on witten s proof of the positive energy theorem this paper gives a mathematically rigorous proof of the positive energy theorem using spinors this completes and simplifies the original argument presented by edward witten we clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant dirac operator

interval censored data a note on the nonparametric maximum likelihood estimator of the distribution function summary gentleman geyer 1994 discuss the analysis of interval censored data and present results based on standard convex optimisation theory here this problem is viewed from the perspective of a mixing problem of indicator functions using an analogy with the problem of mixture distributions a variety of results are easily derived including a characterisation theorem for the maximum likelihood estimator and various reliably convergent algorithms software for the analysis of mixture distributions can be used to find the nonparametric estimate of the distribution function of the interval censored survival time examples are provided to demonstrate the theory

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

survival curve estimation for informatively coarsened discrete event time data interval censored or more generally coarsened event time data arise when study participants are observed at irregular time periods and experience the event of interest in between study observations such data are often analysed assuming non informative censoring which can produce biased results if the assumption is wrong this paper extends the standard approach for estimating survivor functions to allow informatively interval censored data by incorporating various assumptions about the censoring mechanism into the model we include a bayesian extension in which final estimates are produced by mixing over a distribution of assumed censoring mechanisms we illustrate these methods with a natural history study of hiv infected individuals using assumptions elicited from an aids expert

eigenvalue boundary problems for the dirac operator on a compact riemannian spin manifold with mean convex boundary we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental dirac operator including the global aps condition and a riemannian version of the local mit bag condition we show that friedrich s inequality for the eigenvalues of the dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems more precisely we prove that for both the aps and the mit conditions the equality cannot be achieved and for the other two conditions the equality characterizes respectively half spheres and domains bounded by minimal hypersurfaces in manifolds carrying non trivial real killing spinors

estimation of a failure time distribution based on imperfect diagnostic tests summary sequentially administered diagnostic tests used to determine the occurrence of a silent event are sometimes subject to error leading to false positive and false negative test results in such cases standard methods for interval censored data do not give valid estimates of the distribution of the time to the event we present methods for estimating the distribution of the time to the event that account for multiple types of imperfect diagnostic test as well as differing periods at risk we illustrate the methods with simulated data and results from a clinical trial for the prevention of mother to infant transmission of hiv in tanzania

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

hazard based nonparametric survivor function estimation summary a representation is developed that expresses the bivariate survivor function as a function of the hazard function for truncated failure time variables this leads to a class of nonparametric survivor function estimators that avoid negative mass the transformation from hazard function to survivor function is weakly continuous and compact differentiable so that such properties as strong consistency weak convergence to a gaussian process and bootstrap applicability for a hazard function estimator are inherited by the corresponding survivor function estimator the set of point mass assignments for a survivor function estimator is readily obtained by using a simple matrix calculation on the set of hazard rate estimators special cases arise from a simple empirical hazard rate estimator and from an empirical hazard rate estimator following the redistribution of singly censored observations within strips the latter is shown to equal van der laan s repaired nonparametric maximum likelihood estimator for which a greenwood like variance estimator is given simulation studies are presented to compare the moderate sample performance of various nonparametric survivor function estimators

locally efficient estimation with bivariate right censored data estimation of the survival curve for independently right censored bivariate failure time data is a problem that has been studied extensively over the past 20 years in this article we propose a new class of estimators for the bivariate survivor function based on locally efficient le estimation the le estimator takes bivariate estimators fn and gn of the distributions of the time variables t1 t2 and the censoring variables c1 c2 and maps them to the resulting estimator le if fn and gn are appropriate consistent estimators of f and g then le will be nonparametrically efficient thus the term locally efficient however if either fn or gn but not both is not a consistent estimator of f or g then le will still be consistent and asymptotically normally distributed we propose a locally efficient estimator that uses a consistent nonparametric estimator for g and allows the user to supply lower dimensional semiparametric or parametric working model for f because the estimator that we cho

nonparametric estimation of a markov illness death process from interval censored observations with application to diabetes survival data summary the nonparametric estimation of the cumulative transition intensity functions in a threestate time nonhomogeneous markov process with irreversible transitions an illness death model is considered when times of the intermediate transition e g onset of a disease are interval censored the times of death are assumed to be known exactly or to be rightcensored in addition the observed process may be left truncated data of this type arise when the process is sampled periodically for example when the patients are monitored through periodic examinations the observations on times of change in their disease status will be interval censored under the sampling scheme considered here the nelson aalen estimator aalen 1978 for a cumulative transition intensity is not applicable in the proposed method the maximum likelihood estimators of some of the transition intensities are derived from the estimators of the corresponding subdistribution functions the maximum likelihood estimators are shown to have a self consistency property the self consistency algorithm is developed for the computation of the estimators this approach generalises the results from turnbull 1976 and frydman 1992 the methods are illustrated with diabetes survival data

a kernel method for incorporating information on disease progression in the analysis of survival summary this paper considers incorporating information on disease progression in the analysis of survival a three state model is assumed with the distribution of each transition estimated separately the distribution of survival following progression can depend on the time of progression kernel methods are used to give consistent estimators under general forms of dependence the estimators for the individual transitions are then combined into an overall estimator of the survival distribution a test statistic for equality of survival between treatment groups is proposed based on the tests of pepe fleming 1989 1991 in simulations the kernel method successfully incorporated dependence on the time of progression in some reasonable settings but under extreme forms of dependence the tests had substantial bias if survival beyond progression can be predicted fairly accurately then gains in power over standard methods that ignore progression can be substantial but the gains are smaller when survival beyond progression is more variable the methodology is illustrated with an application to a breast cancer clinical trial

alternative em methods for nonparametric finite mixture models summary this research focuses on a general class of maximum likelihood problems in which it is desired to maximise a nonparametric mixture likelihood with finitely many known component densities over the set of unknown weight parameters convergence of the conventional em algorithm for this problem is extremely slow when the component densities are poorly separated and when the maximum likelihood estimator requires some of the weights to be zero as the algorithm can never reach such a boundary point alternative methods based on the principles of em are developed using a two stage approach first a new data augmentation scheme provides improved convergence rates in certain parameter directions secondly two cyclic versions of this data augmentation are created by changing the missing data formulation between the em steps these extend the acceleration directions to the whole parameter space giving another order of magnitude increase in convergence rate examples indicate that the new cyclic versions of the data augmentation schemes can converge up to 500 times faster than the conventional em algorithm for fitting nonparametric finite mixture models

tests of independence for bivariate data with random censoring a contingency table approach procedures are proposed for testing the hypothesis of independence between two discrete or discretized random variables one or both of which may be randomly censored the censoring mechanisms can be either independent bivariate censoring or identical univariate censoring the proposed tests reduce to goodness of fit tests for log linear models applied to either complete or incomplete contingency tables three examples are analyzed for illustrative purposes

non solvability for a class of left invariant second order differential operators on the heisenberg group we study the question of local solvability for second order left invariant differential operators on the heisenberg group h n of the form formula math where a ij is a complex n x n matrix such operators never satisfy a cone condition in the sense of sjostrand and hormander we may assume that p a cannot be viewed as a differential operator on a lower dimensional heisenberg group under the mild condition that re im and their commutator are linearly independent we show that p is not locally solvable even in the presence of lower order terms provided that n 7 in the case n 3 we show that there are some operators of the form described above that are locally solvable this result extends to the heisenberg group h 3 a phenomenon first observed by karadzhov and muller in the case of h 2 it is interesting to notice that the analysis of the exceptional operators for the case n 3 turns out to be more elementary than in the case n 2 when 3 n 6 the analysis of these operators seems to become quite complex from a technical point of view and it remains open at this time

computer assisted analysis of mixtures c a man statistical algorithms summary this paper presents various algorithmic approaches for computing the maximum likelihood estimator of the mixing distribution of a one parameter family of densities and provides a unifying computeroriented concept for the statistical analysis of unobserved heterogeneity i e observations stemming from different subpopulations in a univariate sample the case with unknown number of population subgroups as well as the case with known number of population subgroups with emphasis on the first is considered in the computer package c a man computer assisted mixture analysis it includes an algorithmic menu with choices of the em algorithm the vertex exchange algorithm a combination of both as well as the vertex direction method to ensure reliable convergence a steplength menu is provided for the three latter methods each achieving monotonicity for the direction of choice c a man has the option to work with restricted support size that is the case when the number of components is known a priori in the latter case the em algorithm is used applications of mixture modelling in medical problems are discussed

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

a non parametric maximum likelihood estimator for bivariate interval censored data we derive a non parametric maximum likelihood estimator for bivariate interval censored data using standard techniques for constrained convex optimization our approach extends those taken for univariate interval censored data we illustrate the estimator with bivariate data from an aids study

a conformal lower bound for the smallest eigenvalue of the dirac operator and killing spinors on a riemannian spin manifold we give a lower bound for the square of the eigenvalues of the dirac operator by the smallest eigenvalue of the conformal laplacian the yamabe operator we prove in the limiting case that the eigenspinor field is a killing spinor i e parallel with respect to a natural connection in particular if the scalar curvature is positive the eigenspinor field is annihilated by harmonic forms and the metric is einstein

modification of the greenwood formula for correlated response times life table methodology for interval censored survival times is used to estimate marginal survival probabilities from data consisting of independent cohorts of correlated responses we restrict our attention to situations where response times within cohorts are exchangeable and the marginal survival distributions are the same for all individuals within any cohort a modification of the greenwood formula is derived for estimating variances and covariances of the estimates of the marginal survival probabilities simulation results show that this formula provides accurate adjustments for within cohort correlations in finite samples comparisons with a nonparametric maximum likelihood estimator indicate negligible loss in efficiency from using simple life table estimators of marginal survival probabilities that ignore the within cohort correlations data from an angioplasty study where more than one procedure is performed on some patients are used to illustrate this methodology

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

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

analysis of a randomized trial to prevent vertical transmission of hiv 1 abstract we analyze data from a randomized trial of breastfeeding versus formula feeding in hiv 1 infected mothers and their infants estimates of infant hiv 1 free survival and the timing of vertical transmission of hiv 1 are complicated by the fact that the infants hiv 1 status at death may be unknown the time of hiv 1 infection may be interval censored and the tests used to detect hiv 1 infection in the infants may be less than 100 sensitive or specific to address these problems we develop nonparametric and semiparametric approaches to estimating the joint distribution of hiv 1 infection time and death time the nonparametric estimator is based on a self consistency algorithm and in the current application provides a unique estimate of the hiv 1 free survival at 24 months old however the estimate of the cumulative probability of hiv 1 infection over time is nonunique to resolve the nonuniqueness of the nonparametric estimator a semiparametric approach is proposed in which the joint distributi

maximum likelihood for interval censored data consistency and computation summary standard convex optimization techniques are applied to the analysis of interval censored data these methods provide easily verifiable conditions for the self consistent estimator proposed by turnbull 1976 to be a maximum likelihood estimator and for checking whether the maximum likelihood estimate is unique a sufficient condition is given for the almost sure convergence of the maximum likelihood estimator to the true underlying distribution function

likelihood inference for mixtures geometrical and other constructions of monotone step length algorithms summary this paper considers algorithms for the maximum likelihood estimator of the mixing distribution of a family of parametric densities specific interest is devoted to the monotonicity of the step length of the solution in 2 we introduce the idea of estimating the area above the second derivative curve and in theorem 1 we relate area overestimation to the monotonicity of the associated algorithm in 3 we apply theorem 1 to the monotonicity analysis of well known algorithms as well as to the construction of a new class of monotone step length choices which is particularly simple in the mixture setting numerical comparisons and refinements in keeping the monotonic step length from becoming too conservative are given theorem 3 characterizes conditions under which the newton raphson step is monotonic and if these conditions fail to hold states that the regula falsi step is monotonic

self consistent nonparametric maximum likelihood estimator of the bivariate survivor function summary as usually formulated the nonparametric likelihood for the bivariate survivor function is overparameterized resulting in uniqueness problems for the corresponding nonparametric maximum likelihood estimator here the estimation problem is redefined to include parameters for marginal hazard rates and for double failure hazard rates only at informative uncensored failure time grid points where there is pertinent empirical information double failure hazard rates at other grid points in the risk region are specified rather than estimated with this approach the nonparametric maximum likelihood estimator is unique and can be calculated using a two step procedure the first step involves setting aside all doubly censored observations that are interior to the risk region the nonparametric maximum likelihood estimator from the remaining data turns out to be the dabrowska 1988 estimator the omitted doubly censored observations are included in the procedure in the second stage using self consistency resulting in a noniterative nonparametric maximum likelihood estimator for the bivariate survivor function simulation evaluation and asymptotic distributional results are provided moderate sample size efficiency for the survivor function nonparametric maximum likelihood estimator is similar to that for the dabrowska estimator as applied to the entire dataset while some useful efficiency improvement arises for the corresponding distribution function estimator presumably due to the avoidance of negative mass assignments

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

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

positive mass conjecture for five dimensional lorentzian manifolds we extend witten s proof on positive mass conjecture to five dimensional lorentzian manifolds with a rigorous treatment in mathematics

identifiability of bivariate survival curves from censored data abstract we show that the survival curve is identifiable in bivariate censored data problems under weaker independence assumptions than have commonly been made the common assumption has been mutual independence of t 1 t 2 and z 1 z 2 where t 1 t 2 is the true survival vector z 1 z 2 is a nuisance censoring vector and bivariate right censored data is observed we show that the distribution of t 1 t 2 is identifiable under weaker conditional independence assumptions for distributions with full support bivariate survival analysis is a more powerful analysis tool than univariate analysis if multiple possibly related times are of interest the mutual independence model has become popular as a nonparametric way of analyzing such data analysis of the bivariate problem and analogy with univariate models are used to show that the conditional independence model is more widely applicable as a general nonparametric model for bivariate survival data

regularized estimation in the accelerated failure time model with high dimensional covariates we consider two regularization approaches the lasso and the threshold gradient directed regularization for estimation and variable selection in the accelerated failure time model with multiple covariates based on stute s weighted least squares method the stute estimator uses kaplan meier weights to account for censoring in the least squares criterion the weighted least squares objective function makes the adaptation of this approach to multiple covariate settings computationally feasible we use v fold cross validation and a modified akaike s information criterion for tuning parameter selection and a bootstrap approach for variance estimation the proposed method is evaluated using simulations and demonstrated on a real data example

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

solvability for a class of doubly characteristic differential operators on 2 step nilpotent groups since h lewy s le discovery of a smooth linear partial differential operator without solution the problem of solvability of linear differential operators has been investigated intensively by various authors this has meanwhile led most notably through the work of hdrmander maslov egorov nirenberg erves and beals fefferman but with many important contributions also by others to a very satisfactory theory in the case of principal type operators see h6 much less is known about operators whose principal symbol vanishes of second order on part of its characteristic variety i e those which admit double characteristics even in the case of real principal symbols the main objective of this article is to present a model class of such operators for which we are able to discuss solvability in a complete way our class being composed of rather special operators will certainly not capture all phenomena which might arise in the study of general doubly characteristic operators with real symbol as the results in st already indicate but it is still rich enough to exhibit several new features if one compares with operators of principal type the operators that we shall consider will be left invariant differential operators on connected simply connected 2 step nilpotent lie groups g of the form m 0 1 le ajkvjvk u j k l

covariance and survivor function estimation using censored multivariate failure time data summary the covariance between counting process martingales is used to characterize the dependence between two failure time variates a representation of the bivariate survivor function is obtained in terms of the marginal survivor functions and this covariance function a closely related representation expresses the bivariate survivor function in terms of marginal survivor functions and a conditional covariance function leading to a new nonparametric survivor function estimator generalizations to higher dimensional failure time variates are also given simulation evaluations of the survivor function estimator are presented and generalizations to regression problems are outlined

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

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

a remarkable class of second order differential operators on the heisenberg group mathbb h _2 let l be a differential operator on mathbb r 5 whose principal part is of the form sum j k 1 2 alpha jk x j y k where x j partial x j and y j partial y j x j partial u j 1 2 are the usual vector fields generating the lie algebra of the heisenberg group mathbb h 2 we study the problem of local solvability of these doubly characteristic operators the whole class of operators splits into three subclasses depending on the sign of a respective determinant the operators in the first subclass when the determinant is negative are generically non solvable the operators in the second subclass when the determinant is positive are solvable for arbitrary left invariant lower order terms provided that the coefficient matrix alpha j k is non degenerate this fact seems remarkable since many of these operators have the property that the values taken by their principal symbol are not contained in any proper subcone of the complex plane under suitable conditions solvability even holds in the presence of non invariant lower order terms

positive mass theorems for higher dimensional lorentzian manifolds we extend witten s proof a new proof of the positive energy theorem commun math phys 80 381 402 1981 of the positive mass theorem to a rigorous proof for lorentzian manifolds of any dimension this includes the original higher dimensional positive mass theorem proved by spinors parker t and taubes c on witten s proof of the positive energy theorem commun math phys 84 223 238 1982 for dimension 4 and zhang x positive mass conjecture for five dimensional lorentzian manifolds j math phys 40 3540 3552 1999 for dimension 5 stimulated by the article of zhang positive mass theorem for hypersurface in 5 dimensional lorentzian manifolds commun anal geom 8 635 652 2000 we weaken the spin condition on the spacelike hypersurface to require only a spinc structure and give a modified positive mass theorem for lorentzian manifolds for dimensions 4 5 and 6

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

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

on the solvability of homogeneous left invariant differential operators on the heisenberg group abstract we discuss the problem of solvability for the class of homogenoeus left invariant operators s on the heisenberg group h n introduced by f de mari m m peloso and f ricci 1995 j reine angew math 464 67 96 where s sp c n has a degenerated semi definite real part and n 2 l l 0 1 we completely characterize the solvable operators in this class as a consequence we obtain that certain operators s are solvable while their transposes t s are not this is the case for instance for the operator yz on h 1 where z x iy is the unsolvable lewy operator

recent developments in computer assisted analysis of mixtures this paper reviews recent developments in the area of computer assisted analysis of mixture distributions c a man given a biometric situation of interest in which under homogeneity assumptions a certain parametric density occurs such as the poisson the binomial the geometric the normal and so forth then it is argued that this situation can easily be enlarged to allow a variation of the scalar parameter in the population this situation is called unobserved heterogeneity this naturally leads to a specific form of nonparametric mixture distribution that can then be assumed to be the standard model in the biometric application of interest since it also incorporates the homogeneous situations as a special case besides developments in theory and algorithms the work focuses on developments in biometric applications such as meta analysis fertility studies estimation of prevalence under clustering and estimation of the distribution function of survival time under interval censoring the approach is nonparametric for the mixing distribution including leaving the number of components subpopulations of the mixing distribution unknown

local solvability of linear differential operators with double characteristics i necessary conditions this is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators l defined say in an open set omega subset mathbb r n suppose the principal symbol pk of l vanishes to second order at x 0 xi 0 in t omega setminus 0 and denote by q mathcal h the hessian form associated to pk on t x 0 xi 0 t omega as the main result of this paper we show under some rank conditions and some mild additional conditions that a necessary condition for local solvability of l at x0 is the existence of some theta in mathbb r such that rm re e i theta q mathcal h ge 0 we apply this result in particular to operators of the form l sum limits m j k 1 alpha jk x x j x k rm lower rm order rm terms quad 0 1 where the xj are smooth real vector fields and the jk are smooth complex coefficients forming a symmetric matrix mathcal a x alpha jk x j k we say that l is essentially dissipative at x0 if there is some theta in mathbb r such that ei l is dissipative at x0 in the sense that rm re big e i theta mathcal a x 0 big ge 0 for a large class of doubly characteristic operators l of this form our main result implies that a necessary condition for local solvability at x0 is essential dissipativity of l at x0 by means of hormander s classical necessary condition for local solvability the proof of the main result can be reduced to the following question suppose that qa and qb are two real quadratic forms on a finite dimensional symplectic vector space and let qc qa qb be given by the poisson bracket of qa and qb then qc is again a quadratic form and we may ask when can we find a common zero of qa and qb at which qc does not vanish the study of this question occupies most of the paper and the answers may be of independent interest in the second paper of this series building on joint work with f ricci m peloso and others we shall study local solvability of essentially dissipative left invariant operators of the form 0 1 on heisenberg groups in a fairly comprehensive way various examples exhibiting a kind of exceptional behaviour from previous joint works e g with g karadzhov have shown that there is little hope for a complete characterization of locally solvable operators on heisenberg groups however the generic scheme of what rules local solvability of second order operators on heisenberg groups becomes evident from our work

a simple nonparametric estimator of the bivariate survival function under univariate censoring summary in the presence of univariate censoring the bivariate survival function of paired failure times can be expressed as the ratio of the bivariate at risk probability to the survival function of the censoring time the use of this natural representation yields a very simple nonparametric estimator for the bivariate survival curve the estimator is strongly consistent and upon proper normalization converges weakly to a zero mean gaussian process with an easily estimated covariance function numerical studies demonstrate that both the survival curve estimator and its covariance function estimator perform markedly well for practical sample sizes applications to the correlation problem and to the interval estimation of the difference in median survival times are also studied

robust nonparametric estimation of monotone regression functions with interval censored observations nonparametric estimation of monotone regression functions is a classical problem of practical importance robust estimation of monotone regression functions in situations involving interval censored data is a challenging yet unresolved problem herein we propose a nonparametric estimation method based on the principle of isotonic regression using empirical process theory we show that the proposed estimator is asymptotically consistent under a specific metric we further conduct a simulation study to evaluate the performance of the estimator in finite sample situations as an illustration we use the proposed method to estimate the mean body weight functions in a group of adolescents after they reach pubertal growth spurt

the dirac witten operator on pseudo riemannian manifolds we study the dirac witten operator for spacelike spin submanifolds in pseudo riemannian manifolds when the normal bundles are spin and odd dimensional we derive new eigenvalue lower bounds we also prove the generalized positive mass theorem using this operator

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

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

models for interval censoring and simulation based inference for lifetime distributions summary interval censored lifetime data arise when individuals in a study are inspected inter mittently so that a lifetime is observed to lie between two successive times in settings where only these two times are available methods exist for nonparametric or parametric estimation of lifetime distributions however there has been virtually no discussion of how inspection processes may be estimated or identified such estimates are needed if one is to generate interval censored data by simulation this paper identifies which aspects of an independent inspection process are estimable from interval censored data and shows how to obtain nonparametric estimates the results allow interval censored data from any specified distribution to be generated and give new simulation procedures for estimation or testing a new omnibus goodness of fit test is introduced

estimation of survival curves from dependent censorship models via a generalized self consistent property with nonparametric bayesian estimation application on presente une famille d estimateurs obtenue par generalisation de la propriete autoconsistante efron 1967 de l estimateur de kaplan meier 1958

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

a semiparametric model for regression analysis of interval censored failure time data left right and interval censored response time data arise in a variety of settings including the analyses of data from laboratory animal carcinogenicity experiments clinical trials and longitudinal studies for such incomplete data the usual regression techniques such as the cox 1972 journal of the royal statistical society series b 34 187 220 proportional hazards model are inapplicable in this paper we present a method for regression analysis which accommodates interval censored data we present applications of this methodology to data sets from a study of breast cancer patients who were followed for cosmetic response to therapy a small animal tumorigenicity study and a clinical trial

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