survival analysis using a scale change random effects model abstract frailty models are effective in broadening the class of survival models and inducing dependence in multivariate survival distributions in proportional hazards the random effect multiplies the hazard function the scale change model incorporates unobserved heterogeneity through a random effect that enters the baseline hazard function to change the time scale we interpret this random effect as frailty or other unobserved risks that create heterogeneity in the population this model produces a wide range of shapes for univariate survival and hazard functions we extend this model to multivariate survival data by assuming that members of a group share a common random effect this structure induces association among the survival times in a group and provides alternative association structures to the proportional hazards frailty model we present parametric and semiparametric estimation techniques and illustrate these methods with an example

graphical models for marked point processes based on local independence summary a new class of graphical models capturing the dependence structure of events that occur in time is proposed the graphs represent so called local independences meaning that the intensities of certain types of events are independent of some but not necessarilly all events in the past this dynamic concept of independence is asymmetric similar to granger non causality so the corresponding local independence graphs differ considerably from classical graphical models hence a new notion of graph separation which is called separation is introduced and implications for the underlying model as well as for likelihood inference are explored benefits regarding facilitation of reasoning about and understanding of dynamic dependences as well as computational simplifications are discussed

noncausality in continuous time in this paper we define different concepts of noncausality for continuous time processes using conditional independence and decomposition of semi martingales these definitions extend the ones already given in the case of discrete time processes as in the discrete time setup continuous time noncausality is a property concerned with the prediction horizon global versus instantaneous noncausality and the nature of the prediction strong versus weak noncausality relations between the resulting continuous time noncausality concepts are then studied for the class of decomposable semi martingales for which in general the weak instantaneous noncausality does not imply the strong global noncausality the paper then characterizes these different concepts of noncausality in the cases of counting processes and markov processes

the asymptotic properties of nonparametric tests for comparing survival distributions the asymptotic distribution under alternative hypotheses is derived for a class of statistics used to test the equality of two survival distributions in the presence of arbitrary and possibly unequal right censoring the test statistics include equivalents to the log rank statistic the modified wilcoxon statistic and the class of rank invariant test procedures introduced by peto peto when there are equal censoring distributions and the hazard functions are proportional the sample size formula for the f test used to compare exponential samples is shown to be valid for the log rank test in certain situations the power of the log rank test falls as the amount of censoring decreases

estimating local and global measures of association for bivariate interval censored data with a smooth estimate of the density measures of association for bivariate interval censored data have not yet been studied extensively betensky and finkelstein statist med 1999 18 3101 3109 proposed to calculate kendall s coefficient of concordance using a multiple imputation technique but their method becomes computer intensive for moderate to large data sets we suggest a different approach consisting of two steps firstly a bivariate smooth estimate of the density of log event times is determined the smoothing technique is based on a mixture of gaussian densities fixed on a grid with weights determined by a penalized likelihood approach secondly given the smooth approximation several local and global measures of association can be estimated readily the performance of our method is illustrated by an extensive simulation study and is applied to tooth emergence data of permanent teeth measured on 4468 children from the signal tandmobiel study

the effects of random and discrete sampling when estimating continuous time diffusions high frequency financial data are not only discretely sampled in time but the time separating successive observations is often random we analyze the consequences of this dual feature of the data when estimating a continuous time model in particular we measure the additional effects of the randomness of the sampling intervals over and beyond those due to the discreteness of the data we also examine the effect of simply ignoring the sampling randomness we find that in many situations the randomness of the sampling has a larger impact than the discreteness of the data

non causality in bivariate binary time series in this paper we develop a dynamic discrete time bivariate probit model in which the conditions for granger non causality can be represented and tested the conditions for simultaneous independence are also worked out the model is extended in order to allow for covariates representing individual as well as time heterogeneity the proposed model can be estimated by maximum likelihood granger non causality and simultaneous independence can be tested by likelihood ratio or wald tests a specialized version of the model aimed at testing granger non causality with bivariate discrete time survival data is also discussed the proposed tests are illustrated in two empirical applications

weighted logrank tests for interval censored data when assessment times depend on treatment we consider weighted logrank tests for interval censored data when assessment times may depend on treatment and for each individual we only use the two assessment times that bracket the event of interest it is known that treating finite right endpoints as observed events can substantially inflate the type i error rate under assessment treatment dependence atd but the validity of several other implementations of weighted logrank tests score tests permutation tests multiple imputation tests has not been studied in this situation with a bounded number of unique assessment times the score test under the grouped continuous model retains the type i error rate asymptotically under atd however although the approximate permutation test based on the permutation central limit theorem is not asymptotically valid under every atd scenario we show through simulation that in many atd scenarios it retains the type i error rate better than the score test we show a case where the approximate permutation test retains the type i error rate when the exact permutation test does not we study and modify the multiple imputation logrank tests of huang lee and yu 2008 statistics in medicine 27 3217 3226 showing that the distribution of the rank like scores asymptotically does not depend on the assessment times we show through simulations that our modifications of the multiple imputation logrank tests retain the type i error rate in all cases studied even with atd and a small number of individuals in each treatment group simulations were performed using the interval r package us government work in the public domain

a markov chain monte carlo em algorithm for analyzing interval censored data under the cox proportional hazards model summary this paper proposes a monte carlo em mcem algorithm for fitting the proportional hazards model for interval censored failure time data the algorithm generates orderings of the failures from their probability distribution under the model we maximize the average of the log likelihoods from these completed data sets to obtain updated parameter estimates as with the standard cox model this algorithm does not require the estimation of the baseline hazard function the performance of the algorithm is evaluated using simulations and the method is applied to data from aids and cancer studies our results indicate that our method produced more precise and unbiased estimates than methods of right and midpoint imputation

radial singular solutions for the n laplace equation with exponential nonlinearities abstract in this paper we consider radial distributional solutions of the quasilinear equation n u f u in the punctured open ball b r 0 r n n 2 we obtain sharp conditions on the nonlinearity f for extending such solutions to the whole domain b r by preserving the regularity for a certain class of nonlinearity f we obtain the existence of singular solutions and deduce upper and lower estimates on the growth rate near the singularity

applying the cox proportional hazards model for analysis of latency data with interval censoring the latency time of an infectious disease is defined as the time from infection to disease onset this paper applies the proportional hazards model to estimate the effect of covariates on latency when the time of disease onset is exact or right censored but the time of infection is interval censored we use a monte carlo em algorithm to estimate parameters of the joint distribution of infection times and latency times at each em iteration exact infection times are multiply imputed from the density determined by the parameters of the infection and latency time distributions the methodology is tested using a simulation study and is applied to data from a cohort of haemophiliacs with hiv disease

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

positive solutions of elliptic equations involving supercritical growth positive radial solutions of elliptic equation involving supercritical growth are analysed as their supremum norm tends to infinity it is shown that they converge uniformly away from the origin as well as in h 1 to the unique singular solution

then laplacian elliptic equation variational versus entropy solutions abstract the n laplacian equation u n 2 u f is a kind of limit case in the existence theory of nonlinear elliptic equations when the second member is assumed to be merely integrable l 1 theory here we consider for definiteness the solutions of the homogeneous dirichlet problem in a bounded domain r n and compare the standard concept of a variational or energy solution with the recently introduced concept of an entropy solution which seems natural in the l 1 theory for our equation both concepts have slightly different domains of application we also discuss the conditions on f under which the solutions are bounded a slightly smaller class

maximum likelihood estimation for the proportional hazards model with partly interval censored data summary the maximum likelihood estimator mle for the proportional hazards model with partly interval censored data is studied under appropriate regularity conditions the mles of the regression parameter and the cumulative hazard function are shown to be consistent and asymptotically normal two methods to estimate the variance covariance matrix of the mle of the regression parameter are considered based on a generalized missing information principle and on a generalized profile information procedure simulation studies show that both methods work well in terms of the bias and variance for samples of moderate size an example illustrates the methods

the geometry of mixture likelihoods a general theory in this paper certain fundamental properties of the maximum likelihood estimator of a mixing distribution are shown to be geometric properties of the likelihood set the existence support size likelihood equations and uniqueness of the estimator are revealed to be directly related to the properties of the convex hull of the likelihood set and the support hyperplanes of that hull it is shown using geometric techniques that the estimator exists under quite general conditions with a support size no larger than the number of distinct observations analysis of the convex dual of the likelihood set leads to a dual maximization problem a convergent algorithm is described the defining equations for the estimator are compared with the usual parametric likelihood equations for finite mixtures sufficient conditions for uniqueness are given part ii will deal with a special theory for exponential family mixtures

causality effects in return volatility measures with random times we provide a structural approach to identify instantaneous causality effects between durations and stock price volatility so far in the literature instantaneous causality effects have either been excluded or cannot be identified separately from granger type causality effects by giving explicit moment conditions for observed returns over random duration intervals we are able to identify an instantaneous causality effect the documented causality effect has significant impact on inference for tick by tick data we find that instantaneous volatility forecasts for e g ibm stock returns must be decreased by as much as 40 when not having seen the next quote change before its conditionally median time also instantaneous volatilities are found to be much higher than indicated by standard volatility assessment procedures using tick by tick data for ibm a naive assessment of spot volatility based on observed returns between quote changes would only account for 60 of the actual volatility for less liquidly traded stocks at nyse this effect is even stronger

nonparametric analysis of a duration model with stochastic unobserved heterogeneity abstract this paper develops nonparametric identification and estimation results for a single spell hazard model where the unobserved heterogeneity is specified as a levy subordinator the identification approach solves a nonlinear volterra integral equation of the first kind with an unknown kernel function both the kernel of the integral operator which models the distribution of the unobserved heterogeneity and the functions that enter it are identified given regularity conditions and minimal variation in the observed covariates the paper proposes a shape constrained nonparametric two step sieve minimum distance estimator rates of convergence are derived and monte carlo experiments show the finite sample performance of the estimator

a test for comparing current status survival data with crossing hazard functions and its application to immunogenicity of biotherapeutics several tests have been recently implemented in the nonparametric comparison of current status survival data however they are not suited for the situation of crossing hazards in this setting we propose a new test specifically designed for crossing hazards alternatives the proposed test is compared to classical implemented tests through simulations mimicking crossing hazards situations with various schemes of censoring the results show that the proposed test has a correct type i error and generally outperforms the existing methods the application of the proposed test on a real dataset on immunogenicity of interferon among multiple sclerosis patients highlights the interest of the proposed test

a general dynamical statistical model with causal interpretation summary we develop a general dynamical model as a framework for causal interpretation we first state a criterion of local independence in terms of measurability of processes that are involved in the doob meyer decomposition of stochastic processes then we define direct and indirect influence we propose a definition of causal influence using the concepts of a physical system this framework makes it possible to link descriptive and explicative statistical models and encompasses quantitative processes and events one of the features of the paper is the clear distinction between the model for the system and the model for the observation we give a dynamical representation of a conventional joint model for human immunodeficiency virus load and cd4 cell counts we show its inadequacy to capture causal influences whereas in contrast known mechanisms of infection by the human immunodeficiency virus can be expressed directly through a system of differential equations

a weighted logistic regression model for estimation of recurrence of adenomas in a colorectal polyp prevention trial some participants might have their follow up colonoscopy conducted before the scheduled time i e at the end of the trial this results in variable follow up lengths for participants and the data of recurrence status at the end of the trial can be considered as current status data in this paper we use a weighted logistic regression model to estimate recurrence rate of adenoma data at the end of the trial the weights are used to adjust for variable follow up we show that logistic regression tends to underestimate recurrence rate in a simulation study we show that kaplan meier estimator derived from the right endpoint of the current status data tends to overestimate recurrence rate in contrast to logistic regression and the weighted logistic regression method can produce reasonable estimates of recurrence rate even under a high non compliance rate compared to conventional logistic regression and kaplan meier estimator the method described here is illustrated with an example from a colon cancer study

the liouville bratu gelfand problem for radial operators abstract we determine precise existence and multiplicity results for radial solutions of the liouville bratu gelfand problem associated with a class of quasilinear radial operators which includes perturbations of k hessian and p laplace operators

a limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth abstract we study radial solutions of the semilinear elliptic equation u f u 0 under rather general growth conditions on f we construct a radial singular solution and study the intersection number between the singular solution and a regular solution an application to bifurcation problems of elliptic dirichlet problems is given to this end we derive a certain limit equation from the original equation at infinity using a generalized similarity transformation through a generalized cole hopf transformation all the limit equations can be reduced into two typical cases i e u u p 0 and u e u 0

smooth estimation of survival functions and hazard ratios from interval censored data using bayesian penalized b splines we discuss the use of bayesian p spline and of the composite link model to estimate survival functions and hazard ratios from interval censored data if one further assumes proportionality of the hazards the proposed strategy provides a smoothed estimate of the baseline hazard along with estimates of global covariate effects the frequentist properties of our bayesian estimators are assessed by an extensive simulation study we further illustrate the methodology by three examples showing that the proportionality of the hazards might also be found inappropriate from interval censored data

a sup inf inequality for some nonlinear elliptic equations involving exponential nonlinearities let u be a solution of the equation u v x eu in a domain r2 where 0 a v b and v is lipschitz continuous we prove that sup u can be controlled in terms of inf u more precisely supku inf u c a b k v l for any compact subset k this extends an earlier result of shafrir who obtained a similar conclusion when v 1

interval censored parametric regression survival models and the analysis of longitudinal trials this paper develops interval censoring likelihood methods in the context of parametric proportional hazard ph and non ph regression models in the longitudinal study setting to reanalyze the medical research council s randomized controlled trial of teletherapy in age related macular degeneration we compare the performance of the interval censoring likelihood with proxy likelihoods that were used to analyze the original data it is shown analytically that the use of such proxy likelihoods in selected ph models leads to biased estimators such estimators are artificially precise further the magnitude of their percentage bias is quantified in a data directed simulation study for non ph models we demonstrate that these results obtained from ph models do not hold uniformly and explain the implications of this finding for the reanalysis of proxy likelihood trial data our final analysis of the age related macular degeneration trial data based on fitting ph and non ph models reassuringly confirms the published findings from the original trial copyright 2013 john wiley sons ltd

effects of mid point imputation on the analysis of doubly censored data doubly censored data arise in some cohort studies of the aids incubation period because the time of infection may be known only up to an interval defined by two successive screening tests for hiv antibody a simple analytic approach is to impute the infection time by the mid point of the interval and then apply standard survival techniques for right censored data the objective of this paper is to investigate the statistical properties of such a mid point imputation approach we investigated the asymptotic bias of the kaplan meier estimate coverage probabilities of associated confidence intervals bias in hazard ratio and the size of the logrank test we show that the statistical properties of mid point imputation depend strongly on the underlying distributions of infection times and the incubation periods and the width of the interval between screening tests in the absence of treatment the median incubation period of hiv infection is approximately 10 years and we conclude that for this situation mid point imputation is a reasonable procedure for interval widths of 2 years or less

a bifurcation theorem for potential operators abstract a sufficient condition for bifurcation of real solutions of g u 0 where g r d e d e is given in terms of the crossing number of the eigenvalue perturbation of the linearization along the trivial solutions 0 this number describes how many eigenvalues leave the negative real axis wen the parameter varies from negative to positive values whereas in general only an odd crossing number entails bifurcation for potential operators in a hilbert space e a nonzero crossing number is sufficient for bifurcation the proof uses a lyapunov schmidt reduction and conley s index theory

multiple imputation for interval censored data with auxiliary variables we propose a non parametric multiple imputation scheme npmle imputation for the analysis of interval censored survival data features of the method are that it converts interval censored data problems to complete data or right censored data problems to which many standard approaches can be used and that measures of uncertainty are easily obtained in addition to the event time of primary interest there are frequently other auxiliary variables that are associated with the event time for the goal of estimating the marginal survival distribution these auxiliary variables may provide some additional information about the event time for the interval censored observations we extend the imputation methods to incorporate information from auxiliary variables with potentially complex structures to conduct the imputation we use a working failure time proportional hazards model to define an imputing risk set for each censored observation the imputation schemes consist of using the data in the imputing risk sets to create an exact event time for each interval censored observation in simulation studies we show that the use of multiple imputation methods can improve the efficiency of estimators and reduce the effect of missing visits when compared to simpler approaches we apply the approach to cytomegalovirus shedding data from an aids clinical trial in which cd4 count is the auxiliary variable

the emden fowler equation on a spherical cap of sn abstract let s n r n 1 n 3 be the unit sphere and let s s n be a geodesic ball with geodesic radius 0 we study the bifurcation diagram u r 2 of the radial solutions of the emden fowler equation on s s n u u p 0 in s u 0 on s u 0 in s where p 1 among other things we prove the following for each p p s n 2 n 2 there exists 0 such that the problem has a radial solution for and has no radial solution for 0 moreover this solution is unique in the space of radial functions if is close to if p s p p jl then there exists such that the problem has infinitely many radial solutions for where p jl 1 4 n 4 2 n 1 if n 11 if 2 n 10 asymptotic behaviors of the bifurcation diagram as p and p 1 are also studied

a bifurcation diagram of solutions to an elliptic equation with exponential nonlinearity in higher dimensions we consider the following semilinear elliptic equation where b 1 is the unit ball in d d 3 0 and p 0 firstly following merle and peletier we show that there exists an eigenvalue p such that has a solution p w p satisfying lim x 0 w p x secondly we study a bifurcation diagram of regular solutions to it follows from the result of dancer that has an unbounded bifurcation branch of regular solutions that emanates from u 0 0 here using the singular solution we show that the bifurcation branch has infinitely many turning points around p when 3 d 9 we also investigate the morse index of the singular solution in the d 11 case

point processes and queues martingale dynamics i martingales 1 histories and stopping times 2 martingales 3 predictability 4 square integrable martingales references solutions to exercises chapter i ii point processes queues and intensities 1 counting processes and queues 2 watanabe s characterization 3 stochastic intensity general case 4 predictable intensities 5 representation of queues 6 random changes of time 7 cryptographic point processes references solutions to exercises chapter ii iii integral representation of point process martingales 1 the structure of internal histories 2 regenerative form of the intensity 3 the representation theorem 4 hilbert space theory of poissonian martingales 5 useful extensions references solutions to exercises chapter iii iv filtering 1 the theory of innovations 2 state estimates for queues and markov chains 3 continuous states and nontrivial prehistory references solutions to exercises chapter iv v flows in markovian networks of queues 1 single station the historical results and the filtering method 2 jackson s networks 3 burke s output theorem for networks 4 cascades and loops in jackson s networks 5 independence and poissonian flows in markov chains references solutions to exercises chapter v vi likelihood ratios 1 radon nikodym derivatives and tests of hypotheses 2 changes of intensities a la girsanov 3 filtering by the method of the probability of reference 4 applications 5 the capacity of a point process channel 6 detection formula187 references189 solutions to exercises chapter vi vii optimal control 1 modeling intensity controls 2 dynamic programming for intensity controls complete observation case 3 input regulation a case study in impulsive control 4 attraction controls 5 existence via likelihood ratio references solutions to exercises chapter vii viii marked point processes 1 counting measure and intensity kernels 2 martingale representation and filtering 3 radon nikodym derivatives 4 towards a general theory of intensity references solutions to exercises chapter viii a1 background in probability and stochastic processes 1 introduction 2 monotone class theorem 3 random variables 4 expectations 5 conditioning and independence 6 convergence 7 stochastic processes 8 markov processes references a2 stopping times and point process histories 1 stopping times 2 changes of time and meyer dellacherie s integration formula 3 point process histories references a3 wiener driven dynamical systems 1 ito s stochastic integral 2 square integrable brownian martingales 3 girsanov s theorem references a4 stieltjes lebesgue calculus 1 the stieltjes lebesgue integral 2 the product and exponential formulas references general bibliography

a linear theory for noncausality abstract different definitions of noncausality according to granger sims haugh and pierce are analyzed in terms of orthogonality in the hilbert space of square integrable variables conditions when necessary are given for their respective equivalence some problems of testability are mentioned finally noncausality is also analyzed in terms of rational expectations extending previous results of sims author