulative sums of martingale based residuals over follow up time and or covariate values the distributions of these stochastic processes under the assumed model can be approximated by zero mean gaussian processes each observed process can then be compared both visually and analytically with a number of simulated realizations from the approximate null distribution these comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are special attention is given to checking the functional form of a covariate the form of the link function and the validity of the proportional hazards assumption an omnibus test consistent against any model misspecification is also studied the proposed techniques are illustrated with two real data sets

the factorization of the ninth fermat number in this paper we exhibit the full prime factorization of the ninth fermat number f9 2 512 1 it is the product of three prime factors that have 7 49 and 99 decimal digits we found the two largest prime factors by means of the number field sieve which is a factoring algorithm that depends on arithmetic in an algebraic number field in the present case the number field used was q fifth root 2 the calculations were done on approximately 700 workstations scattered around the world and in one of the final stages a supercomputer was used the entire factorization took four months

erratum to an optimal gap theorem here we shall use the same notations of the original paper the references to equations lemmas propositions theorems are with respect to the paper unless stated otherwise the argument provided for the proof of proposition 3 1 is not complete it overlooks the fact that the solution 2 form x t of the problem 3 1 may contain non trivial 2 0 and 0 2 parts hence proposition 3 1 needs a correction we thank gilles carron for pointing out this oversight in the original proof of proposition 3 1 and his comments we also thank luen fai tam for helpful related discussions we refer interested reader to 5 for an alternate proof of theorem 1 2 since the preservation of the positivity for solution 2 forms solving the problem 3 1 in the paper without knowing that the 2 form being 1 1 type seems elusive even for the 1 1 part projection of the 2 form solutions we replace 3 1 with the following initial boundary value problem on 1 1 form let be a smooth bounded domain in m for a given point p let be the unit outward normal in a small neighborhood near p we

on the asymptotic properties of the jackknife histogram we study the asymptotic normality of the jackknife histogram for one sample mean it holds if and only if r the number of observations retained and d n r the number of observations deleted both diverge to infinity the best convergence rate n 1 2 is obtained when r o n and d o n for u statistics of degree 2 and nonlinear statistics admitting a certain expansion similar results are obtained under conditions on r and d a second order approximation based on the edgeworth expansion is discussed

estimates of regression coefficients based on the sign covariance matrix the influence function and asymptotic theory are developed to consider robustness and limiting efficiencies of the scm regression estimate the estimate is shown to be consistent with a limiting multinormal distribution the influence function as a function of the length of the contamination vector is shown to be linear in elliptic cases for the least squares ls estimate it is quadratic the asymptotic relative efficiencies with respect to the ls estimate are given in the multivariate normal as well as the t distribution cases the scm regression estimate is highly efficient in the multivariate normal case and for heavy tailed distributions it performs better than the ls estimate simulations are used to consider finite sample efficiencies with similar results the theory is illustrated with an example

pseudomartingale estimating equations for modulated renewal process models summary we adapt martingale estimating equations based on gap time information to a gen eral intensity model for a single realization of a modulated renewal process the consistency and asymptotic normality of the estimators is proved under ergodicity conditions previous work has considered either parametric likelihood analysis or semiparametric multiplicative models using partial likelihood the framework is generally applicable to semiparametric and paramet ric models including additive and multiplicative specifications and periodic models it facilitates a semiparametric extension of a popular parametric earthquake model simulations and empir ical analyses of taiwanese earthquake sequences illustrate the methodology s practical utility

partly functional temporal process regression with semiparametric profile estimating functions summary marginal mean models of temporal processes in event time data analysis are gaining more attention for their milder assumptions than the traditional intensity models recent work on fully functional temporal process regression tpr offers great flexibility by allowing all the regression coefficients to be nonparametrically time varying the existing esti mation procedure however prevents successive goodness of fit test for covariate coefficients in comparing a sequence of nested models this article proposes a partly functional tpr model in the line of marginal mean models some covari ate effects are time independent while others are completely unspecified in time this class of models is very rich includ ing the fully functional model and the semiparametric model as special cases to estimate the parameters we propose semiparametric profile estimating equations which are solved via an iterative algorithm starting at a consistent estimate from a fully functional model in the existing work no smoothing is needed in contrast to other varying coefficient meth ods the weak convergence of the resultant estimators are developed using the empirical process theory successive tests of time varying effects and backward model selection procedure can then be carried out the practical usefulness of the methodology is demonstrated through a simulation study and a real example of recurrent exacerbation among cystic fibrosis patients

first occurrence prime gaps an ongoing search for first occurrence prime gaps continues an ongoing search for first occurrence prime gaps is being carried out which extends all previous work done on this subject to date this search has found all such gaps for primes up to 7 263 x 1013 first occurrence prime gaps had previously been known for primes less than 4 444 x 1012 2 several gaps larger than the previously largest gap of 682 not a first occurrence found by weintraub 4 have been found computer programs were written in fortran and cal cray assembly language on a cray 2 supercomputer the computation was conducted as follows a sufficient number of primes were generated to perform a sieve odd numbers beginning with 3 were sieved one block at a time where each block was chosen to contain 40 000 000 numbers based on system resource availability the even numbers were eliminated during initialization of each block one number was stored per 64 bit computer word after sieving each block the differences between consecutive primes were calculated and stored this was accomplished by loading 64 elements of the sieved block at a time into a vector register a 64 bit vector mask was created containing l s for corresponding nonzero values in the vector register if the vector mask was zero the next 64 values of the sieved block were loaded into the vector register if the mask was nonzero an instruction to count the number of leading zeros was executed to get the offset from the beginning of the vector register for the next prime a subtraction of the previous prime was done thus arriving at the gap the leftmost 1 of the vector mask was cleared and the method repeated beginning with checking if the vector mask was zero this was repeated until the entire block was processed the last prime in each block was saved in order to calculate the difference between that prime and the first prime in the next block to make sure no gaps were missed the time to perform the sieve for each block and to calculate all gaps generated was about 10 5 seconds for numbers in the range of 7 2 x 1013 the largest prime in the last block processed to date is 72635119999997 so the table of first occurrence prime gaps is complete to that prime the following table lists all the first occurrence prime gaps found this table agrees with all previously published results 2 3 1 the maximal first occurrence prime gaps are marked with an asterisk received june 8 1987 revised november 16 1987 1980 mathematics subject classification 1985 revision primary 11a41 1989 american mathematical society 0025 5718 89 1 00 s 25 per page

semiparametric transformation models for causal inference in time to event studies with all or nothing compliance we consider causal inference in randomized survival studies with right censored outcomes and all or nothing compliance using semiparametric transformation models to estimate the distribution of survival times in treatment and control groups conditional on covariates and latent compliance type estimands depending on these distributions for example the complier average causal effect cace the complier effect on survival beyond time t and the complier quantile effect are then considered maximum likelihood is used to estimate the parameters of the transformation models using a specially designed expectation maximization em algorithm to overcome the computational difficulties created by the mixture structure of the problem and the infinite dimensional parameter in the transformation models the estimators are shown to be consistent asymptotically normal and semiparametrically efficient inferential procedures for the causal parameters are developed a simulation study is conducted to evaluate the finite sample performance of the estimated causal parameters we also apply our methodology to a randomized study conducted by the health insurance plan of greater new york to assess the reduction in breast cancer mortality due to screening

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

nonparametric k sample tests with panel count data we study the nonparametric k sample test problem with panel count data the asymptotic normality of a smooth functional of the nonparametric maximum pseudolikelihood estimator wellner zhang 2000 is established under some mild conditions we construct a class of easy to implement nonparametric tests for comparing mean functions of k populations based on this asymptotic normality we conduct various simulations to validate and compare the tests the simulations show that the tests perform quite well and generally have good power to detect differences among the mean functions the method is illustrated with a real life example

inference from stratified samples second order analysis of three methods for nonlinear statistics abstract for stratified samples and nonlinear statistics that can be expressed as functions of estimated totals second order asymptotic expansions of the linearization jackknife and balanced repeated replication variance estimators are obtained based on these comparisons are made in terms of their biases some higher order asymptotic equivalence results are also established the special case of a combined ratio estimator is investigated in detail some results on bias reduction achieved by the jack knife and balanced repeated replication estimators of a nonlinear function of totals are also given

uniqueness and frechet differentiability of functional solutions to maximum likelihood type equations solutions of simultaneous equations of the maximum likelihood type or m estimators can be represented as functionals existence and uniqueness of a root in a local region of the parameter space are proved under conditions that are easy to check if only one root of the equation exists the resulting statistical functional is frechet differentiable and robust when several solutions exists conditions on the loss criterion used to select the root for the statistic ensure frechet differentiability an interesting example of a frechet differentiable functional is the solution of the maximum likelihood equations for location and scale parameters in a cauchy distribution the estimator is robust and asymptotically efficient

speeding the pollard and elliptic curve methods of factorization since 1974 several algorithms have been developed that attempt to factor a large number n by doing extensive computations module n and occasionally taking gcds with n these began with pollard s p 1 and monte carlo methods more recently williams published a p 1 method and lenstra discovered an elliptic curve method ecm we present ways to speed all of these one improvement uses two tables during the second phases of p 1 and ecm looking for a match polynomial preconditioning lets us search a fixed table of size n with n 2 o n multiplications a parametrization of elliptic curves lets step 1 of ecm compute the x coordinate of np from that of p in about 9 3 1og2 n multiplications for arbitrary p

plurisubharmonic functions and the structure of complete kahler manifolds with nonnegative curvature in this paper we study global properties of continuous plurisubharmonic functions on complete noncompact kahler manifolds with nonnegative bisectional curvature and their applications to the structure of such manifolds we prove that continuous plurisubharmonic functions with reasonable growth rate on such manifolds can be approximated by smooth plurisubharmonic functions through the heat flow deformation optimal liouville type theorem for the plurisubharmonic functions as well as a splitting theorem in terms of harmonic functions and holomorphic functions are established the results are then applied to prove several structure theorems on complete noncompact kahler manifolds with nonnegative bisectional or sectional curvature

towards a general asymptotic theory for cox model with staggered entry a general asymptotic theory is established for the two parameter cox score process with staggered entry data it extends in several directions the existing theory developed by sellke and siegmund slud and gu and lai an essential tool employed here is a modern empirical process theory as elucidated in a recent monograph by pollard

estimation of direct effects for survival data by using the aalen additive hazards model summary we extend the definition of the controlled direct effect of a point exposure on a survival outcome other than through some given time fixed intermediate variable to the additive hazard scale we propose two stage estimators for this effect when the exposure is dichotomous and randomly assigned and when the association between the intermediate variable and the survival outcome is confounded only by measured factors which may themselves be affected by the exposure the first stage of the estimation procedure involves assessing the effect of the intermediate variable on the survival outcome via aalen s additive regression for the event time given exposure intermediate variable and confounders the second stage involves applying aalen s additive model given the exposure alone to a modified stochastic process i e a modification of the observed counting process based on the first stage estimates we give the large sample properties of the estimator proposed and investigate its small sample properties by monte carlo simulation a real data example is provided for illustration

semiparametric modelling and estimation of covariate adjusted dependence between bivariate recurrent events a time dependent measure termed the rate ratio was proposed to assess the local dependence between two types of recurrent event processes in one sample settings however the one sample work does not consider modeling the dependence by covariates such as subject characteristics and treatments received the focus of this paper is to understand how and in what magnitude the covariates influence the dependence strength for bivariate recurrent events we propose the covariate adjusted rate ratio a measure of covariate adjusted dependence we propose a semiparametric regression model for jointly modeling the frequency and dependence of bivariate recurrent events the first level is a proportional rates model for the marginal rates and the second level is a proportional rate ratio model for the dependence structure we develop a pseudo partial likelihood to estimate the parameters in the proportional rate ratio model we establish the asymptotic properties of the estimators and evaluate the finite sample performance via simulation studies we illustrate the proposed models and methods using a soft tissue sarcoma study that examines the effects of initial treatments on the marginal frequencies of local distant sarcoma recurrence and the dependence structure between the two types of cancer recurrence

tables of fibonacci and lucas factorizations we list the known prime factors of the fibonacci numbers fn for n 999 and lucas numbers ln for n 500 we discuss the various methods used to obtain these factorizations and primality tests and give some history of the subject

semiparametric regression analysis of longitudinal data with informative observation times statistical analysis of longitudinal data has been discussed by many authors and a number of methods have been proposed most of the research have focused on situations where observation times are independent of or carry no information about the response variable and therefore rely on conditional inference procedures given the observation times this article considers a different situation where the independence assumption may not hold that is the observation times may carry information about the response variable for inference estimating equation approaches are proposed and both large sample and final sample properties of the proposed methods are established the methodology is applied to a bladder cancer study that motivated this investigation

a lie algebraic approach to ricci flow invariant curvature conditions and harnack inequalities we consider a subset s of the complex lie algebra so n c and the cone c s of curvature operators which are nonnegative on s we show that c s defines a ricci flow invariant curvature condition if s is invariant under adso n c the analogue for kahler curvature operators holds as well although the proof is very simple and short it recovers all previously known invariant nonnegativity conditions as an application we reprove that a compact kahler manifold with positive orthogonal bisectional curvature evolves to a manifold with positive bisectional curvature and is thus biholomorphic to cp moreover the methods can also be applied to prove harnack inequalities we consider a lie algebra g endowed with a scalar product which is invariant under the adjoint representation of the lie algebra the reader should think of g either as the space so n of skew adjoint endomorphism of r with the scalar product a b 1 2 tr ab or of the lie subalgebra u n so 2n corresponding to the unitary group u n so 2n endowed with the induced scalar product we consider the space of selfadjoint endomorphisms of s g every selfadjoint endomorphism r s g is determined by the corresponding bilinear form x y 7 rx y the extension of this form to a complex bilinear form r g r c g r c c will be denoted with the same letter r notice that for any x g r c the number r x x is real where x 7 x is complex conjugation also recall that the space of algebraic curvature operators s b so n is a linear subspace of s so n similarly the space of algebraic kahler curvature operators s k u n is a linear subspace of s u n the subspaces are also invariant under the ricci flow ode on s g r r r where rx y 12 tr adx r ady r for x y g we have the following basic result theorem 1 let s be a subset of the complex lie algebra g r c and let gc denote a lie group with lie algebra g r c if s is invariant under the adjoint representation of gc then for h r the set c s h r s g r v v h for all v s is invariant under the ode r r r in many cases s is scaling invariant and then h 0 is the only meaningful choice for h 0 the set c s c s 0 is a cone and the curvature condition c s can be thought of as a nonnegativity condition we recall that for a o n invariant subset c s so n we say that a manifold satisfies c if the curvature operator

some new estimators for cox regression new estimators for cox regression are considered their asymptotic properties both on and off the model are established corollaries include conditions under which the maximum partial likelihood estimator defines a parameter in the population and the asymptotics of the case cohort estimator robust estimators that minimize the asymptotic variance subject to a bound on the maximal bias on infinitesimal neighborhoods are discussed the estimators are illustrated with medical data

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

goodness of fit analysis for the cox regression model based on a class of parameter estimators abstract in this article we propose a class of estimation functions for the vector of regression parameters in the cox proportional hazards model with possibly time dependent covariates by incorporating the weight functions commonly used in weighted log rank tests into the partial likelihood score function the resulting estimators behave much like the conventional maximum partial likelihood estimator in that they are consistent and asymptotically normal when the cox model is inappropriate however the estimators with different weight functions generally converge to nonidentical constant vectors for example the magnitude of the parameter estimator using the kaplan meier survival estimator as the weight function will be stochastically larger than that of the maximum partial likelihood estimator if covariate effects diminish over time such facts motivate us to develop goodness of fit methods for the cox regression model by comparing parameter estimators with different weight functions under the assume

jackknife bootstrap and other resampling methods in regression analysis motivated by a representation for the least squares estimator we propose a class of weighted jackknife variance estimators for the least squares estimator by deleting any fixed number of observations at a time they are unbiased for homoscedastic errors and a special case the delete one jackknife is almost unbiased for heteroscedastic errors the method is extended to cover nonlinear parameters regression m estimators nonlinear regression and generalized linear models interval estimators can be constructed from the jackknife histogram three bootstrap methods are considered two are shown to give biased variance estimators and one does not have the bias robustness property enjoyed by the weighted delete one jackknife a general method for resampling residuals is proposed it gives variance estimators that are bias robust several bias reducing estimators are proposed some simulation results are reported

a monotonicity formula on complete kahler manifolds with nonnegative bisectional curvature in y yau proposed to study the uniformization of complete kahler manifolds with nonnegative curvature in particular one wishes to determine whether or not a complete kahler manifold m with positive bisectional curvature is biholomorphic to cm see also gw si for this sake it was further asked in y whether or not the ring of the holomorphic functions with polynomial growth which we denote by op m is finitely generated and whether or not the dimension of the spaces of holomorphic functions of polynomial growth is bounded from above by the dimension of the corresponding spaces of polynomials on cm this paper addresses the latter questions we denote by 0 m the space of holomorphic functions of polynomial growth with degree d see section 3 for the precise definition then op m jd 0 od m in this paper we show that

on subsampling estimators with unknown rate of convergence abstract politis and romano have put forth a general subsampling methodology for the construction of large sample confidence regions for a general unknown parameter associated with the probability distribution generating the stationary sequence x 1 x n the subsampling methodology hinges on approximating the large sample distribution of a statistic t n t n x 1 x n that is consistent for at some known rate n although subsampling has been shown to yield confidence regions for of asymptotically correct coverage under very weak assumptions the applicability of the methodology as it has been presented so far is limited if the rate of convergence n happens to be unknown or intractable in a particular setting in this article we show how it is possible to circumvent this limitation by a using the subsampling methodology to derive a consistent estimator of the rate n and b using the estimated rate to construct asymptotically correct confidence regions for based on subsampling

estimates of regression coefficients based on lift rank covariance matrix we introduce a new equivariant estimation method of the parameters of the multivariate regression model with q responses and p regressors the estimate matrix is derived from the lift rank covariance matrix lrcm where the lift rank vectors are based on the oja criterion function the k p q variate ranks and k 1 variate lift ranks are constructed using hyperplanes or fits going through k observations the new lrcm regression estimate and the least squares ls estimate are shown to be weighted sums of the elemental estimates based on these hyperplanes the lrcm regression estimate is equivariant and convergent has a limiting multinormal distribution and is highly efficient in the multivariate normal case for heavy tailed distributions it performs better than the standard ls estimate estimation of the variance covariance matrix of the lrcm estimate is briefly discussed the theory is illustrated by simulations and a real data example

an optimal gap theorem by solving the cauchy problem for the hodge laplace heat equation for d closed positive 1 1 forms we prove an optimal gap theorem for kahler manifolds with nonnegative bisectional curvature which asserts that the manifold is flat if the average of the scalar curvature over balls of radius r centered at any fixed point o is a function of o r 2 furthermore via a relative monotonicity estimate we obtain a stronger statement namely a positive mass type result asserting that if m g is not flat then liminf r to infty frac r 2 v o r int b o r mathcal s y d mu y 0 for any o m

model selection via multifold cross validation a natural extension of the simple leave one out cross validation cv method is to allow the deletion of more than one observations in this article several notions of the multifold cross validation mcv method have been discussed in the context of variable selection under a linear regression model we show that the delete d mcv criterion is asymptotically equivalent to the well known fpe criterion two computationally more feasible methods the r fold cross validation and the repeated learning testing criterion are also studied the performance of these criteria are compared with the simple leave one out cross validation method simulation results are obtained to gain some understanding on the small sample properties of these methods

on complete noncompact kahler manifolds with positive bisectional curvature we prove that a complete noncompact kahler manifold m n of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of c n and we show that the manifold is topologically r 2 n in particular when m n is a kahler surface of positive bisectional curvature satisfying certain natural geometric growth conditions it is biholomorphic to c 2

sharp differential estimates of li yau hamilton type for positive p p forms on kahler manifolds in this paper we study the heat equation of hodge laplacian deformation of p p forms on a kahler manifold after identifying the condition and establishing that the positivity of a p p form solution is preserved under such an invariant condition we prove the sharp differential harnack in the sense of liyau hamilton estimates for the positive solutions of the hodge laplacian heat equation we also prove a nonlinear version coupled with the kahler ricci flow and some interpolating matrix differential harnack type estimates for both the kahler ricci flow and the ricci flow 2011 wiley periodicals inc

a semiparametric recurrent events model with time varying coefficients we consider a recurrent events model with time varying coefficients motivated by two clinical applications a random effects gaussian frailty model is used to describe the intensity of recurrent events the model can accommodate both time varying and time constant coefficients the penalized spline method is used to estimate the time varying coefficients laplace approximation is used to evaluate the penalized likelihood without a closed form the smoothing parameters are estimated in a similar way to variance components we conduct simulations to evaluate the performance of the estimates for both time varying and time independent coefficients we apply this method to analyze two data sets a stroke study and a child wheeze study

asymptotic semismoothness probabilities we call an integer semismooth with respect to y and z if each of its prime factors is y and all but one are z such numbers are useful in various factoring algorithms including the quadratic sieve let g be the asymptotic probability that a random integer n is semismooth with respect to n and n we present new recurrence relations for g and related functions we then give numerical methods for computing g tables of g and estimates for the error incurred by this asymptotic approximation

an fft extension to the p 1 factoring algorithm j m pollard in 1974 presented the p integer factoring al gorithm his paper couched the algorithm in theoretical terms based upon use of fast fourier transform techniques but he was unable to say whether the method could be made practical we discuss the mathematical basis of the al gorithm and show how it can work in practice the practical implementation depends for its success upon the use of residue number systems we also present an open problem as to how the method could be made to work for the elliptic curve factoring algorithm

on the dependence structure of bivariate recurrent event processes inference and estimation bivariate or multivariate recurrent event processes are often encountered in longitudinal studies in which more than one type of event is of interest there has been much research on regression analysis for such data but little has been done to measure the dependence between recurrent event processes we propose a time dependent measure termed the rate ratio to assess the local dependence between two types of recurrent event processes we model the rate ratio as a parametric function of time and leave unspecified all other aspects of the distribution we develop a composite likelihood procedure for model fitting and parameter estimation we show that the proposed estimator is consistent and asymptotically normal its finite sample performance is evaluated by simulation and illustrated by an application to a soft tissue sarcoma study

orthogonal decomposition of finite population statistics and its applications to distributional asymptotics we study orthogonal decomposition of symmetric statistics based on samples drawn without replacement from finite populations several applications to finite population statistics are given we establish one term edge worth expansions for general asymptotically normal symmetric statistics prove an efron stein inequality and the consistency of the jackknife estimator of variance our expansions provide second order a s approximations to wu s jackknife histogram

bias correction with jackknife bootstrap and taylor series we analyze bias correction methods using jackknife bootstrap and taylor series we focus on the binomial model and consider the problem of bias correction for estimating f p where f in c 0 1 is arbitrary we characterize the supremum norm of the bias of general jackknife and bootstrap estimators for any continuous functions and demonstrate the in delete d jackknife different values of d may lead to drastically different behaviors in jackknife we show that in the binomial model iterating the bootstrap bias correction infinitely many times may lead to divergence of bias and variance and demonstrate that the bias properties of the bootstrap bias corrected estimator after r 1 rounds are of the same order as that of the r jackknife estimator if a bounded coefficients condition is satisfied

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

simple robust tests for scale differences in paired data summary the classical statistical test for assessing the difference between the two scale parameters in paired data due to pitman 1939 and morgan 1939 is not robust this paper explores robust alternatives to pitman s test using the framework of the one sample t test the asymptotic behaviour of the tests under the null hypothesis is examined in general it is easy to construct tests that are asymptotically distribution free provided the data come from a symmetric bivariate distribution versions of these tests that do not require symmetry are derived monte carlo simulation experiments are done to assess small sample behaviour and power characteristics the tests are applied to a small data set comparing cancerous and noncancerous lungs

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

linear model selection by cross validation abstract we consider the problem of selecting a model having the best predictive ability among a class of linear models the popular leave one out cross validation method which is asymptotically equivalent to many other model selection methods such as the akaike information criterion aic the c p and the bootstrap is asymptotically inconsistent in the sense that the probability of selecting the model with the best predictive ability does not converge to 1 as the total number of observations n we show that the inconsistency of the leave one out cross validation can be rectified by using a leave n v out cross validation with n v the number of observations reserved for validation satisfying n v n 1 as n this is a somewhat shocking discovery because nv n 1 is totally opposite to the popular leave one out recipe in cross validation motivations justifications and discussions of some practical aspects of the use of the leave n v out cross validation method are provided and results