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

complete solution of the electrostatic equilibrium problem for classical weights abstract in this paper we complete the solution of the electrostatic equilibrium problem for all classical weight functions in particular we solve this problem for classical generalized bessel jacobi on 0 and pseudo jacobi weights in addition we present an elementary unified way of dealing with the electrostatic equilibrium problem which depends on the weights satisfying the pearson differential equation

a stationary pairwise independent absolutely regular sequence for which the central limit theorem fails a strictly stationary finite state non degenerate random sequence is constructed which satisfies pairwise independence and absolute regularity but fails to satisfy a central limit theorem the mixing rate for absolute regularity is only slightly slower than that in a corresponding central limit theorem of ibragimov

on the blockwise bootstrap for empirical processes for stationary sequences in this paper we study the weak convergence to an appropriate gaussian process of the empirical process of the block based bootstrap estimator proposed by k unsch for stationary sequences the classes of processes investigated are weak dependent and associated sequences we also prove that differently from the independent situation the bootstrapped estimator of the mean of certain dependent sequences satisfies the central limit theorem while the mean of the original sequence does not

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

a glivenko cantelli lemma and weak convergence for empirical processes of associated sequences with the help of an extension of hoeffding s equality we develop a way for estimating the covariance structures for empirical functions of associated sequences in terms of covariances of the original random variables based on these estimations a glivenko cantelli lemma for associated sequences and weak convergence for empirical processes of stationary associated sequences are obtained all under the conditions on covariances of the original random variables

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

a bootstrap based on the estimating equations of the linear model summary this paper presents a method of bootstrap estimation for the linear model rather than resampling from the original sample as is conventional the proposed method resamples summands in an estimating function used to produce the original estimate the method is computationally simpler then existing competitors however its main advantage lies in the robustness to heteroscedasticity of the estimators it produces these estimators are shown to have desirable asymptotic properties under fairly general conditions a simulation study provides small sample support for the new method

cox s periodic regression model le modele de cox peut etre adapte pour modeliser un processus ponctuel ayant une intensite periodique

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

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

bootstrap critical values for tests based on generalized method of moments estimators monte carlo experiments have shown that tests based on generalized method ofmoments estimators often have true levels that differ greatly from their nominal levels when asymptotic critical values are used this paper gives conditions under which the bootstrap provides asymptotic refinements to the critical values of t tests and the test of overidentifying restrictions particular attention is given to the case of dependent data it is shown that with such data the bootstrap must sample blocks of data and that the formulae for the bootstrap versions of test statistics differ from the formulae that apply with the original data the results of monte carlo experiments on the numerical performance of the bootstrap show that it usually reduces the errors in level that occur when critical values based on first order asymptotic theory are used the bootstrap also provides an indication of the accuracy of critical values obtained from first order asymptotic theory

on the convergence rate in the central limit theorem for associated processes on donne des taux de convergence uniforme dans le theoreme central limite pour des processus associes au troisieme moment fini la stationarite n etant pas requise

the functional law of the iterated logarithm for stationary strongly mixing sequences let x i i z be a strictly stationary and strongly mixing sequence of real valued mean zero random variables let a n n 0 be the sequence of strong mixing coefficients we define the strong mixing function by t t and we denote by q the quantile function of x 0 assume that 0 1 1 t q 2 t dt where f 1 denotes the inverse of the monotonic function f the main result of this paper is that the functional law of the iterated logarithm lil holds whenever x i i z satisfies moreover it follows from doukhan massart and rio that for any positive there exists a stationary sequence x i i z with strong mixing coefficients n of the order of n such that the bounded lil does not hold if condition is violated the proof of the functional lil is mainly based on new maximal exponential inequalities for strongly mixing processes which are of independent interest

jackknifing linear estimating equations asymptotic theory and applications in stochastic processes let x1 x2 x be a vector of possibly dependent random variables having distribution f x 0 let g x 0 e 1 gi x 0 0 be an estimating equation for 0 e g the score function or the maximum pseudolikelihood estimating equation in spatial processes let o n be the estimator obtained from g such that 0on 0o in probability and n 1 2 0 00 n o v in distribution in many situations it is difficult to derive an analytical expression for v e g for maximum pseudolikelihood estimators for the spatial processes in this paper we give a jackknife estimator of v and show that it is weakly consistent the method consists of deleting one estimating equation instead of one observation at a time and thus obtaining the pseudovalues the method of proof and conditions are similar to those of reeds with some modifications the method applies equally to independent and identically distributed random variables independent but not identically distributed random variables timeor space dependent stochastic processes our conditions are less severe than carlstein s who deals with a similar problem of estimating v for dependent observations we also give some simulation results

algebraic description of the finite stieltjes moment problem abstract the stieltjes problem of moments seeks for a nondecreasing positive distribution function on the semi axis 0 so that its moments match a given infinite sequence of positive real numbers m 0 m 1 in his seminal paper investigations on continued fractions published in 1894 stieltjes gave a complete solution including the conditions for the existence and uniqueness in relation to his main goal the convergence theory of continued fractions one can also reformulate the stieltjes problem of moments as looking for a sequence of positive distribution functions 1 2 where the nth distribution function has n points of increase and m 0 m 1 m 2 n 1 represent its first 2n moments i e as the sequence of the finite stieltjes moment problems this view can be linked to iterative solution of large linear algebraic systems providing that m 0 m 1 are moments of some linear self adjoint and coercive operator a on a hilbert space with respect to a given vector f the finite stieltjes moment problems determine the iterations of the conjugate gradient method applied for solving a u f and vice versa here the existence and uniqueness is guaranteed by the properties of the operator a reformulation for finite sequences matrices and finite vectors is obvious this fundamental link raises a question on how the solution of the finite stieltjes moment problem can be described purely algebraically this has motivated the presented exposition built upon ideas published previously by several authors since the description uses matrices of moments it is not intended for numerical computations

dual birth and death processes and orthogonal polynomials we discuss connections between birth and death processes and orthogonal polynomials we survey certain spectral properties of orthogonal polynomials and spectra of birth and death processes birth and death processes with linear birth and death rates are studied in some detail we also mention some results concerning birth and death processes whose transition rates are rational functions

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

a maximal inequality and dependent marcinkiewicz zygmund strong laws this paper contains some extension of kolmogorov s maximal inequality to dependent sequences next we derive dependent marcinkiewicz zygmund type strong laws of large numbers from this inequality in particular for stationary strongly mixing sequences x i i z with sequence of mixing coefficients n n 0 the marcinkiewicz zygmund slln of order p holds if 0 1 1 t p 1 q p t dt where 1 denotes the inverse function of the mixing rate function t t and q denotes the quantile function of x 0 the condition is obtained by an interpolation between the condition of doukhan massart and rio implying the clt p 2 and the integrability of x 0 implying the usual slln p 1 moreover we prove that this condition cannot be improved for stationary sequences and power type rates of strong mixing

a functional limit theorem for dependent sequences with infinite variance stable limits under an appropriate regular variation condition the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non gaussian stable random variable a functional version of this is known to be true as well the limit process being a stable levy process the main result in the paper is that for a stationary regularly varying sequence for which clusters of high threshold excesses can be broken down into asymptotically independent blocks the properly centered partial sum process still converges to a stable levy process due to clustering the levy triple of the limit process can be different from the one in the independent case the convergence takes place in the space of cadlag functions endowed with skorohod s m 1 topology the more usual j 1 topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values collapsing in the limit to a single jump the result rests on a new limit theorem for point processes which is of independent interest the theory is applied to moving average processes squared garch 1 1 processes and stochastic volatility models

indeterminate moment problems and the theory of entire functions the set of solutions to an indeterminate hamburger moment problem is given by the nevanlinna parametrization involving four entire functions a b c d we review the properties of these functions and discuss recent progress in the still open problem of characterizing the class of entire functions which occur in this way they belong to the krein class and are of minimal exponential type we deduce the analogous parametrization of the set of solutions to an indeterminate stieltjes moment problem from the corresponding symmetric hamburger problem the survey ends with a discussion of three special cases where the functions a b c d are known explicitly

a functional central limit theorem for strongly mixing sequences of random variables we prove a functional central limit theorem for a class of strongly mixing sequences of random variables stationarity is not assumed but the variances of the partial sums must grow linearly our theorem extends previous results by supplying sufficient conditions for the weak convergence of the partial sum process to the wiener process under less restrictive moment assumptions for instance if mathop sup limits n text ex n 2 log x n parallel 1 varepsilon 0 and the mixing rate is exponential then this functional c l t holds under the weaker assumption with e 0 the c l t may fail to hold and it is possible that the c l t is satisfied but the sequence of partial sum processes is not tight

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

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

the impact of stieltjes work on continued fractions and orthogonal polynomials additional material abstract in the recent new edition of the collected works of t j stieltjes one of us gave an impression of the impact of stieltjes work a century after his death 43 in this paper we give an update and mention some observations which were missing from 43 and some results which appeared during the last two years and which are directly related to stieltjes work in particular there is a large section on special polynomials which were already considered by stieltjes but which were rediscovered later often without the knowledge of stieltjes work

discrete charges on a two dimensional conductor we investigate the electrostatic equilibria of n discrete charges of size 1 n on a two dimensional conductor domain we study the distribution of the charges on symmetric domains including the ellipse the hypotrochoid and various regular polygons with an emphasis on understanding the distributions of the charges as the shape of the underlying conductor becomes singular we find that there are two regimes of behavior a symmetric regime for smooth conductors and a symmetry broken regime for singular domains for smooth conductors the locations of the charges can be determined to within o left sqrt log n mathord left vphantom n n 2 right kern nulldelimiterspace n 2 right by an integral equation due to pommerenke math ann 179 212 218 1969 we present a derivation of a related but different integral equation which has the same solutions we also solve the equation to obtain asymptotic solutions which show universal behavior in the distribution of the charges in conductors with somewhat smooth cusps conductors with sharp cusps and singularities show qualitatively different behavior where the symmetry of the problem is broken and the distribution of the discrete charges does not respect the symmetry of the underlying domain we investigate the symmetry breaking both theoretically and numerically and find good agreement between our theory and the numerics we also find that the universality in the distribution of the charges near the cusps persists in the symmetry broken regime although this distribution is very different from the one given by the integral equation

electrostatic interpretation for the zeros of certain polynomials and the darboux process abstract the purpose of this paper is to extend the electrostatic interpretation of stieltjes to the zeros of a larger class of important polynomials to keep things simple i consider polynomials that are obtained from those considered by stieltjes by means of the so called darboux process applied to the difference operator featuring in their three term recursion relation it will turn out that some of the beauty in the results of stieltjes is preserved in this process it would be interesting to interpret the darboux process at the electrostatic level and this paper gives experimental data that might help in this direction

likelihood based scoring rules for comparing density forecasts in tails abstract we propose new scoring rules based on conditional and censored likelihood for assessing the predictive accuracy of competing density forecasts over a specific region of interest such as the left tail in financial risk management these scoring rules can be interpreted in terms of kullback leibler divergence between weighted versions of the density forecast and the true density existing scoring rules based on weighted likelihood favor density forecasts with more probability mass in the given region rendering predictive accuracy tests biased toward such densities using our novel likelihood based scoring rules avoids this problem

a general resampling scheme for triangular arrays of alpha mixing random variables with application to the problem of spectral density estimation in 1989 kunsch introduced a modified bootstrap and jackknife for a statistic which is used to estimate a parameter of the m dimensional joint distribution of stationary and alpha mixing observations the modification amounts to resampling whole blocks of consecutive observations or deleting whole blocks one at a time liu and singh independently proposed in 1988 the same technique for observations that are m dependent however many time series statistics notably estimators of the spectral density function involve parameters of the whole infinite dimensional joint distribution and hence do not fit in this framework in this report we generalize the moving blocks resampling scheme of kunsch and liu and singh a still modified version of the nonparametric bootstrap and jackknife is seen to be valid for general linear statistics that are asymptotically normal and consistent for a parameter of the whole joint distribution we then apply this result to the problem of estimation of the spectral density

central limit theorems for mixing sequences of random variables under minimal conditions soit xj j 1 une suite strictement stationnaire de variables aleatoires de moyenne nulle de variance finie et satisfaisant une condition de melange forte soit s n la nieme somme partielle et supposons var s n variant regulierement et d ordre 1 on demontre que si s n var s n 1 2 ne converge pas vers 0 en l 1 alors xj j 1 est dans le domaine d attraction partielle d une loi gaussienne

on semiparametric inference for modulated renewal processes summary modulated renewal processes suggested by d r cox give a flexible way to introduce dependencies into point processes we discuss the asymptotics of partial likelihood inference for modulated renewal processes when the random covariate for the process involves its history we show in some generality that the estimators of the regression parameter and the cumulative hazard have the same asymptotic distributions that they would have under the usual proportional hazards model even though the martingale justification for partial likelihood no longer applies because of a reordering of the time scale an example is given to illustrate the ideas a simulation study is presented to confirm the theoretical results

robust analysis of semiparametric renewal process models a rate model is proposed for a modulated renewal process comprising a single long sequence where the covariate process may not capture the dependencies in the sequence as in standard intensity models we consider partial likelihood based inferences under a semiparametric multiplicative rate model which has been widely studied in the context of independent and identical data under an intensity model gap times in a single long sequence may be used naively in the partial likelihood with variance estimation utilizing the observed information matrix under a rate model the gap times cannot be treated as independent and studying the partial likelihood is much more challenging we employ a mixing condition in the application of limit theory for stationary sequences to obtain consistency and asymptotic normality the estimator s variance is quite complicated owing to the unknown gap times dependence structure we adapt block bootstrapping and cluster variance estimators to the partial likelihood simulation studies and an analysis of a semiparametric extension of a popular model for neural spike train data demonstrate the practical utility of the rate approach in comparison with the intensity approach

rational approximants associated with measures supported on the unit circle and the real line abstract as a continuation of the well known connection between the theory of orthogonal polynomials on the unit circle and the interval 1 1 in this paper properties concerning error and convergence of certain rational approximants associated with the measures d t and d d cos supported on 1 1 and the unit circle respectively are deduced numerical illustrations are also given

blockwise bootstrapped empirical process for stationary sequences we apply the bootstrap for general stationary observations proposed by kunsch to the empirical process for p dimensional random vectors it is known that the empirical process in the multivariate case converges weakly to a certain gaussian process we show that the bootstrapped empirical process converges weakly to the same gaussian process almost surely assuming that the block length l for constructing bootstrap replicates satisfies l n o n 1 2 varepsilon 0 varepsilon frac 1 2 and l n rightarrow infty an example where the multivariate setup arises are the robust gm estimates in an autoregressive model we prove the asymptotic validity of the bootstrap approximation by showing that the functional associated with the gm estimates is frechet differentiable

the functional central limit theorem under the strong mixing condition we prove a central limit theorem for strongly mixing sequences under a sharp sufficient condition which combines the rate of the strong mixing coefficient with the quantile function the result improves on all earlier central limit theorems for this type of dependence and answers a conjecture raised by bradley in 1997 moreover we derive the corresponding functional central limit theorem

reconstructing the topology of clones function clones are sets of functions on a fixed domain that are closed under composition and contain the projections they carry a natural algebraic structure provided by the laws of composition which hold in them as well as a natural topological structure provided by the topology of pointwise convergence under which composition of functions becomes continuous inspired by recent results indicating the importance of the topological ego of function clones even for originally algebraic problems we study questions of the following type in which situations does the algebraic structure determine the topological structure of a functional clone we pay particular attention to function clones which contain an oligomorphic permutation group and discuss applications of this situation in model theory and theoretical computer science

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