estimating medical costs with censored data incompleteness of follow up data is a common problem in estimating medical costs naive analysis using summary statistics on the collected data can result in severely misleading statistical inference this paper focuses on the problem of estimating the mean medical cost from a sample of individuals whose medical costs may be right censored a class of weighted estimators which account appropriately for censoring are introduced our estimators are shown to be consistent and asymptotically normal with easily estimated variances the efficiency of these estimators is studied with the goal of finding as efficient an estimator for the mean medical cost as is feasible extensive simulation studies are used to show that our estimators perform well in finite samples even with heavily censored data for a variety of circumstances the methods are applied to a set of cost data from a cardiology trial conducted by the duke university medical center extensions to other censored data problems are also discussed

eigenvalue accumulation for dirac operators with spherically symmetric potential we consider dirac operators h in with spherically symmetric potentials the main result is a criterion for eigenvalue accumulation and non accumulation at the endpoints 1 and 1 of the essential spectrum under rather weak assumptions on the potential this result is proved by showing an analogous criterion for the associated radial dirac operators h and by proving that for sufficiently large each h does not have any eigenvalues in the interval 1 0 and 0 1 respectively of the gap 1 1 of the essential spectrum for the latter properties of solutions of certain riccati differential equations depending on the parameter and the spectral parameter are used

local conditional lyapunov exponent characterization of consistency of dynamical response of the driven lorenz system local conditional lyapunov exponents are introduced and plotted in dynamical phase space to visualize the local susceptibility of the lorenz system driven by chaos and noise signals when the system produces consistent response outputs the dependence of the distribution of the local conditional lyapunov exponent on changes in the drive signal is investigated we consider the existence of multiple basins of consistent response for the same drive signal and the dependence of the basins on the drive signal

quantile regression for duration models with time varying regressors abstract since the seminal work of koenker and bassett 1978 quantile regression has become a widely used tool in duration analysis the existing literature however has focused on time invariant regressors even though time varying regressors are common in practice in this paper we introduce a quantile regression framework with time varying regressors and develop quantile regression estimators our estimators are motivated by manski s 1975 1985 maximum score estimator and chen s 2010 integrated maximum score estimator our estimators are consistent and asymptotically normal under some regularity conditions and perform well in finite samples our method is illustrated with an unemployment duration data set

plots and tests for goodness of fit with randomly censored data summary in this paper we consider nonparametric procedures for assessing goodness of fit when the data may be subject to random censoring we introduce two classes of large sample tests based respectively on the maximum and average weighted difference between the specified survival function and the kaplan meier estimate of the true survival function the procedures in the first class are inverted to obtain simultaneous confidence bands for the survival and cumulative hazard functions these bands provide simultaneous bands associated with the percentage p p quantile q q and hazard plots for censored data some key word8 hazard function nonparametric procedure probability plot shift function survival function

convergence rates for the bootstrapped product limit process on etablit des taux d approximation forte pour le processus limite produit du bootstrap et le processus quantile correspondant

an elementary approach to weak convergence for quantile processes with applications to censored survival data abstract let be a continuously differentiable function with positive derivative and let n be a sequence of right continuous increasing processes we show that if n 1 2 n w where w is continuous then n 1 2 1 n 1 w 1 1 this result is applied to classical processes such as the empirical distribution function the kaplan meier estimator and some other situations we also prove an analogous result for the bootstrapped version of n 1 2 n and show how this allows one to obtain confidence bands for the quantile function 1 based on the bootstrap several examples are given

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 brief biography and appreciation of calyampudi radhakrishna rao with a bibliography of his books and papers we present a brief biography and appreciation of calyampudi radhakrishna rao to whom this fifth special issue on linear algebra and statistics of linear algebra and its applications is dedicated in celebration of his 75th birthday a bibliography of his books and papers is also included

a lil type result for the product limit estimator let x1 x2 xn be i i d r v s with p x u f u and y1 y2 yn be i i d p y u g u where both f and g are unknown continuous survival functions for i 1 2 n set i 1 if xi yi and 0 if xi yi and zi min itxi yi one way to estimate f from the observations zi i i l n is by means of the product limit p l estimator fn kaplan meier 1958 6

strong approximations of some biometric estimates under random censorship the empirical distribution function has been widely used as an estimator for the distribution function of the elements of a random sample it is not however appropriate when the observations are incomplete this is the case for example when the observations have been randomly censored on the right in the latter case the so called product limit estimate of kaplan and meier 16 has been generally accepted as a substitute for the empirical distribution function and it possesses many of the same properties cf meier 19 developing the corresponding statement of efron 8 weak convergence over a finite interval of the kaplan meier product limit estimate to a gaussian process when there is censorship on the right has been obtained by breslow and crowley 3 cf also meier 19 breslow and crowley obtained their result via the limiting distribution of the closely related cumulative hazard process under random censorship a generalisation of the efron breslow crowley theorem was formulated by yang 24 for the case when censorship on the right is performed by more competing risks these results have numerous statistical applications in areas such as medical follow up studies life testing actuarial sciences and demography koziol and green 18 have computed tables for the limiting distribution of the cram6r von mises statistic based on the kaplan meier product limit estimate under a specific random censorship model a point will be stressed in this paper however that by a certain transformation which seems to have escaped many researchers attention and is due to efron 8 it is always possible to obtain the standard wiener process in the limit the distributions of many functionals of which are well known this transformation was only recently used in a stimulating note by gillespie and fisher 14 to construct asymptotic confidence bands for the unknown distribution function in the kaplan meier model a general scheme of random censorship can be defined in the following way let x be a real random variable with distribution function f t

bootstrapping the kaplan meier estimator abstract randomly censored data consist of iid pairs of observations xi i i 1 n if i 0 xi denotes a censored observation and if i 1 xi denotes an exact survival time which is the variable of interest for estimating the distribution f of the survival times the product limit estimator proposed by kaplan and meier 1958 has been studied extensively and it has been shown to enjoy a number of optimality properties wellner 1982 see gill 1980 chaps 1 4 1983 for a modern treatment and for references with censored data bootstrapping can be carried out using two different resampling plans introduced by efron 1981 and reid 1981 respectively with efron s plan one takes a random sample with replacement from x 1 xn n whereas with reid s plan one takes a random sample from the kaplan meier estimator the purpose of this article is to study the asymptotic behavior of the bootstrapped kaplan meier estimator with both resampling plans the approach adopted uses the

defect enhanced anomaly in frequency synchronization of asymmetrically coupled spatially extended systems we analytically establish and numerically show that anomalous frequency synchronization occurs in a pair of asymmetrically coupled chaotic space extended oscillators the transition to anomalous behaviors is crucially dependent on asymmetries in the coupling configuration while the presence of phase defects has the effect of enhancing the anomaly in frequency synchronization with respect to the case of merely time chaotic oscillators

the rate of strong uniform consistency for the product limit estimator the maximal deviation of the product limit estimate from the estimated distribution function is investigated as a consequence of a functional law of the iterated logarithm the log log law is proved for this deviation on appropriate half lines with the precise constants this result implies that the log log law need not hold in general for the maximal deviation on the whole line then a general asymptotic order of magnitude is determined for the latter deviation this order depends on the joint behaviour of the censoring and censored distributions in a well defined way corresponding to specific joint behaviours several limsup results are deduced as consequence including all the previously known log log type laws in improved form the results are also extended to the variable censoring model

synchronization of uncoupled oscillators by common gamma impulses from phase locking to noise induced synchronization nonlinear oscillators can mutually synchronize when they are driven by common external impulses two important scenarios are i synchronization resulting from phase locking of each oscillator to regular periodic impulses and ii noise induced synchronization caused by the poisson random impulses but their difference has not been fully quantified here we analyze a pair of uncoupled oscillators subject to common random impulses with gamma distributed intervals which can be smoothly interpolated between the regular periodic and the random poisson impulses their dynamics are characterized by phase distributions frequency detuning lyapunov exponents and information theoretic measures which clearly reveal the differences between the two synchronization scenarios

estimating the occurrence rate for prevalent survival data in competing risks models abstract the problem of interest is the estimation of occurrence probabilities based on prevalent data in competing risks models in the literature the development of nonparametric methods has relied heavily on the independent competing risks assumption the primary purpose of this article is to establish a statistical framework without excluding the possibility of dependence among competing risks this is done through the use of crude hazard functions the crude hazard functions not only are estimable regardless of whether the competing risks are independent but also are mathematically more tractable in this article we show that there is a one to one correspondence between the crude hazard functions and the occurrence probabilities the general inversion formulas for the occurrence probabilities are presented from which various representations can be derived under different sampling techniques maximum likelihood estimators are derived using these representations for nonparametric and length bias dat

characterizing the dynamics of stochastic bistable systems by measures of complexity the dynamics of noisy bistable systems is analyzed by means of lyapunov exponents and measures of complexity we consider both the classical kramers problem with additive white noise and the case when the barrier fluctuates due to additional external colored noise in the case of additive noise we calculate the lyapunov exponents and all measures of complexity analytically as functions of the noise intensity or the mean escape time respectively for the problem of a fluctuating barrier the usual description of the dynamics with the mean escape time is not sufficient the application of the concept of measures of complexity allows us to describe the structures of motion in more detail most complexity measures indicate the value of the correlation time at which the phenomenon of resonant activation occurs with an extremum

accurate bootstrap confidence limits for the cumulative hazard and survivor functions under random censoring abstract in clinical trials and other settings confidence intervals for survival probabilities at a fixed point in time are often required typically such intervals are constructed on the cumulative hazard scale the quantiles required for computing these intervals are generally obtained using a first order normal theory approximation the appropriateness of such intervals for small sample sizes or under moderate to heavy censoring is suspect we demonstrate that significant improvement is possible for randomly censored data using the nonparametric bootstrap in particular using edgeworth expansion it is shown that the percentile t and bc a bootstrap methodologies based on the studentized nelson aalen estimator yield second order correct confidence limits for the cumulative hazard function at a fixed time point an explicit formula for the correction term of the edgeworth expansion is found easy to compute estimators for the components of the correction term are also found and lead to simple analytic

reduction of the synchronization time in random logistic maps we report on the effects of additive noises in a nonchaotic logistic map in this system the lyapunov exponent changes from negative to positive as the noise intensity is increased when the lyapunov exponent is negative the synchronization of orbits with different initial conditions occurs we find that the synchronization time cannot be determined solely by the lyapunov exponent when the noise intensity is greater than a point at which the lyapunov exponent is minimum we show that this reduction of the synchronization time is attributed to initial nonstationary behaviors where the critical point of the logistic map plays an important role

dense point spectrum for the one dimensional dirac operator with an electrostatic potential for the one dimensional dirac operator examples of electrostatic potentials with decay behaviour arbitrarily close to coulomb decay are constructed for which the operator has a prescribed set of eigenvalues dense in the whole or part of its essential spectrum a simple proof that the essential spectrum of one dimensional dirac operators with electrostatic potentials is never empty is given in the appendix

quantile regression under random censoring censored regression models have received a great deal of attention in both the theoretical and applied econometric literature most of the existing estimation procedures for either cross sectional or panel data models are designed only for models with fixed censoring in this paper a new procedure for adapting these estimators designed for fixed censoring to models with random censoring is proposed this procedure is then applied to the clad and quantile estimators of powell j econom 25 1984 303 32 1986a 143 to obtain an estimator of the coefficients under a mild conditional quantile restriction on the error term that is applicable to samples exhibiting fixed or random censoring the resulting estimator is shown to have desirable asymptotic properties and performs well in a small scale simulation study

central limit theorems for l_p distances of kernel estimators of densities under random censorship a sequence of independent nonnegative random variables with common distribution function f is censored on the right by another sequence of independent identically distributed random variables these two sequences are also assumed to be independent we estimate the density function f of f by a sequence of kernel estimators f n t int infty infty k t x h n d hat f n x h n where h n is a sequence of numbers k is kernel density function and hat f n is the product limit estimator of f we prove central limit theorems for int t 0 f n t f t p d mu t 1 leq p infty 0 t leq infty where mu is a measure on the borel sets of the real line the result is tested in monte carlo trials and applied for goodness of fit

on the koziol green model for random censorship summary in the koziol green 1976 model of random censorship the survival distribution of the censoring variables is some power the censoring parameter of the survival distribution of the lifetimes using a strong approximation result of burke csorgo and horvath we construct empirical exact confidence bands for the life distribution of the censored variable and a randomly efron transformed variant of the usual product limit process is shown to converge weakly to the brownian motion process on any interval 0 t when the censoring parameter is also estimated from the sample the technique automatically gives rates of convergence for a wide class of functionals including the cramer von mises functional whose limit theory is worked out in detail the latter goodness of fit statistic is applied to reexamine the hospital data of koziol green 1976 to test whether oestrogen treatment for prostatic cancer was effective or not for the cancer itself while it caused censoring deaths of cardiovascular diseases

laws of the iterated logarithm for censored data first and second order laws of the iterated logarithm are obtained for both the nelson aalen and the kaplan meier estimators in the random censorship model uniform up to a large order statistic of the censored data the rates for the first order processes are exact except for constants the lil for the second order processes where one subtracts a linear empirical process term from the difference between the original process and the estimator uniform over fixed intervals is also proved somewhat surprisingly there is a certain degree of proof unification for fixed and variable intervals in the second order results for the nelson aalen estimator no assumptions are made on the distribution of the censoring variables and only continuity of the distribution function of the original variables is assumed for the results on the kaplan meier estimator

generalized synchronization of chaos in directionally coupled chaotic systems synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time we explore a generalization of this condition which equates dynamical variables from one subsystem with a function of the variables of another subsystem this means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space we explore this idea in systems where a response system bold y ital t is driven with the output of a driving system bold x ital t but there is no feedback to the driver we lose generality but gain tractability with this restriction to investigate the existence of the synchronization condition bold y ital t phi bold x ital t we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space the synchronization condition also implies that the response dynamics is determined by the drive alone and we provide tests for this as well examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits determining the presence of generalized synchronization will be quite important when one has only scalarmore observations from the drive and from the response systems since the use of time delay or other embedding methods will produce imperfect coordinates in which strict equality of the synchronized variables is unlikely to transpire less

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

nonparametric estimation of specific occurrence exposure rate in risk and survival analysis abstract a cohort of individuals exposed to some risk is followed up to a point of time m and observations on two random variables y are recorded for each individual the variable refers to one of the four possible events that can occur for an individual in the period 0 m i dies of a specific disease say cancer ii dies of a natural cause iii withdraws from the study and iv is alive and still under study at time m the variable y refers to the time at which an event occurs based on such data for n individuals we consider the problem of estimation of a specific occurrence exposure rate soer which is a risk ratio defined as the ratio of probability of death due to cancer in the interval 0 m to the mean lifetime of all individuals up to the time point m the asymptotic distribution of a nonparametric estimator of soer is shown to be normal and the asymptotic variance involves unknown parameters various ways of bootstrapping are discussed for construction of confidence interva

on the rate of uniform convergence of the product limit estimator strong and weak laws by approximating the classical product limit estimator of a distribution function with an average of iid random variables we derive sufficient and necessary conditions for the rate of both strong and weak uniform convergence of the product limit estimator over the whole line these findings somehow fill a longstanding gap in the asymptotic theory of survival analysis the result suggests a natural way of estimating the rate of convergent we also prove a related conjecture raised by gill and discuss its application to the construction of a confidence interval for a survival function near the endpoint

the product limit estimator and the bootstrap some asymptotic representations the product limit estimator and its quantile process are represented as i i d mean processes with a remainder of ordern 3 4 logn 3 4 a s corresponding bootstrap versions of these representations are given which can help one visualize how the bootstrap procedure operates in this set up

two sided bias bound of the kaplan meier estimator we prove in this paper that the bias of the kaplan meier estimator functional which always under estimates the true value is decreasing at an exponential rate an application of the inequality is also given

confidence interval estimation for the ratio of simple and standardized rates in cohort studies summary computer simulation has been used to compare four methods for calculating confidence intervals for simple rate ratios estimated from cohort studies the method proposed by cornfield 1956 in proceedings of the third berkeley symposium on mathematical statistics and probability vol iv 135148 for interval estimation for the odds ratio in a 2 x 2 table adapted to the ratio of two poisson variables gives reasonable and conservative results and is recommended in view of its relative computational simplicity three methods are compared for interval estimation for the ratio of standardized rates such as the standardized morbidity or mortality ratio when the data are stratified by some confounding factor the adapted cornfield method again is the recommended procedure an examination of the use of a summary chi square test with and without continuity correction indicates that the test size closest to the nominal is given by excluding such a correction when the test statistic is computed cohort studies of the possible association between an exposure factor and the risk of a disease are conducted by following groups of individuals with different exposures for a period of time the morbidity or mortality experience of the cohort is monitored and the rates of disease occurrence among the different groups are compared a parameter of particular interest in comparing the disease experience of two groups of individuals with different exposures is the simple rate ratio rr estimated by rr x2 m2 1 rr xi ml where xi and x2 are the numbers of events cases or deaths from disease occurring in the two groups and ml and m2 are the respective denominators the denominator usually used is person years of observation in order to allow for individuals being under observation for different lengths of time when the relative rate rr in the population of which the cohort is a sample remains constant during the time period of the study rr is a consistent estimator of rr if rr is not constant then t1rr will be a consistent estimator only when both exposed and nonexposed subjects are followed during the same time interval and when the sampling fraction of both groups remains constant i e when there is no loss to follow up in order to allow for possible confounding factors such as age with distributions that may differ between the two groups a parameter of more general interest is the standardized rate ratio srr estimated by

paucity of attractors in nonlinear systems driven with complex signals we study the probability of multistability in a quadratic map driven repeatedly by a random signal of length n where n is taken as a measure of the signal complexity we first establish analytically that the number of coexisting attractors is bounded above by n we then numerically estimate the probability p of a randomly chosen signal resulting in a multistable response as a function of n interestingly with increasing drive signal complexity the system exhibits a paucity of attractors that is almost any drive signal beyond a certain complexity level will result in a single attractor response p 0 this mechanism may play a role in allowing sensitive multistable systems to respond consistently to external influences

choosing the resampling scheme when bootstrapping a case study in reliability abstract often when dealing with complex data structures there is no unique way to bootstrap if the data can be viewed as u 1 u n iid from some distribution p then one can bootstrap by resampling the u s alternately one can resample in a more model based way that is by making use of the structure of the model p a typical example of this is linear regression in which the data is y i x i i 1 n one can resample the pairs y i x i or one can resample the residuals from a fitted model this phenomenon arises over quite a wide spectrum of problems and in many cases the different methods of bootstrapping can give substantially different results it seems hopeless to come up with a general theory that compares the different ways of bootstrapping in this article we study in some detail a certain model that arises in reliability theory in which there are two natural ways to bootstrap this model is described as follows available for testing is a sample of n iid systems each having t

stochastic calculus as a tool in survival analysis a review this paper gives a brief survey of the uses of stochastic calculus in survival analysis the role played by martingale central limit theory in deriving asymptotic distributions of estimators and test statistics is described the nelson aalen estimator the kaplan meier estimator cox s proportional hazards model aalen s additive risk model and a goodness of fit test for cox s model are discussed sketches of the proofs of the main results are included

survival analysis with median regression models abstract the median is a simple and meaningful measure for the center of a long tailed survival distribution to examine the covariate effects on survival a natural alternative to the usual mean regression model is to regress the median of the failure time variable or a transformation thereof on the covariates in this article we propose semiparametric procedures to make inferences for such median regression models with possibly censored observations our proposals can be implemented efficiently using a simulated annealing algorithm numerical studies are conducted to show the advantage of the new procedures over some recently developed methods for the accelerated failure time model a special type of mean regression models in the survival analysis the proposals discussed in the article are illustrated with a lung cancer data set

functional laws of the iterated logarithm for the product limit estimator of a distribution function under random censorship or truncation functional laws of the iterated logarithm are established for a modified version of the classical product limit estimator of a distribution function when the data are subject to random censorship or truncation making use of stochastic integral representations and empirical process theory strong approximations involving i i d continuous parameter martingales are developed for the product limit estimator and these strong approximations are then applied to derive the functional laws

strong uniform consistency for nonparametric survival curve estimators from randomly censored data let p x t f t p y t g t and p z t h t h t f t g t an important problem of survival analysis is the estimation of the distribution function f recently the properties of two types of estimators were investigated one of them is the product limit p l estimator of kaplan and meier 8 the other is the nonparametric bayesian estimator introduced by v susarla and j van ryzin 11 in an earlier paper 13 b b winter and the authors proved the uniform consistency of the p l in a second joint paper 6 there was proved the uniform consistency of the p l with rate of o log n l12 nll4 the results of the above two papers were proved without any continuity conditions on f and g in case of arbitrary f and discrete g having finitely many jump points the uniform consistency of the p l with rate factor 0 log nl t was proved in 5 in 10 susarla and van rjzin proved for the bayesian estimator the pointwise consistency with rate factor o log n and the pointwise mean square consistency with rate factor