root n consistent semiparametric regression on considere une variable aleatoire x z dans 7b r p 7b r q on construit un estimateur generalisant l estimateur des moindres carres ordinaires en inserant des estimateurs non parametriques de la regression dans la projection orthogonale non lineaire sur z

nonparametric estimation of the sojourn time distributions for a multipath model summary we use a multipath multistate model to describe data with multiple end points statistical inference based on the intermediate end point is challenging because of the problems of nonidentifiability and dependent censoring we study nonparametric estimation for the path probability and the sojourn time distributions between the states the methodology proposed can be applied to analyse cure models which account for the competing risk of death asymptotic properties of the estimators proposed are derived simulation shows that the methods proposed have good finite sample performance the methodology is applied to two data sets

nonparametric estimation of a multivariate distribution in the presence of censoring summary this paper presents examples of situations in which one wishes to estimate a multivariate distribution from data that may be right censored a distinction is made between what we term homogeneous and heterogeneous censoring it is shown how a multivariate empirical survivor function must be constructed in order to be considered a nonparametric maximum likelihood estimate of the underlying survivor function a closed form solution similar to the product limit estimate of kaplan and meier is possible with homogeneous censoring but an iterative method such as the em algorithm is required with heterogeneous censoring an example is given in which an anomaly is produced if censored multivariate data are analyzed as a series of univariate variables this anomaly is shown to disappear if

complete integrability of the kortweg de vries equation under perturbation around its solution lie backlund symmetry approach it is shown that when the korteweg de vries equation is perturbed about a particular solution the resulting evolution equations to all order of perturbations admit infinitely many lie backlund symmetries the corresponding commuting constants of motion are derived and thereby complete integrability can be established

the incidental parameter problem since 1948 this paper was written to mark the 50th anniversary of neyman and scott s econometrica paper defining the incidental parameter problem it surveys the history both of the paper and of the problem in the statistics and econometrics literature

nonparametric estimation of successive duration times under dependent censoring summary this paper considers nonparametric inference for duration times of two successive events since the second duration process becomes observable only if the first event has occurred the length of the first duration affects the probability of the second duration being censored dependent censoring arises if the two duration times are correlated which is often the case standard approaches to this problem fail because of dependent censoring mechanism a new product limit estimator for the second duration variable and a pathdependent joint survival function estimator are proposed both modified for the dependent censoring properties of the estimators are discussed an example from lawless 1982 is studied for illustrative purposes as well as a simulation study

nonparametric tests for the gap time distributions of serial events based on censored data summary this article deals with the problem of comparing two populations with respect to the distribution of the gap time between two successive events when each subject can experience a series of events and when the event times are potentially right censored several families of nonparametric tests are developed all of which allow arbitrary distributions and dependence structures for the serial events the asymptotic and small sample properties of the proposed tests are investigated an illustration with data taken from a colon cancer study is provided the related problem of testing the independence of two successive gap times is also studied

density estimation under random censorship and order restrictions from asymptotic to small samples do a random censorship and or order restrictions e g nonnegativity monotonicity convexity affect estimation of a smooth density under mean integrated squared error mise under mild assumptions the known asymptotic results which are concerned only with rates answer no this answer especially for censored data contradicts practical experience and statistical intuition so what can be said about constants of mise convergence it is shown that asymptotically a censorship does affect the constant and this allows one to find a relationship between sample sizes of directly observed and censored datasets that implies the same precision of estimation and b an order restriction does not affect the constant and thus no isotonic estimation is needed intensive monte carlo simulations show that the lessons of the sharp asymptotics are valuable for small sample sizes also the estimator developed is illustrated both on simulated data and a dataset of lifetimes of conveyer blades used at wastewate

potential conservation laws we prove that potential conservation laws have characteristics depending only on local variables if and only if they are induced by local conservation laws therefore characteristics of pure potential conservation laws have to essentially depend on potential variables this statement provides a significant generalization of results of the recent paper by bluman et al j math phys 47 113505 2006 moreover we present extensions to gauged potential systems abelian and general coverings and general foliated systems of differential equations an example illustrating possible applications of these results is given a special version of the hadamard lemma for fiber bundles and the notions of weighted jet spaces are proposed as new tools for the investigation of potential conservation laws

estimation of the density of regression errors estimation of the density of regression errors is a fundamental issue in regression analysis and it is typically explored via a parametric approach this article uses a nonparametric approach with the mean integrated squared error mise criterion it solves a long standing problem formulated two decades ago by mark pinsker about estimation of a nonparametric error density in a nonparametric regression setting with the accuracy of an oracle that knows the underlying regression errors the solution implies that under a mild assumption on the differentiability of the design density and regression function the mise of a data driven error density estimator attains minimax rates and sharp constants known for the case of directly observed regression errors the result holds for error densities with finite and infinite supports some extensions of this result for more general heteroscedastic models with possibly dependent errors and predictors are also obtained in the latter case the marginal error density is estimated in all considered cases a blockwise shrinking efromovich pinsker density estimate based on plugged in residuals is used the obtained results imply a theoretical justification of a customary practice in applied regression analysis to consider residuals as proxies for underlying regression errors numerical and real examples are presented and discussed and the s plus software is available

estimation of additive regression models with known links summary we consider estimation of an additive nonparametric regression model with known link function the asymptotic distribution of this regression estimate is given the practical performance is investigated via an application to the study of migration between east and west germany

root n consistent estimation of fixed effect panel data transformation models with censoring this paper considers semiparametric n consistent estimation of the parameters of the generalized panel data transformation model with fixed effects under various forms of censoring without parametric specification for the transformation function or the error distribution while the approach in abrevaya 1999 is n consistent it is not applicable when censoring is present for the case with fixed censoring existing approaches such as those of manski 1987 and abrevaya 2000 apply but their estimators converge at rates slower than n thus possessing zero efficiency compared with n consistent estimators while the approaches by honore 1992 and ridder and tunali 1999 do produce n consistent estimators under fixed and independent censoring respectively they require either the error distribution or the transformation function to be completely known our n consistent estimator for the fixed censoring case could be extended to the cases with independent and dependent censoring under dependent censoring in contrast to our method the existing approaches e g horowitz and lee 2003 lee 2008 and das and ying 2005 require parametric specification for the transformation function or the error distribution large sample properties of the proposed estimators are presented we also provide a simulation study to illustrate our estimation methods in finite samples

exact mean integrated squared error an exact and easily computable expression for the mean integrated squared error mise for the kernel estimator of a general normal mixture density is given for gaussian kernels of arbitrary order this provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis the family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes a number of applications of this method giving important new insights into kernel density estimation are presented among these is the discovery that the usual asymptotic approximations to the mise can be quite inaccurate especially when the underlying density contains substantial fine structure and also strong evidence that the practical importance of higher order kernels is surprisingly small for moderate sample sizes 1 introduction substantial research has been devoted to kernel density estimation this is because it provides a simple yet appealing context in which to study problems and issues that arise in all types of nonparametric curve estimation this includes regression spectral density and hazard estimation and also a variety of other estimators including histograms splines and orthogonal series three important and useful tools for understanding the behavior of nonparametric curve estimators are asymptotic analysis simulation and numerical calculation of error criteria each of these methods provides many useful insights into the complicated structure present in the study of curve estimation however each has its limitations as well rhe strength of asymptotic analysis is that it frequently allows simultaneous study of many different specific examples through general results applying to entire classes of settings the weakness of asymptotics is that they only describe behavior in the limit this is still very useful in many situations because the asymptotics describe the actual situation quite well however it is less useful when the asymptotics have not yet kicked in that is in studying situations where the asymptotically dominant effect has not taken over yet perhaps the biggest drawback to asymptotics is that it is very difficult to determine in a given situation which of these possibilities is occurring

linearizing mappings for certain nonlinear diffusion equations in this paper we consider the nonlinear diffusion equations of the type it is shown that linearizing point transformations do not exist this equation can be equivalently written as a system of two and three equations respectively linearizing point transformations are sought for these two auxiliary systems and the complete list is presented these in turn may be employed to construct contact transformations which map the nonlinear diffusion equation to a linear partial differential equation such linearizing point transformations exist only if n 2 m and n 2 m

estimating the quality of life adjusted gap time distribution of successive events subject to censoring summary when treatment effects are studied in the context of successive or recurrent life events separate analyses of the quality of life scores and of the inter event gap times might lead to possibly contradictory conclusions in an attempt to reconcile this we propose a unitary and more comprehensive nonparametric analysis that combines the two separate analyses by introducing the quality of life adjusted gap time concept inverse probability of censoring estimators of the quality of life adjusted gap time joint and con ditional distributions are proposed and are shown to be consistent and asymptotically normal simulations performed in a variety of scenarios indicate that the joint and conditional quality of life adjusted gap time distribution estimators are virtually unbiased with properly estimated standard errors and asymptotic normality features an example from the international breast cancer study group trial v illustrates the use of the proposed estimators

regression methods for gap time hazard functions of sequentially ordered multivariate failure time data sequentially ordered multivariate failure time data are often observed in biomedical studies and inter event or gap times are often of interest generally standard hazard regression methods cannot be applied to the gap times because of identifiability issues and induced dependent censoring we propose estimating equations for fitting proportional hazards regression models to the gap times model parameters are shown to be consistent and asymptotically normal simulation studies reveal the appropriateness of the asymptotic approximations in finite samples the proposed methods are applied to renal failure data to assess the association between demographic covariates and both time until wait listing and time from wait listing to kidney transplantation

stratified partial likelihood estimation when multiple durations are generated by a single unit they may be related in a way that is not fully captured by the regressors the omitted unit specific variables might vary over the durations they might also be correlated with the variables in the regression component we propose an estimator that responds to these concerns and develop a specification test for detecting unobserved unit specific effects data from malaysia reveal that concentration of child mortality in some families is imperfectly explained by observed explanatory variables and that failure to control for unobserved heterogeneity seriously biases the parameter estimates

semiparametric analysis of the additive risk model in contrast to the proportional hazards model the additive risk model specifies that the hazard function associated with a set of possibly time varying covariates is the sum of rather than the product of the baseline hazard function and the regression function of covariates this formulation describes a different aspect of the association between covariates and the failure time than the proportional hazards model and is more plausible than the latter for many applications in the present paper simple procedures with high efficiencies are developed for making inference about the regression parameters under the additive risk model with an unspecified baseline hazard function

direct semiparametric estimation of single index models with discrete covariates abstract others have developed average derivative estimators of the parameter in the model e y x x g x where g is an unknown function and x is a random vector these estimators are noniterative and easy to compute but require that x be continuously distributed this article develops a noniterative easily computed estimator of for models in which some components of x are discrete the estimator is n consistent and asymptotically normal an application to data on product innovation by german manufacturers illustrates the estimator s usefulness

curve fitting by polynomial trigonometric regression summary we consider a method of estimating an unknown regression curve by regression on a combination of low order polynomial terms and trigonometric terms estimation based on trigonometric functions alone is known to suffer from bias problems at the boundaries due to the periodic nature of the fitted functions we show that these boundary problems are alleviated by adding low order polynomial terms the utility of the method is illustrated with examples and asymptotic results show that the estimators are competitive with other nonparametric procedures applications to estimation in semiparametric models are also considered with particular emphasis on the case of analysis of covariance when the relationship between the response and the covariate is difficult to model parametrically

semiparametric estimation of a proportional hazard model with unobserved heterogeneity the proportional hazard model with unobserved heterogeneity gives the hazard function of a random variable conditional on covariates and a second random variable representing unobserved heterogeneity this paper shows how to estimate the baseline hazard function and the distribution of the unobserved heterogeneity nonparametrically the baseline hazard function and heterogeneity distribution are assumed to satisfy smoothness conditions but are not assumed to belong to known finite dimensional parametric families existing estimators assume that the baseline hazard function or heterogeneity distribution belongs to a known parametric family thus the estimators presented here are more general than existing ones

copula inference under censoring this paper discusses copula model selection procedures and goodness of fit tests under censoring the proposed methodology is based on a comparison of nonparametric and model based estimators of the probability integral transformation k new weighted estimators for k are introduced the resulting tests are compared to an existing approach by simulation and illustrated with an example involving bleeding changes in a woman s reproductive history

partial rank estimation of duration models with general forms of censoring abstract in this paper we propose estimators for the regression coefficients in censored duration models which are distribution free impose no parametric specification on the baseline hazard function and can accommodate general forms of censoring the estimators are shown to have desirable asymptotic properties and monte carlo simulations demonstrate good finite sample performance among the data features the new estimators can accommodate are covariate dependent censoring double censoring and fixed individual or group specific effects we also examine the behavior of the estimator in an empirical illustration

estimating a density under order restrictions nonasymptotic minimax risk on considere la classe de toutes les densites unimodales definies sur un intervalle de longueur l et borne par h on etudie le risque minimax sur cette classe lorsqu on estime en utilisant n observations i i d la perte etant mesuree par la distance l 1 entre l estimateur et la densite reelle

estimation of a bivariate distribution function under random censorship summary estimators are proposed of the underlying distribution function of independent bivariate random vectors which are subject to random right censorship in each coordinate the estimators satisfy the monotonicity requirements of a distribution function are uniformly strongly consistent at a rate equal to that of the empirical distribution function and are fairly simple to compute

identification of a competing risks model with unknown transformations of latent failure times this paper is concerned with identification of a competing risks model with unknown transformations of latent failure times the model includes as special cases competing risks versions of proportional hazards mixed proportional hazards and accelerated failure time models it is shown that covariate effects on latent failure times cause specific link functions and the joint survivor function of the disturbance terms can be identified without relying on modelling the dependence between latent failure times parametrically nor using an exclusion restriction among covariates as a result the paper provides an identification result about the joint survivor function of the latent failure times conditional on covariates

a nonparametric test for panel count data summary panel count data arise when a recurrent event is under investigation and each study subject is observed only at discrete time points in this situation observed data include only the numbers of occurrences of the event of interest between observation time points and no information is available on subjects between their observation time points we propose a nonparametric test for comparing the point processes characterising the recurrent event when only panel count data are available the asymptotic distribution of the test statistic is derived and a simulation study is conducted to evaluate its performance the method is illustrated using data from a medical follow up study

locally efficient estimation of a multivariate survival function in longitudinal studies we consider estimation of the joint distribution of multivariate survival times t t1 tk which are subject to right censoring by a common censoring variablec two estimators are proposed an initial inverse probability of censoring weighted ipcw estimator and a one step estimator both estimators incorporate information on available time independent and time dependent prognostic factor covariate data the ipcw estimator is consistent and asymptotically normal can under coarsening at random car and a correct specification of a model for the hazard of censoring given the past covariate and failure data the one step estimator is a locally efficient doubly robust estimator that is i it is can under the assumption of car and either but not necessarily both correct specification of a model for the hazard of censoring given the past or correct specification of a model for the conditional distribution of t given past failure and covariate information and ii it is efficient when both these

solution of second order difference equations via the τ method abstract classically the solution of many functional equations is sought in the form of an infinite series of some kind in practice to get numerical results the series must be truncated in recent times alternative finite series approximations have been sought as solutions of a functional equation slightly perturbed from the given one this concept was first introduced by lanczos who named it the method in previous studies the author has applied the method to get polynomial and rational approximations for differential equations mostly of hypergeometric type in the present paper we apply the method concept to get like approximations for solutions of difference equations in particular we study the second order difference equation satisfied by the bessel function of the first kind except for some recent work on the gamma function by the author wimp and ting the literature is void on use of the method idea for solutions of difference equations

non parametric inference of adverse events under informative censoring in long term treatments or in treatments associated with frequent severe adverse events aes such as those for oncology it is important to know the probability of occurrence of aes over time and their severity however some patients discontinue treatment and drop out of the clinical trial assumption that the drop outs are non informative is not always true and are not validated by data we propose a method of applying competing risk analysis by defining events of occurrence of ae and drop out prior to ae we focus on one ae at a time we distinguish obvious non informative censoring from other censorings that may not be non informative therefore our approach does not need an independent assumption for drop outs the cumulative incidence function estimator cife for the ae by severity can be obtained by treating the degree of severity as a competing risk within the ae we also propose a non parametric estimator of cif for sequential occurrence of the same ae by forming a subset of subjects with prior occurrence s of the same ae and by applying wang and wells estimator we give a very simple formulation of the cumulative joint incidence function estimator cjife for subjects who drop out of the clinical trial after having suffered from the ae at least one time we evaluate the performance of pepe s variance estimator for cjife with small samples by simulations we find that it works well with a sample size more than 100 a useful graphical presentation for cifes and cjife is shown

non parametric estimation and model checking procedures for marginal gap time distributions for recurrent events for recurrent events there is evidence that misspecification of the frailty distribution can cause severe bias in estimated regression coefficients am j epidemiol 1998 149 404 411 statist med 2006 25 1672 1684 in this paper we adapt a procedure originally suggested in biometrika 1999 86 381 393 for parallel data for checking the gamma frailty to recurrent events to apply the model checking procedure a consistent non parametric estimator for the marginal gap time distributions is needed this is in general not possible due to induced dependent censoring in the recurrent events setting however in biometrika 1999 86 59 70 a non parametric estimator for the joint gap time distributions based on the principle of inverse probability of censoring weights is suggested here we attempt to apply this estimator in the model checking procedure and the performance of the method is investigated with simulations and applied to danish registry data the method is further investigated using the usual kaplan meier estimator and a marginalized estimator for the marginal gap time distributions we conclude that the procedure only works when the recurrent event is common and when the intra individual association between gap times is weak

nonparametric estimation of the gap time distributions for serial events with censored data summary in many follow up studies each subject can potentially experience a series of events which may be repetitions of essentially the same event or may be events of entirely diverent natures this paper provides a simple nonparametric estimator for the multivariate distribution function of the gap times between successive events when the follow up time is subject to right censoring the estimator is consistent and upon proper normalisation converges weakly to a zero mean gaussian process with an easily estimated covariance function numerical studies demonstrate that both the distribution function estimator and its covariance function estimator perform well for practical sample sizes an application to a colon cancer study is presented

smooth optimum kernel estimators of densities regression curves and modes several criteria concerning the choice of kernels for the nonparametric estimation of functions and their derivatives are discussed a specific optimality criterion is described which applies to kernels of compact support and of different orders of smoothness the solutions of the corresponding variational problems are explicitly given many kernels discussed previously are obtained as special cases

backlund transformations for diffusion equations abstract a class of backlund transformations is deduced for the diffusion equation s u t c u u x x by taking advantage of the conservative form of the equation the transformations are used for obtaining new relations connecting various equations of the type u t u a u x x and u t u a u xx

nonparametric analysis of bivariate gap time with competing risks this article considers nonparametric methods for studying recurrent disease and death with competing risks we first point out that comparisons based on the well known cumulative incidence function can be confounded by different prevalence rates of the competing events and that comparisons of the conditional distribution of the survival time given the failure event type are more relevant for investigating the prognosis of different patterns of recurrence disease we then propose nonparametric estimators for the conditional cumulative incidence function as well as the conditional bivariate cumulative incidence function for the bivariate gap times that is the time to disease recurrence and the residual lifetime after recurrence to quantify the association between the two gap times in the competing risks setting a modified kendall s tau statistic is proposed the proposed estimators for the conditional bivariate cumulative incidence distribution and the association measure account for the induced dependent censoring for the second gap time uniform consistency and weak convergence of the proposed estimators are established hypothesis testing procedures for two sample comparisons are discussed numerical simulation studies with practical sample sizes are conducted to evaluate the performance of the proposed nonparametric estimators and tests an application to data from a pancreatic cancer study is presented to illustrate the methods developed in this article

nonparametric bivariate estimation with randomly censored data summary the estimation of a bivariate distribution function with randomly censored data is considered it is assumed that the censoring occurs independently of the lifetimes and that deaths and losses which occur simultaneously can be separated two estimators are developed a reduced sample estimator and a self consistent one it is shown that the latter estimator satisfies a nonparametric likelihood function and is unique up to the final uncensored values in any dimension it jumps at the points of double deaths in both dimensions some key word8 censored data kaplan meier estimator life table product limit estimator survival estimation

group transformations and the nonlinear heat diffusion equation a nonlinear diffusion equation with a power law diffusion coefficient is studied self similar and partially invariant solutions are shown to be identical the theory of homology is introduced and used to generate classes of equations the solutions of which are related through backlund transformations an example of an application is given which generalizes the solution of storm j appl phys 22 1951 pp 940 951 on heat conduction in metals

quasi linear wavelet estimation abstract the main paradigm of the modern wavelet theory of spatial adaptation formulated by donoho and johnstone is that there is a divergence between the linear minimax adaptation theory and the heuristic guiding algorithm development that leads to the necessity of using strongly nonlinear adaptive thresholded methods on the other hand it is well known that linear adaptive estimates are the best whenever an estimated function is smooth is it possible to suggest a quasi linear wavelet estimate by adding to a linear adaptive estimate a minimal number of nonlinear terms on finest scales that offers advantages of linear adaptive estimates and at the same time matches asymptotic properties of strongly nonlinear procedures like the benchmark sureshrink the answer is yes and we discuss quasi linear estimation both theoretically and via a monte carlo study in particular i show that asymptotically a quasi linear procedure not only matches properties of sureshrink over the besov scale but also allows

spline smoothing and optimal rates of convergence in nonparametric regression models linear estimation is considered in nonparametric regression models of the form yi f xi ei xi e a b where the zero mean errors are uncorrelated with common variance a2 and the response function f is assumed only to have a bounded square integrable qth derivative the linear estimator which minimizes the maximum mean squared error summed over the observation points is derived and the exact minimax rate of convergence is obtained for practical problems where bounds on 11 f2q 11 and a2 may be unknown generalized cross validation is shown to give an adaptive estimator which achieves the minimax optimal rate under the additional assumption of normality 1 the model consider the nonparametric regression model

a note on nonparametric estimators of the bivariate survival function under univariate censoring summary a simple nonparametric estimator is proposed for the underlying bivariate distribution of paired failure times in the presence of univariate right censoring the estimator is shown to be a particular path dependent estimator which is a generalisation of an estimator proposed by campbell foldes 1982 under univariate censoring this path dependent estimator has the smallest variance among all path dependent estimators which includes the estimator proposed by lin ying 1993

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

analysis of episodic data with application to recurrent pulmonary exacerbations in cystic fibrosis patients we consider a special type of recurrent event data termed recurrent episode data arising in episodic illness studies when an event occurs it lasts for a random length of time a naive recurrent event analysis disregards the length of the episodes which may contain important information about the severity of the disease the associated medical costs and quality of life bivariate gap time models have been suggested in which length of episodes and time between episodes are modeled jointly these models are useful but may obscure the overall effects of treatment and other prognostic factors the analysis can be further complicated if covariate effects change over time as may occur when the effects vary across episodes this article reviews the existing methods applied to recurrent episode data and approaches the problem using the recently developed temporal process regression novel endpoints are constructed that summarize both episode frequency and the length of episodes and time between episodes

jackknife and analytical bias reduction for nonlinear panel models fixed effects estimators of panel models can be severely biased because of the well known incidental parameters problem we show that this bias can be reduced by using a panel jackknife or an analytical bias correction motivated by large t we give bias corrections for averages over the fixed effects as well as model parameters we find large bias reductions from using these approaches in examples we consider asymptotics where t grows with n as an approximation to the properties of the estimators in econometric applications we show that if t grows at the same rate as n the fixed effects estimator is asymptotically biased so that asymptotic confidence intervals are incorrect but that they are correct for the panel jackknife we show t growing faster than n 1 3 suffices for correctness of the analytic correction a property we also conjecture for the jackknife

integrals of jacobi functions in this paper we study f 1 x a 1 x bp a x dx and its partial derivatives with respect to a and b where p a x is the jacobi function our expressions generalize the work of blue gautschi and gatteschi the results are useful to derive integration formulas for integrands with algebraic and logarithmic singularities

the central limit theorem under random censorship let f n be the kaplan meier estimator of a distribution function f computed from randomly censored data we show that under optimal integrability assumptions on a function the kaplan meier integral df n when properly standardized is asymptotically normal

hierarchy of conservation laws of diffusion convection equations we introduce notions of equivalence of conservation laws with respect to lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems we also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials to construct conservation laws we develop and apply the most direct method which is effective to use in the case of two independent variables admitting possibility of dependence of conserved vectors on a number of potentials we generalize the iteration procedure proposed by bluman and doran wu for finding nonlocal potential conservation laws as an example we completely classify potential conservation laws including arbitrary order local ones of diffusion convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations

bivariate estimation with right truncated data abstract bivariate estimation with survival data has received considerable attention recently however most of the work has focused on random censoring models another common feature of survival data random truncation is considered in this study truncated data may arise if the time origin of the events under study precedes the observation period in a random right truncation model one observes the iid samples of y t only if y t where y is the variable of interest and t is an independent variable that prevents the complete observation of y suppose that y x is a bivariate vector of random variables where y is subject to right truncation in this study the bivariate reverse hazard vector is introduced and a nonparametric estimator is suggested an estimator for the bivariate survival function is also proposed weak convergence and strong consistency of this estimator are established via a representation by iid variables an expression for the limiting covariance function is provided and a

inverse probability of censoring weighted estimates of kendall s τ for gap time analyses summary in life history studies interest often lies in the analysis of the interevent or gap times and the association between event times gap time analyses are challenging however even when the length of follow up is determined independently of the event process because associations between gap times induce dependent censoring for second and subsequent gap times this article discusses nonparametric estimation of the association between consecutive gap times based on kendall s in the presence of this type of dependent censoring a nonparametric estimator that uses inverse probability of censoring weights is provided estimates of conditional gap time distributions can be obtained following specification of a particular copula function simulation studies show the estimator performs well and compares favorably with an alternative estimator generalizations to a piecewise constant clayton copula are given several simulation studies and illustrations with real data sets are also provided

nonparametric inference on bivariate survival data with interval sampling association estimation and testing in many biomedical applications interest focuses on the occurrence of two or more consecutive failure events and the relationship between event times such as age of disease onset and residual lifetime bivariate survival data with interval sampling arise frequently when disease registries or surveillance systems collect data based on disease incidence occurring within a specific calendar time interval the initial event is then retrospectively confirmed and the subsequent failure event may be observed during follow up in life history studies the initial and two consecutive failure events could correspond to birth disease onset and death the statistical features and bias of observed data in relation to interval sampling were discussed by zhu wang 2012 here we propose nonparametric estimation of the association between bivariate failure times based on kendall s tau for data collected with interval sampling a nonparametric estimator is given where the contribution of each comparable and orderable pair is weighted by the inverse of the associated selection probability analysis methods for bivariate survival data with interval sampling rely on the assumption of quasi independence i e that bivariate failure times and the time of the initial event are independent in the observable region this paper develops a nonparametric test of quasi independence based on a bivariate conditional kendall s tau for such data simulation studies demonstrate that the association estimator and testing procedure perform well with moderate sample sizes illustrations with two real datasets are provided

conditional density estimation in a regression setting regression problems are traditionally analyzed via univariate characteristics like the regression function scale function and marginal density of regression errors these characteristics are useful and informative whenever the association between the predictor and the response is relatively simple more detailed information about the association can be provided by the conditional density of the response given the predictor for the first time in the literature this article develops the theory of minimax estimation of the conditional density for regression settings with fixed and random designs of predictors bounded and unbounded responses and a vast set of anisotropic classes of conditional densities the study of fixed design regression is of special interest and novelty because the known literature is devoted to the case of random predictors for the aforementioned models the paper suggests a universal adaptive estimator which i matches performance of an oracle that knows both an underlying model and an estimated conditional density ii is sharp minimax over a vast class of anisotropic conditional densities iii is at least rate minimax when the response is independent of the predictor and thus a bivariate conditional density becomes a univariate density iv is adaptive to an underlying design fixed or random of predictors

analysing bivariate survival data with interval sampling and application to cancer epidemiology in biomedical studies ordered bivariate survival data are frequently encountered when bivariate failure events are used as outcomes to identify the progression of a disease in cancer studies interest could be focused on bivariate failure times for example time from birth to cancer onset and time from cancer onset to death this paper considers a sampling scheme termed interval sampling in which the first failure event is identified within a calendar time interval the time of the initiating event can be retrospectively confirmed and the occurrence of the second failure event is observed subject to right censoring in a cancer data application the initiating first and second events could correspond to birth cancer onset and death the fact that the data are collected conditional on the first failure event occurring within a time interval induces bias interval sampling is widely used for collection of disease registry data by governments and medical institutions though the interval sampling bias is frequently overlooked by researchers this paper develops statistical methods for analysing such data semiparametric methods are proposed under semi stationarity and stationarity numerical studies demonstrate that the proposed estimation approaches perform well with moderate sample sizes we apply the proposed methods to ovarian cancer registry data

maximum likelihood estimation of smooth monotone and unimodal densities we study the nonparametric estimation of univariate monotone and unimodal densities using the maximum smoothed likelihood approach the monotone estimator is the derivative of the least concave majorant of the distribution corresponding to a kernel estimator we prove that the mapping on distributions with density the derivative of the least concave majorant of is a contraction in all l p norms 1 p and some other distances such as the hellinger and kullback leibler distances the contractivity implies error bounds for monotone density estimation almost the same error bounds hold for unimodal estimation

model selection and semiparametric inference for bivariate failure time data abstract we propose model selection procedures for bivariate survival models for censored data generated by the archimedean copula family in route to constructing the selection methodology we develop estimates of some time dependent association measures including estimates of the local and global kendall s tau local odds ratio and other measures defined throughout the literature we propose a goodness of fit based model selection methodology as well as a graphical approach we show that the proposed methods have desirable asymptotic properties and perform well in finite samples

semiparametric time to event modeling in the presence of a latent progression event summary in cancer research interest frequently centers on factors influencing a latent event that must precede a terminal event in practice it is often impossible to observe the latent event precisely making inference about this process difficult to address this problem we propose a joint model for the unobserved time to the latent and terminal events with the two events linked by the baseline hazard covariates enter the model parametrically as linear combinations that multiply respectively the hazard for the latent event and the hazard for the terminal event conditional on the latent one we derive the partial likelihood estimators for this problem assuming the latent event is observed and propose a profile likelihood based method for estimation when the latent event is unobserved the baseline hazard in this case is estimated nonparametrically using the em algorithm which allows for closed form breslow type estimators at each iteration bringing improved computational efficiency and stability compared with maximizing the marginal likelihood directly we present simulation studies to illustrate the finite sample properties of the method its use in practice is demonstrated in the analysis of a prostate cancer data set

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

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

nonparametric estimation of the bivariate survival function with an application to vertically transmitted aids summary in this paper we propose a nonparametric estimator of the bivariate survival function when the two durations are subject to right censoring we consider the situation where the duration variables are successive and are always observed in a particular order and the censoring mechanism bears on their sum rather than on each separately thus if they are correlated the second duration is right censored by a dependent variable a nonparametric maximum likelihood estimator is derived that takes the dependent censoring into account we also derive the asymptotic variance of the estimator and show how it can be estimated the methods are applied on a vertically acquired mother to child transmission aids data set the first duration corresponds to the hiv incubation period and the second to the period of aids

solution of the difference equation for gauss hypergeometric function by use of the tau method abstract in two previous papers the idea of the tau method to get approximate solutions of differential equations was extended to get like results for difference equations the models treated were the difference equation for the reciprocal of the gamma function and the difference equation for the entire part of the bessel function of the first kind in this paper we apply the tau method as described in the title of this paper some numerics are presented which indicate that the process is convergent in this connection an a posteriori error analysis for the bessel function case is presented how this analysis could be extended to the model of the present paper is discussed but further analysis is deferred to future research

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

maximum likelihood estimation in semiparametric regression models with censored data summary semiparametric regression models play a central role in formulating the effects of covariates on potentially censored failure times and in the joint modelling of incomplete repeated measures and failure times in longitudinal studies the presence of infinite dimensional parameters poses considerable theoretical and computational challenges in the statistical analysis of such models we present several classes of semiparametric regression models which extend the existing models in important directions we construct appropriate likelihood functions involving both finite dimensional and infinite dimensional parameters the maximum likelihood estimators are consistent and asymptotically normal with efficient variances we develop simple and stable numerical techniques to implement the corresponding inference procedures extensive simulation experiments demonstrate that the inferential and computational methods proposed perform well in practical settings applications to three medical studies yield important new insights we conclude that there is no reason theoretical or numerical not to use maximum likelihood estimation for semiparametric regression models we discuss several areas that need further research

a missing data approach to semi competing risks problems for event time data involving multiple mutually exclusive competing causes of failure classic competing risks results show that marginal survival distributions are not identifiable in a related instance one or more failure modes may be observed provided that the failure events occur in a specific order in such situations sometimes referred to as semi competing risks problems the observations may under realistic assumptions lend information about parameters of interest that would be nonidentifiable in the strict competing risks case here we present an approach that makes use of partially observable multiple modes of failures to obtain an estimate of the marginal distribution of one event type that may occur prior to the occurrence of another event type or be precluded by it we apply the proposed method to the problem of estimating the distribution of time to tumour recurrence at specific sites among breast cancer patients participating in randomized clinical trials

simulation and the asymptotics of optimization estimators on demontre un theoreme de limite centrale general pour des estimateurs definis par minimisation de la longueur d une fonction de critere aleatoire a valeurs vectorielles aucune hypothese de regularite suffisante n est imposee sur la fonction de critere pour que les resultats puissent etre appliques a une classe d estimlateurs de simulation assez large des analyses completes de deux estimateurs de simulation l un introduit par pakes l autre par mcfadden illustrent l application des theoremes generaux

nonparametric estimation of the bivariate recurrence time distribution summary this article considers statistical models in which two different types of events such as the diagnosis of a disease and the remission of the disease occur alternately over time and are observed subject to right censoring we propose nonparametric estimators for the joint distribution of bivariate recurrence times and the marginal distribution of the first recurrence time in general the marginal distribution of the second recurrence time cannot be estimated due to an identifiability problem but a conditional distribution of the second recurrence time can be estimated nonparametrically in the literature statistical methods have been developed to estimate the joint distribution of bivariate recurrence times based on data on the first pair of censored bivariate recurrence times these methods are inefficient in the model considered here because recurrence times of higher orders are not used asymptotic properties of the proposed estimators are established numerical studies demonstrate the estimators perform well with practical sample sizes we apply the proposed method to the south verona italy psychiatric case register pcr data set for illustration of the methods and theory

flexible maximum likelihood methods for bivariate proportional hazards models this article presents methodology for multivariate proportional hazards ph regression models the methods employ flexible piecewise constant or spline specifications for baseline hazard functions in either marginal or conditional ph models along with assumptions about the association among lifetimes because the models are parametric ordinary maximum likelihood can be applied it is able to deal easily with such data features as interval censoring or sequentially observed lifetimes unlike existing semiparametric methods a bivariate clayton model 1978 biometrika 65 141 151 is used to illustrate the approach taken because a parametric assumption about association is made efficiency and robustness comparisons are made between estimation based on the bivariate clayton model and working independence methods that specify only marginal distributions for each lifetime variable

kernel estimation of hazard functions when observations have dependent and common covariates abstract we propose a hazard model where dependence between events is achieved by assuming dependence between covariates this model allows for correlated variables specific to observations as well as macro variables which all observations share this setup better fits many economic and financial applications where events are not independent nonparametric estimation of the hazard function is then studied kernel estimators proposed in nielsen and linton 1995 and linton et al 2003 are shown to have similar asymptotic properties compared with the i i d case mixing conditions ensure the asymptotic results follow these results depend on adjustments to bandwidth conditions simulations are conducted which verify the impact of dependence on estimators bandwidth selection accounting for dependence is shown to improve performance in an empirical application trade intensity in high frequency financial data is estimated

a hot deck multiple imputation procedure for gaps in longitudinal recurrent event histories summary we propose a regression based hot deck multiple imputation method for gaps of missing data in longitudinal studies where subjects experience a recurrent event process and a terminal event examples are repeated asthma episodes and death or menstrual periods and menopause as in our motivating application research interest concerns the onset time of a marker event defined by the recurrent event process or the duration from this marker event to the final event gaps in the recorded event history make it difficult to determine the onset time of the marker event and hence the duration from onset to the final event simple approaches such as jumping gap times or dropping cases with gaps have obvious limitations we propose a procedure for imputing information in the gaps by substituting information in the gap from a matched individual with a completely recorded history in the corresponding interval predictive mean matching is used to incorporate information on longitudinal characteristics of the repeated process and the final event time multiple imputation is used to propagate imputation uncertainty the procedure is applied to an important data set for assessing the timing and duration of the menopausal transition the performance of the proposed method is assessed by a simulation study

semiparametric estimation of a panel data proportional hazards model with fixed effects this paper considers a panel duration model that has a proportional hazards specification with fixed effects the paper shows how to estimate the baseline and integrated baseline hazard functions without assuming that they belong to known finite dimensional families of functions existing estimators assume that the baseline hazard function belongs to a known parametric family therefore the estimators presented here are more general than existing ones this paper also presents a method for estimating the parametric part of the proportional hazards model with dependent right censoring under which the partial likelihood estimator is inconsistent the paper presents some monte carlo evidence on the small sample performance of the new estimators

the theil sen estimator with doubly censored data and applications to astronomy abstract the theil sen estimator of the slope parameter in simple linear regression is extended to data with both the response and the covariate subject to censoring based on inverting a suitable version of kendall s statistic this estimator requires weak assumptions and is simple to compute and a simple estimate of its asymptotic variance is obtained a second extension of the theil sen estimator based on a direct estimation of the median of pairwise slopes is given these estimators are compared numerically with versions of schmitt s estimator and applied to two data sets from the recent astronomical literature

potential equivalence transformations for nonlinear diffusion convection equations potential equivalence transformations pets are effectively applied to a class of nonlinear diffusion convection equations for this class all possible potential symmetries are classified and a theorem on their connection with point symmetries via pets is also proved it is shown that the known nonlocal transformations between equations under consideration are nothing but pets the action of pets on sets of exact solutions of a fast diffusion equation is investigated

nonparametric estimation of a bivariate survival function in the presence of censoring a new family of estimators of a bivariate survival function based on censored vectors is obtained from a decomposition of the bivariate survival function these estimators are uniformly consistent under bivariate censoring and are self consistent under univariate censoring

marginal estimation for multi stage models waiting time distributions and competing risks analyses we provide non parametric estimates of the marginal cumulative distribution of stage occupation times waiting times and non parametric estimates of marginal cumulative incidence function proportion of persons who leave stage j for stage j within time t of entering stage j using right censored data from a multi stage model we allow for stage and path dependent censoring where the censoring hazard for an individual may depend on his or her natural covariate history such as the collection of stages visited before the current stage and their occupation times additional external time dependent covariates that may induce dependent censoring can also be incorporated into our estimates if available our approach requires modelling the censoring hazard so that an estimate of the integrated censoring hazard can be used in constructing the estimates of the waiting times distributions for this purpose we propose the use of an additive hazard model which results in very flexible robust estimates examples based on data from burn patients and simulated data with tracking are also provided to demonstrate the performance of our estimators

efficient estimation in the bivariate censoring model and repairing npmle the npmle in the bivariate censoring model is not consistent the problem is caused by the singly censored observations in this paper we prove that the npmle based on interval censoring the singly censored observations is ecient for this reduced data and moreover if we let the width of the interval converge to zero slowly enough then the npmle is also ecient for the original data we are able to determine a lower bound for the rate at which the bandwidth should converge to zero the eciency proof uses the general identity which holds for npmle of a linear parameter in convex models as proved in van der laan 1993a

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

smoothing spline anova frailty model for recurrent event data summary gap time hazard estimation is of particular interest in recurrent event data this article proposes a fully nonparametric approach for estimating the gap time hazard smoothing spline analysis of variance anova decompositions are used to model the log gap time hazard as a joint function of gap time and covariates and general frailty is introduced to account for between subject heterogeneity and within subject correlation we estimate the nonparametric gap time hazard function and parameters in the frailty distribution using a combination of the newton raphson procedure the stochastic approximation algorithm saa and the markov chain monte carlo mcmc method the convergence of the algorithm is guaranteed by decreasing the step size of parameter update and or increasing the mcmc sample size along iterations model selection procedure is also developed to identify negligible components in a functional anova decomposition of the log gap time hazard we evaluate the proposed methods with simulation studies and illustrate its use through the analysis of bladder tumor data

nonparametric estimation of a generalized additive model with an unknown link function mean of a random scalar y conditional on a random vector x e rd let h x e yix x denote the conditional mean function this paper is concerned with estimating h under weak assumptions about its functional form when x is a multidimensional continuous random variable the paper describes a model that nests single index additive and multiplicative models and that can be estimated with a faster rate of convergence in probability than can a fully nonparametric model nonparametric estimation of h is usually unattractive when x is multidimensional the rate of convergence of nonparametric estimators decreases rapidly as the dimension of x increases and nonparametric estimates can be very imprecise with samples of practical size when x is multidimensional the curse of dimensionality this problem can be overcome through the use of dimension reduction methods they reduce the effective dimension of the estimation problem by making assumptions about the functional form of h that are stronger than those of a fully nonparametric estimator but weaker than those of a finite dimensional parametric model one method of dimension reduction is semiparametric single index modeling the best known and most easily estimated semiparametric single index model assumes that 1 1 h x g 3 x where 3 is an unknown d x 1 vector and g is an unknown function the vector 3 can be estimated with an n 1 2 rate of convergence in probability which is the rate that would be obtained if g were known han 1987 hardle and stoker 1989 horowitz and hardle 1996 ichimura 1993 klein and spady 1993 newey and stoker 1993 powell stock and stoker 1989 sherman 1993 stoker 1991 g can be estimated with the rate of convergence n r 2r 1 where r is an integer that depends on the smoothness of g but not on d dimension reduction can also be achieved by using a nonparametric generalized additive model in such a model