a numerical algorithm for a class of fractional bvps with p laplacian operator and singularity the convergence and dependence analysis abstract in the paper we establish the uniqueness of positive solutions for a model of higher order singular fractional boundary value problems with p laplacian operator the equation includes the caputo and the riemann liouville fractional derivative the boundary conditions contain riemann stieltjes integrals and nonlocal infinite point boundary conditions the nonlinear terms f and h may be singular on the time variable and space variables the uniqueness result is obtained by the theory of mixed monotone operators we also discuss the dependence of solutions upon a parameter furthermore two examples illustrate our main results via numerical analysis

discontinuous inverse eigenvalue problems on considere des problemes de sturm liouville inverses dans lesquels les fonctions propres ont une discontinuite dans un point interieur on montre que si le potentiel est connu sur la moitie de l intervalle et si une condition limite est donnee alors le potentiel et l autre condition limite sont determines de facon unique par les valeurs propres

inverse wave scattering with discontinuous wave speed the inverse scattering problem on the line is studied for the generalized schrodinger equation d2 dx2 k2h x 2 q x where h x is a positive piecewise continuous function with positive limits h as x this equation in the frequency domain describes the wave propagation in a nonhomogeneous medium where q x is the restoring force and 1 h x is the variable wave speed changing abruptly at various interfaces a related riemann hilbert problem is formulated and the associated singular integral equation is obtained and proved to be uniquely solvable the solution of this integral equation leads to the recovery of h x in terms of the scattering data consisting of q x a reflection coefficient either of h and the bound state energies and norming constants some explicitly solved examples are provided

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

impedance profile inversion via the first transport equation abstract the subject of this paper is the inverse reflection problem for a stratified elastic half space that is a linear elastic medium whose elastic properties depend only on depth from a planar free surface is stimulated at t 0 by a plane wave impulsive source the motion of a typical surface element is recorded for 0 t 2t it is shown that this surface trace determines the acoustic impedance of the medium as a function of travel time to travel time depth t moreover we give a precise characterization of those functions which may appear as surface traces and show uniqueness existence and continuous dependence of the logarithm of the impedance as a function of the surface trace in the sobolev h1 topology

eigenvalue problems of second order impulsive differential equations abstract this paper is concerned with an eigenvalue problem for second order differential equations with impulse the existence of a countably infinite set of eigenvalues and eigenfunctions is proved

symmetric positive solutions to singular system with multi point coupled boundary conditions in this paper we study the existence and multiplicity of symmetric positive solutions for a nonlinear system with multi point coupled boundary conditions the arguments are based upon a specially constructed cone and the fixed point index theorem in cones an example is then given to demonstrate the applicability of our results

on the existence of solutions for fractional differential inclusions with sum and integral boundary conditions in this paper by using the endpoint result for multifunctions we investigate the existence of solutions for a boundary value problem for fractional differential inclusions with sum and integral boundary conditions finally an example is also given to illustrate the validity of our main result

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

stability analysis by krasnoselskii s fixed point theorem for nonlinear fractional differential equations this paper is concerned with the stability analysis of nonlinear fractional differential equations of order 1 2 our main results are obtained by using krasnoselskii s fixed point theorem in a weighted banach space an example and its corresponding simulation are presented to illustrate the main results

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

multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions abstract in this paper we study the existence of multiple solutions for the integral boundary value problems of fractional differential equations by the method of upper and lower solutions and leray schauder degree theory the sufficient conditions about the existence of at least three solutions are obtained moreover it is proved that the integral boundary value problem has at least three positive solutions under the conditions of m 0 and f is nonnegative by given two upper and lower solutions which can be easily obtained through our methods we can present the existence theorem of at least three solutions two examples are also included to illustrate the effectiveness of the proposed results

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 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

on sturm liouville operators with discontinuity conditions inside an interval abstract in this study transformation operator some properties of kernel properties of spectral characteristics and also uniquiness theorems for the inverse problems are studied for a class of sturm liouville differential operators which have discontinuity conditions inside a finite interval

large sample behaviour of the product limit estimator on the whole line weak convergence results are proved for the product limit estimator on the whole line applications are given to confidence band construction estimation of mean lifetime and to the theory of q functions the results are obtained using stochastic calculus and in probability linear bounds for empirical processes

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

empirical likelihood for right censored lifetime data abstractwhen the empirical likelihood el of a parameter is constructed with right censored data literature shows that 2log empirical likelihood ratio typically has an asymptotic scaled chi squared distribution where the scale parameter is a function of some unknown asymptotic variances therefore the el construction of confidence intervals for requires an additional estimation of the scale parameter additional estimation would reduce the coverage accuracy for by using a special influence function as an estimating function we prove that under very general conditions 2log empirical likelihood ratio has an asymptotic standard chi squared distribution with one degree of freedom this eliminates the need for estimating the scale parameter as well as eases some of the often demanding computations of the el method our estimating function yields a smaller asymptotic variance than those of wang and jing 2001 and qin and zhao 2007 thus it is not surprising that confidence intervals usi

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

functional limit laws for the increments of kaplan meier product limit processes and applications we prove functional limit laws for the increment functions of empirical processes based upon randomly right censored data the increment sizes we consider are classified into different classes covering the whole possible spectrum we apply these results to obtain a description of the strong limiting behavior of a series of estimators of local functionals of lifetime distributions in particular we treat the case of kernel density and hazard rate estimators

non strange weird resampling for complex survival data this paper introduces a new data dependent multiplier bootstrap for nonparametric analysis of survival data possibly subject to competing risks the nw procedure includes the general wild bootstrap and the weird bootstrap as special cases the data may be subject to independent right censoring and left truncation the asymptotic correctness of the proposed resampling procedure is proven under standard assumptions simulation results on time simultaneous inference suggest that the weird bootstrap performs better than the standard normal multiplier approach

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

eigenvalues of sturm liouville problems with discontinuity conditions inside a finite interval abstract in this work we use the regularized sampling method to compute the eigenvalues of sturm liouville problems with discontinuity conditions inside a finite interval we work out an example by computing a few eigenvalues and their corresponding eigenfunctions

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

estimating the median survival time summary two methods are proposed for setting confidence intervals for the median survival time it is shown how these are related to bootstrap methods for censored data and some numerical comparisons are presented in many medical follow up studies the survival experience of a group of patients is summarized by the product limit or kaplan meier estimate of the survival distribution this may follow an analysis of the relation between survival time and some explanatory variables by for example the proportional hazards model the median of the estimated survival distribution is sometimes a convenient summary statistic and may be preferred to the mean because it is less sensitive to long tails in the estimated survival distribution this paper is concerned with estimating the distribution of the estimated median and providing confidence intervals for the true median two methods are considered in 2 they are both based in different ways on generalizations of the solution for the uncensored case these methods turn out to be connected to two ways of extending bootstrap methods for use with censored data this connexion is discussed in 3 thinking about extensions of the bootstrap to censored data highlights the need for careful consideration of the role of the censoring mechanism and how it should be incorporated into the data analysis section 4 provides some examples and monte carlo simulations the work is closely related to that of efron 1981 and emerson 1981

existence results for a coupled system of caputo type sequential fractional differential equations with nonlocal integral boundary conditions this paper is concerned with the existence and uniqueness of solutions for a coupled system of caputo type sequential fractional differential equations supplemented with nonlocal riemann liouville integral boundary conditions the existence of solutions is derived by applying leray schauder s alternative while the uniqueness of solution is established via banach s contraction principle an illustrative example is also included the paper concludes with some interesting observations

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

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

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

asymptotically optimal bandwidth selection for kernel density estimators from randomly right censored samples abstract this paper makes two important contributions to the theory of bandwidth selection for kernel density estimators under right censorship first an asymptotic representation of the integrated squared error into easily understood variance and squared bias components is given second it is shown that if the bandwidth is chosen by the data based method of least squares cross validation then it is asymptotically optimal in a compelling sense a by product of the first part is an interesting comparison of the two most popular kernel estimators keywords nonparametric density estimation smoothing parameter

lattice fractional diffusion equation in terms of a riesz caputo difference abstract a fractional difference is defined by the use of the right and the left caputo fractional differences the definition is a two sided operator of riesz type and introduces back and forward memory effects in space difference then a fractional difference equation method is suggested for anomalous diffusion in discrete finite domains a lattice fractional diffusion equation is proposed and the numerical simulation of the diffusion process is discussed for various difference orders the result shows that the riesz difference model is particularly suitable for modeling complicated dynamical behaviors on discrete media

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

positive solutions for a system of fractional differential equations with coupled integral boundary conditions we investigate the existence of positive solutions for a system of nonlinear riemann liouville fractional differential equations with coupled integral boundary conditions

regression on quantile residual life a time specific log linear regression method on quantile residual lifetime is proposed under the proposed regression model any quantile of a time to event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring consistency and asymptotic normality of the regression estimator are established an asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point evaluation of the test statistic does not require estimation of the variance covariance matrix of the regression estimators which involves the probability density function of the survival distribution with censoring simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic the new regression method is applied to a breast cancer data set with long term follow up to estimate the patients median residual lifetimes adjusting for important prognostic factors

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

sturm liouville problems with impulse effects abstract this paper deals with the computation of the eigenvalues of sturm liouville problems with several discontinuity conditions inside a finite interval and parameter dependent boundary conditions using the regularized sampling method a few examples are worked out by computing several eigenvalues and their corresponding eigenfunctions

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

uniqueness theorems for differential pencils with eigenparameter boundary conditions and transmission conditions abstract inverse spectral problems are considered for differential pencils with boundary conditions depending polynomially on the spectral parameter and with a finite number of transmission conditions we give formulations of the associated inverse problems such as titchmarsh weyl theorem hochstadt lieberman theorem and mochizuki trooshin theorem and prove corresponding uniqueness theorems the obtained results are generalizations of the similar results for the classical sturm liouville operator on a finite interval

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

positive solutions of three point boundary value problems for systems of nonlinear second order ordinary differential equations abstract in this paper we study the three point boundary value problems for systems of nonlinear second order ordinary differential equations of the form u f t v t 0 1 v g t u t 0 1 u 0 v 0 0 u u 1 v v 1 under some conditions we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones

fitting semiparametric transformation regression models to data from a modified case cohort design summary we consider the problem of fitting semiparametric transformation regression models to data from a modified case cohort study in which the censoring times of all the censored subjects in the cohort are observed we propose to maximise a conditional profile likelihood to obtain the estimator of the regression parameter under the assumption that the censoring is independent of the covariates the estimator is shown to be consistent and asymptotically normally distributed numerical studies suggest that the relative efficiency of the estimator is very high and that the estimator is often less biased than the estimator from the complete case analysis and more accurate than the pseudolikelihood estimator

analytical methods for recovering coefficients in differential equations from spectral data in this paper analytical methods for recovering coefficients from spectral data are discussed properties of these methods are given and mathematical models for which these methods might apply are presented

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

a kernel type estimator of a quantile function from right censored data abstract based on right censored data from a lifetime distribution f 0 a kernel type estimator of the quantile function qo p inf t f 0 t p 0 p 1 is proposed the estimator is defined by which is smoother than the usual product limit quantile function where denotes the product limit estimator of f 0 from the censored sample under the random censorship model and general conditions on hn k and f 0 it is shown that qn p is strongly consistent in addition an approximation to qn is shown to be asymptotically equivalent to qn with probability one a small monte carlo simulation study shows that for several values of the bandwidth hn qn performs better than in the sense of estimated mean squared errors an optimal bandwidth hn may be estimated by bootstrap methods in some cases the procedure is illustrated by an application to data from a mechanical switch life test

solvability and numerical simulations for bvps of fractional coupled systems involving left and right fractional derivatives abstract we study the three point boundary value problems for the fractional differential coupled systems with both left and right fractional derivatives in the paper firstly the upper lower solutions method for the systems is presented and some existence results to the systems solutions are also obtained secondly we also give out an iterative approach for finding approximate solutions of the systems finally we consider a concrete example where we obtain the existence result of the solution and also give out its numerical solution and the graphical simulation

uniqueness for inverse sturm liouville problems with a finite number of transmission conditions abstract we establish various uniqueness results for inverse spectral problems of sturm liouville operators with a finite number of discontinuities at interior points at which we impose the usual transmission conditions we consider both the cases of classical robin and of eigenparameter dependent boundary conditions

systems of riemannliouville fractional equations with multi point boundary conditions we study the existence and multiplicity of positive solutions for a system of nonlinear riemannliouville fractional differential equations subject to multi point boundary conditions which contain fractional derivatives the nonsingular and singular cases are investigated

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

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

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

inverse spectral problems for the sturm liouville operator with discontinuity abstract in this work we consider the sturm liouville operator on a finite interval 0 1 with discontinuous conditions at 1 2 we prove that if the potential is known a priori on a subinterval b 1 with b 1 2 then parts of two spectra can uniquely determine the potential and all parameters in discontinuous conditions and boundary conditions for the case b 1 2 parts of either one or two spectra can uniquely determine the potential and a part of parameters

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

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

inverse problems for nonabsorbing media with discontinuous material properties one dimensional electromagnetic and elastic inverse problems are formulated for media with discontinuous material properties in addition an impedence mismatch between source and medium is allowed the measured data for either problem is shown to generate a reflection kernel which is used in the solution of the inverse problem the solution algorithm itself is a time domain technique which is a special case of previously obtained results for absorbing media

a law of the logarithm for kernel quantile density estimators received february 1992 revised february 1993 1this manuscript was prepared using computer facilities supported in part by nsf grants dms 89 05292 dms 87 03942 dms 86 01732 and dms 84 04941 and by the university of chicago block fund ams 1991 subject classifications primary 60f15 secondary 62g05 62g30

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