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

edge effect on the power law distribution of granular avalanches many punctuated phenomena in nature are claimed e g by the theory of self organized criticality soc to be power law distributed in our experiments on a three dimensional pile of long grained rice we find that by only changing the boundary condition of the system we switch from such power law distributed avalanche sizes to quasiperiodic system spanning avalanches conversely by removing ledges the incidence of system spanning avalanches is significantly reduced this may offer a perspective on new avalanche prevention schemes in addition our findings may help to explain why the archetype of soc the sandpile was found to have power law distributed avalanches in some experiments while in other experiments quasiperiodic system spanning avalanches were found

continuously varying critical exponents in a sandpile model with internal disorder a sandpile model with an internal disorder is presented the updating of critical sites is done according to a stochastic rule with a probabilistic toppling q using a unified mean field theory and numerical simulations we have shown that the criticality is ensured for any value of q the static critical exponents have been calculated and found to be the same as those obtained for the deterministic sandpile model which is a particular case of the stochastic model they have a universal q independent behavior in the limit of slow driving we have developed a relation between our model and the branching process in order to compute the size exponent tau it presents a continuous variation with the parameter of toppling q

how cells grow and divide mathematical analysis confirms demand for the cell cycle eukaryotes usually grow through cell growth and division how cells grow and divide is essential to life because too small or too large cells cannot function well in order for an organism to survive even under a condition where cell growth and division processes are independent of each other cells must have an appropriate growth factor growth rate and division rate to determine them we derive a time evolution equation for the size distribution of cells from the master equation describing changes in the cell size due to growth and in the total number of cells due to division it is found that long time behaviors of moments of the size distribution divide the parameter space consisting of the growth factor and the ratio of the division rate to the growth rate into infinitely many regions examining the properties of each region we conclude that growth with a small growth factor may be disastrous this demonstrates the demand for the cell cycle consisting of coordinated growth and division processes

weibull type limiting distribution for replicative systems the weibull function is widely used to describe skew distributions observed in nature however the origin of this ubiquity is not always obvious to explain in the present paper we consider the well known galton watson branching process describing simple replicative systems the shape of the resulting distribution about which little has been known is found essentially indistinguishable from the weibull form in a wide range of the branching parameter this can be seen from the exact series expansion for the cumulative distribution which takes a universal form we also find that the branching process can be mapped into a process of aggregation of clusters in the branching and aggregation process the number of events considered for branching and aggregation grows cumulatively in time whereas for the binomial distribution an independent event occurs at each time with a given success probability

proportional hazards regression with missing covariates abstract nonparametric maximum likelihood npml is used to estimate regression parameters in a proportional hazards regression model with missing covariates the npml estimator is shown to be consistent and asymptotically normally distributed under some conditions em type algorithms are applied to solve the maximization problem variance estimates of the regression parameters are obtained by a profile likelihood approach that uses em aided numerical differentiation simulation results indicate that the npml estimates of the regression parameters are more efficient than the approximate partial likelihood estimates and estimates from complete case analysis when missing covariates are missing completely at random and that the proposed method corrects for bias when the missing covariates are missing at random

recurrence time statistics in chaotic dynamics i discrete time maps the dynamics of transitions between the cells of a finite phase space partition in a variety of systems giving rise to chaotic behavior is analyzed with special emphasis on the statistics of recurrence times in the case of one dimensional piecewise markow maps the recurrence problem is cast into a renewal process in the presence of intermittency transitions between cells define a non markovian non renewal process reflected in the presence of power law probability distributions and of divergent variances and mean values

avalanche dynamics surface roughening and self organized criticality experiments on a three dimensional pile of rice we present a two dimensional system that exhibits features of self organized criticality the avalanches that occur on the surface of a pile of rice are found to exhibit finite size scaling in their probability distribution the critical exponents are ensuremath tau 1 21 2 for the avalanche size distribution and d 1 99 2 for the cutoff size furthermore the geometry of the avalanches is studied leading to a fractal dimension of the active sites of d b 1 58 2 using a set of scaling relations we can calculate the roughness exponent ensuremath alpha d ensuremath d b 0 41 3 and the dynamic exponent z d 2 ensuremath ensuremath tau 1 56 8 this result is compared with that obtained from a power spectrum analysis of the surface roughness which yields ensuremath alpha 0 42 3 and z 1 5 1 in excellent agreement with those obtained from the scaling relations

analysis of the proportional hazards model with sparse longitudinal covariates regression analysis of censored failure observations via the proportional hazards model permits time varying covariates that are observed at death times in practice such longitudinal covariates are typically sparse and only measured at infrequent and irregularly spaced follow up times full likelihood analyses of joint models for longitudinal and survival data impose stringent modeling assumptions that are difficult to verify in practice and that are complicated both inferentially and computationally in this article a simple kernel weighted score function is proposed with minimal assumptions two scenarios are considered half kernel estimation in which observation ceases at the time of the event and full kernel estimation for data where observation may continue after the event as with recurrent events data it is established that these estimators are consistent and asymptotically normal however they converge at rates that are slower than the parametric rates that may be achieved with fully observe

estimation for a semimartingale regression model using the method of sieves estimation by the method of sieves for a semimartingale regression model introduced by aalen 1980 is studied it is of interest to estimate functions which describe the influence of the covariates over time an estimator for these functions is introduced and conditions which ensure consistency of the estimator in l 2 norm are given applications to diffusion processes and point processes with censored data are also discussed

clustering of extreme and recurrent events in deterministic chaotic systems we study the nontrivial clustering properties of extreme or recurrent events in the context of deterministic chaotic systems we find that correlations between return times of such events can depend nonmonotonically on the threshold used to define the events which leads to counterintuitive behavior in particular the distribution of the conditional return intervals can indicate clustering as well as repelling of extreme events for the same condition but different thresholds in sharp contrast to what has been observed for stochastic processes with long range correlations as well as for independent and identically distributed random variables this has important implications for the time dependent hazard assessment of extreme events indicating that possible threshold dependencies should always be taken into account

applications of extreme value statistics in physics we present a descriptive review of physical problems dealing with extreme values in several fields of physics we consider different physical situations involving random variables that are correlated or not and study the statistics of extremal variables which is relevant for situations where height fluctuations catastrophic events such as material failure or power outrage occur we describe the general theory and relate the cumulative limit distributions that can be accessible in experiments to microscopic models in many cases however the random variables are correlated in interface problems for example and the characteristics of the interaction are revealed in the asymptotic behavior of the limit distribution

a joint model for survival and longitudinal data measured with error the relationship between a longitudinal covariate and a failure time process can be assessed using the cox proportional hazards regression model we consider the problem of estimating the parameters in the cox model when the longitudinal covariate is measured infrequently and with measurement error we assume a repeated measures random effects model for the covariate process estimates of the parameters are obtained by maximizing the joint likelihood for the covariate process and the failure time process this approach uses the available information optimally because we use both the covariate and survival data simultaneously parameters are estimated using the expectation maximization algorithm we argue that such a method is superior to naive methods where one maximizes the partial likelihood of the cox model using the observed covariate values it also improves on two stage methods where in the first stage empirical bayes estimates of the covariate process are computed and then used as time dependent covariates in a second stage to find the parameters in the cox model that maximize the partial likelihood

a semiparametric estimator for the proportional hazards model with longitudinal covariates measured with error summary a common objective in longitudinal studies is to characterise the relationship between a failure time process and time independent and time dependent covariates timedependent covariates are generally available as longitudinal data collected periodically during the course of the study we assume that these data follow a linear mixed effects model with normal measurement error and that the hazard of failure depends both on the underlying random effects describing the covariate process and other time independent covariates through a proportional hazards relationship a routine assumption is that the random effects are normally distributed however this need not hold in practice within this framework we develop a simple method for estimating the proportional hazards model parameters that requires no assumption on the distribution of the random effects large sample properties are discussed and finite sample performance is assessed and compared to competing methods via simulation

extreme value statistics and return intervals in long range correlated uniform deviates we study extremal statistics and return intervals in stationary long range correlated sequences for which the underlying probability density function is bounded and uniform the extremal statistics we consider e g maximum relative to minimum are such that the reference point from which the maximum is measured is itself a random quantity we analytically calculate the limiting distributions for independent and identically distributed random variables and use these as a reference point for correlated cases the distributions are different from that of the maximum itself i e a weibull distribution reflecting the fact that the distribution of the reference point either dominates over or convolves with the distribution of the maximum the functional form of the limiting distributions is unaffected by correlations although the convergence is slower we show that our findings can be directly generalized to a wide class of stochastic processes we also analyze return interval distributions and compare them to recent conjectures of their functional form

universality classes for rice pile models we investigate sandpile models where the updating of unstable columns is done according to a stochastic rule we examine the effect of introducing nonlocal relaxation mechanisms we find that the models self organize into critical states that belong to three different universality classes the models with local relaxation rules belong to a known universality class that is characterized by an avalanche exponent ensuremath tau ensuremath approx 1 55 whereas the models with nonlocal relaxation rules belong to new universality classes characterized by exponents ensuremath tau ensuremath approx 1 35 and ensuremath tau ensuremath approx 1 63 we discuss the values of the exponents in terms of scaling relations and a mapping of the sandpile models to interface models

solution of the nonlinear theory and tests of earthquake recurrence times we develop an efficient numerical scheme to solve accurately the set of nonlinear integral equations derived previously in a saichev and d sornette j geophys res 112 b04313 2007 which describes the distribution of interevent times in the framework of a general model of earthquake clustering with long memory detailed comparisons between the linear and nonlinear versions of the theory and direct synthetic catalogs show that the nonlinear theory provides an excellent fit to the synthetic catalogs while there are significant biases resulting from the use of the linear approximation we then address the suggestions proposed by some authors to use the empirical distribution of interevent times to obtain a better determination of the so called clustering parameter our theory and tests against synthetic and empirical catalogs find a rather dramatic lack of power for the distribution of interevent times to distinguish between quite different sets of parameters casting doubt on the usefulness of this statistic for the specific purpose of identifying the clustering parameter

generalized extreme value statistics and sum of correlated variables we show that generalized extreme value statistics the statistics of the kth largest value among a large set of random variables can be mapped onto a problem of random sums this allows us to identify classes of non identical and generally correlated random variables with a sum distributed according to one of the three k dependent asymptotic distributions of extreme value statistics namely the gumbel frechet and weibull distributions these classes as well as the limit distributions are naturally extended to real values of k thus providing a clear interpretation to the onset of gumbel distributions with non integer index k in the statistics of global observables this is one of the very few known generalizations of the central limit theorem to non independent random variables finally in the context of a simple physical model we relate the index k to the ratio of the correlation length to the system size which remains finite in strongly correlated systems

modeling left truncated and right censored survival data with longitudinal covariates there is a surge in medical follow up studies that include longitudinal covariates in the modeling of survival data so far the focus has been largely on right censored survival data we consider survival data that are subject to both left truncation and right censoring left truncation is well known to produce biased sample the sampling bias issue has been resolved in the literature for the case which involves baseline or time varying covariates that are observable the problem remains open however for the important case where longitudinal covariates are present in survival models a joint likelihood approach has been shown in the literature to provide an effective way to overcome those difficulties for right censored data but this approach faces substantial additional challenges in the presence of left truncation here we thus propose an alternative likelihood to overcome these difficulties and show that the regression coefficient in the survival component can be estimated unbiasedly and efficiently issues about the bias for the longitudinal component are discussed the new approach is illustrated numerically through simulations and data from a multi center aids cohort study

covariate measurement errors in frailty models for clustered survival data summary we propose a new class of frailty measurement error models for clustered survival data when covariates are measured with error we show that the induced hazard function conditional on the observed covariates also follows a frailty model but of a more complicated form we study the asymptotic bias in regression coefficients and variance components when measurement error is ignored and the impact of censoring on this asymptotic bias we show that the naive estimator of the regression coefficient is attenuated and the naive estimator of the variance component is inflated when measurement error is ignored as the censoring proportion increases the asymptotic bias in the former becomes larger but the asymptotic bias in the latter interestingly becomes smaller we develop a structural approach for parameter estimation using the nonparametric maximum likelihood method where the baseline hazard is estimated nonparametrically we prove model identifiability and the existence of the nonparametric maximum likelihood estimators an em algorithm is developed for calculating the nonparametric maximum likelihood estimates the method is applied to the western kenya parasitaemia data and its performance is evaluated through simulations

asymptotic theory for the frailty model the frailty model is a generalization of cox s proportional hazards model which includes a random effect nielsen gill andersen and sorensen 1992 proposed an em algorithm to estimate the cumulative baseline hazard and the variance of the random effect here the asymptotic distribution of the estimators is given along with a consistent estimator of the asymptotic variance

consistency in a proportional hazards model incorporating a random effect supported in part by the north atlantic treaty organization under a grant awarded in 1990 conducted at the mathematical institute of the university of utrecht the netherlands

size independence of the strength of snow the mechanical reliability of 600 randomly taken snow samples follows weibull distributions if sigma max is the maximum stress present in a specimen of given density the fraction of specimens that fail at stresses below sigma max is p 1 exp sigma max sigma 0 m the scale parameter sigma 0 evaluated by the maximum likelihood method increases nearly quadratically with the density rho of snow but unlike predicted by the weakest link model is independent of size and shape of the specimen there is no size dependence of the strength of snow the weibull parameter m is independent of density size and shape of the snow sample m 1 5 0 5 this implies on the one hand that the results of laboratory scale tests can be used for avalanche prediction but on the other hand that these predictions remain contaminated with large statistical errors snow is a fragile weak and unreliable material

on using the cox proportional hazards model with missing covariates summary we propose two methods for handling missing covariates in using the cox proportional hazards model the maximum partial likelihood estimator based only on study subjects having complete covariates does not utilise all available information also it is biased when the probability of missingness depends on the failure or censoring time our suggestion is to impute the conditional expectation of the statistic involving missing covariates given the available information the proposed method provides a consistent regression parameter estimator when the probability of missingness depends on the failure or censoring time as well as on the observed covariates also the proposed estimator is more efficient than the estimator suggested previously by lin ying 1993 when data are missing completely at random

discriminating between weibull distributions and log normal distributions emerging in branching processes we consider the yule type multiplicative growth and division process and describe the ubiquitous emergence of weibull and log normal distributions in a single framework with the help of the integral transform and series expansion we show that both distributions serve as asymptotic solutions of the time evolution equation for the branching process in particular the maximum likelihood method is employed to discriminate between the emergence of the weibull distribution and that of the log normal distribution further the detailed conditions for the distinguished emergence of the weibull distribution are probed it is observed that the emergence depends on the manner of the division process for the two different types of distribution numerical simulations are also carried out confirming the results obtained analytically

return times for stochastic processes with power law scaling an analytical study of the return time distribution of extreme events for stochastic processes with power law correlation has been carried out the calculation is based on an expansion in the correlation exponent c t ensuremath mid t ensuremath mid ensuremath 1 the fixed point of the theory is associated with stretched exponential scaling of the distribution analytical expressions have been provided in the preasymptotic regime also the permanence time distribution appears to be characterized by stretched exponential scaling the conditions for application of the theory to non gaussian processes have been analyzed and the relations with the issue of return times in the case of multifractal measures have been discussed

recurrence time analysis long term correlations and extreme events the recurrence times between extreme events have been the central point of statistical analyses in many different areas of science simultaneously the poincar e recurrence time has been extensively used to characterize nonlinear dynamical systems we compare the main properties of these statistical methods pointing out their consequences for the recurrence analysis performed in time series in particular we analyze the dependence of the mean recurrence time and of the recurrence time statistics on the probability density function on the interval whereto the recurrences are observed and on the temporal correlations of time series in the case of long term correlations we verify the validity of the stretched exponential distribution which is uniquely defined by the exponent ensuremath gamma at the same time showing that it is restricted to the class of linear long term correlated processes simple transformations are able to modify the correlations of time series leading to stretched exponentials recurrence time statistics with different ensuremath gamma which shows a lack of invariance under the change of observables

asymptotic theory for the cox model with missing time dependent covariate the relationship between a time dependent covariate and survival times is usually evaluated via the cox model time dependent covariates are generally available as longitudinal data collected regularly during the course of the study a frequent problem however is the occurence of missing covariate data a recent approach to estimation in the cox model in this case jointly models survival and the longitudinal covariate however theoretical justification of this approach is still lacking in this paper we prove existence and consistency of the maximum likelihood estimators in a joint model the asymptotic distribution of the estimators is given along with a consistent estimator of the asymptotic variance

memory effects in recurrent and extreme events a dynamical approach to recurrent and extreme events is developed focusing on the role of correlations and memory in the structure of the probability distributions and their low order moments the procedure is illustrated on homogeneous first and second order markov chains non markovian and nonhomogeneous processes and deterministic dynamical systems substantial differences with classical statistical theory as applied to independent identically distributed random variables are identified

predictability of the coherent noise model and its applications we study the threshold distribution function of the coherent noise model for the case of infinite number of agents this function is piecewise constant with a finite number of steps n the latter exhibits a 1 f behavior as a function of the order of occurrence of an avalanche and hence versus natural time an analytic expression of the expectation value mathcal e s for the size s of the next avalanche is obtained and used for the prediction of the next avalanche apart from mathcal e s the number of steps n can also serve for this purpose this enables the construction of a similar prediction scheme which can be applied to real earthquake aftershock data

dynamics of sandpiles physical mechanisms coupled stochastic equations and alternative universality classes we present a set of coupled nonlinear stochastic equations in one space dimension designed to model the surface of an evolving sandpile these include nonlinear couplings to represent the constant transfer between relatively immobile clusters and mobile grains incorporate the presence of tilt and contain representations of inertia and evolving configurational disorder the critical behavior of these phenomenological equations is investigated numerically it is found to be diverse in the sense that different combinations of noise as well as different symmetries lead to nontrivial exponents in the cases most directly comparable with previous studies we find that our equations lead to a surface with a roughness exponent mathrm ensuremath alpha mathrm tilt ensuremath approxeq 0 40 to be compared with the edwards wilkinson and kardar parisi zhang values namely mathrm ensuremath alpha mathrm ew 1 4 and mathrm ensuremath alpha mathrm kpz 1 3 respectively this is in our view directly due to the effect of the tilt term finally we discuss our results as well as possible modifications to our equations textcopyright 1996 the american physical society

return interval distribution of extreme events and long term memory the distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines it is well known that many physical processes in nature and society display long range correlations hence in the last few years considerable research effort has been directed towards studying the distribution of return intervals for long range correlated time series based on numerical simulations it was shown that the return interval distributions are of stretched exponential type in this paper we obtain an analytical expression for the distribution of return intervals in long range correlated time series which holds good when the average return intervals are large we show that the distribution is actually a product of power law and a stretched exponential form we also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events

scaling properties of a rice pile model inertia and friction effects we present a rice pile cellular automaton model that includes inertial and friction effects this model is studied in one dimension where the updating of metastable sites is done according to a stochastic dynamics governed by a probabilistic toppling parameter p that depends on the accumulated energy of moving grains we investigate the scaling properties of the model using finite size scaling analysis the avalanche size the lifetime and the residence time distributions exhibit a power law behavior their corresponding critical exponents respectively ensuremath tau y and y r are not universal they present continuous variation versus the parameters of the system the maximal value of the critical exponent ensuremath tau that our model gives is very close to the experimental one ensuremath tau 2 02 frette et al nature london 379 49 1996 and the probability distribution of the residence time is in good agreement with the experimental results we note that the critical behavior is observed only in a certain range of parameter values of the system which correspond to low inertia and high friction

numerical determination of the avalanche exponents of the bak tang wiesenfeld model we consider the bak tang wiesenfeld sandpile model on a two dimensional square lattice of lattice sizes up to l 4096 a detailed analysis of the probability distribution of the size area duration and radius of the avalanches will be given to increase the accuracy of the determination of the avalanche exponents we introduce a new method for analyzing the data which reduces the finite size effects of the measurements the exponents of the avalanche distributions differ slightly from previous measurements and estimates obtained from a renormalization group approach

auxiliary covariate data in failure time regression summary we consider the problem of missing covariate data in the context of censored failure time relative risk regression auxiliary covariate data which are considered informative about the missing data but which are not explicitly part of the relative risk regression model may be available full covariate information is available for a validation set an estimated partial likelihood method is proposed for estimating relative risk parameters this method is an extension of the estimated likelihood regression analysis method for uncensored data pepe 1992 pepe fleming 1991 a key feature of the method is that it is nonparametric with respect to the association between the missing and observed including auxiliary covariate components asymptotic distribution theory is derived for the proposed estimated partial likelihood estimator in the case where the auxiliary or mismeasured covariates are categorical asymptotic efficiencies are calculated for exponential failure times using an exponential relative risk model the estimated partial likelihood estimator compares favourably with a fully parametric maximum likelihood analysis comparisons are also made with a standard partial likelihood analysis which ignores the incomplete observations important efficiency gains can be made with the estimated partial likelihood method small sample properties are investigated through simulation studies

cox regression with incomplete covariate measurements abstract this article provides a general solution to the problem of missing covariate data under the cox regression model the estimating function for the vector of regression parameters is an approximation to the partial likelihood score function with full covariate measurements and reduces to the pseudolikelihood score function of self and prentice in the special setting of case cohort designs the resulting parameter estimator is consistent and asymptotically normal with a covariance matrix for which a simple and consistent estimator is provided extensive simulation studies show that the large sample approximations are adequate for practical use the proposed approach tends to be more efficient than the complete case analysis especially for large cohorts with infrequent failures for case cohort designs the new methodology offers a variance covariance estimator that is much easier to calculate than the existing ones and allows multiple subcohort augmentations to improve efficiency real data taken f

calculation of the transition matrix and of the occupation probabilities for the states of the oslo sandpile model the oslo sandpile model or if one wants to be precise ricepile model is a cellular automaton designed to model experiments on granular piles displaying self organized criticality we present an analytic treatment that allows the calculation of the transition probabilities between the different configurations of the system from here using the theory of markov chains we can obtain the stationary occupation distribution which tells us that the phase space is occupied with probabilities that vary in many orders of magnitude from one state to another our results show how the complexity of this simple model is built as the number of elements increases and allow for small system sizes the exact calculation of the avalanche size distribution and other properties related to the profile of the pile