local full likelihood estimation for the proportional hazards model summary in this paper a new approach to local likelihood estimation for censored data is proposed this method employs the full likelihood and alternates between estimating the baseline hazard and estimating the covariate effect the proposed methodology incorporates multidimensional data via additive models some results regarding inference for the covariate effect are also presented in this paper we propose new methodology for estimating a general covariate effect when the data arise from a proportional hazards model the methodology uses the full likelihood rather than the partial likelihood and is iterative in nature we note that techniques relying on the partial likelihood do not agree with the modeling assumptions and hence may be less desirable to begin some notation is introduced the data are observed as independent identically distributed triples ti bi zi where t is the minimum of the true failure time and the true censoring time 6 is an indicator of failure and z is a covariate that may be vector valued the true possibly unobserved failure time is assumed to have distribution function f the survivor function is denoted x the hazard x t and the integrated hazard a t further it is assumed that the data arise from a proportional hazards model that is the hazard x t z is assumed to be the product of a baseline hazard xo t and a covariate effect v z

local em estimation of the hazard function for interval censored data summary we propose a smooth hazard estimator for interval censored survival data using the method of local likelihood the model is fit using a local em algorithm the estimator is more descriptive than traditional empirical estimates in regions of concentrated information and takes on a parametric flavor in regions of sparse information we derive two different standard error estimates for the smooth curve one based on asymptotic theory and the other on the bootstrap we illustrate the local em method for times to breast cosmesis deterioration finkelstein 1986 biometrics42 845 854 and for times to hiv 1 infection for individuals with hemophilia kroner et al 1994 journal of aids7 279 286 our hazard estimates for each of these data sets show interesting structures that would not be found using a standard parametric hazard model or empirical survivorship estimates

global partial likelihood for nonparametric proportional hazards models as an alternative to the local partial likelihood method of tibshirani and hastie and fan gijbels and king a global partial likelihood method is proposed to estimate the covariate effect in a nonparametric proportional hazards model t x exp x 0 t the estimator x reduces to the cox partial likelihood estimator if the covariate is discrete the estimator is shown to be consistent and semiparametrically efficient for linear functionals of x moreover breslow type estimation of the cumulative baseline hazard function using the proposed estimator x is proved to be efficient the asymptotic bias and variance are derived under regularity conditions computation of the estimator involves an iterative but simple algorithm extensive simulation studies provide evidence supporting the theory the method is illustrated with the stanford heart transplant data set the proposed global approach is also extended to a partially linear proportional hazards model and found to provide efficien

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

similarity preserving linear maps on b h we determine kernels of similarity preserving bounded linear maps on b h and give characterizations for elementary operators of length 1 to be similarity preserving

local partial likelihood estimation in proportional hazards regression fan gijbels and king ann statist 25 1997 1661 1690 considered the estimation of the risk function x in the proportional hazards model their proposed estimator is based on integrating the estimated derivative function obtained through a local version of the partial likelihood they proved the large sample properties of the derivative function but the large sample properties of the estimator for the risk function itself were not established in this paper we consider direct estimation of the relative risk function x 2 x 1 for any location normalization point x 1 the main novelty in our approach is that we select observations in shrinking neighborhoods of both x 1 and x 2 when constructing a local version of the partial likelihood whereas fan gijbels and king ann statist 25 1997 1661 1690 only concentrated on a single neighborhood resulting in the cancellation of the risk function in the local likelihood function the asymptotic properties of our estimator are rigorously established and the variance of the estimator is easily estimated the idea behind our approach is extended to estimate the differences between groups a simulation study is carried out

relative risk trees for censored survival data a method is developed for obtaining tree structured relative risk estimates for censored survival data the first step of a full likelihood estimation procedure is used in a recursive partitioning algorithm that adopts most aspects of the widely used classification and regression tree cart algorithm of breiman et al 1984 classification and regression trees belmont california wadsworth the performance of the technique is investigated through stimulation and compared to the tree structured survival methods proposed by davis and anderson 1989 statistics in medicine 8 947 961 and therneau grambsch and fleming 1990 biometrika 77 147 160

diagnostic plots to reveal functional form for covariates in multiplicative intensity models abstract we show how plots based on the residuals from a proportional hazards model may be used to reveal the correct functional form for covariates in the model a smoothed plot of the martingale residues was suggested for this purpose by therneau grambsch and fleming 1990 biometrika 77 147 160 however its consistency required that the covariates be independent they also noted that the plot could be biased for large covariate effects we introduce two refinements which overcome these difficulties the first is based on a ratio of scatter plot smooths where the numerator is the smooth of the observed count plotted against the covariate and the denominator is a smooth of the expected count this is related to the arjas goodness of fit plot 1988 journal of the american statistical association 83 204 212 the second technique smooths the martingale residuals divided by the expected count using expected count as a weight this latter approach is related to a glm partial residual plot as well as to the iterative methods of hastie and tibshirani 1990 biometrics 46 1005 1016 and gentleman and crowley 1991 biometrics 47 1283 1296 applications to survival data sets are given

functional anova modeling for proportional hazards regression the logarithm of the relative risk function in a proportional hazards model involving one or more possibly time dependent covariates is treated as a specified sum of a constant term main effects and selected interaction terms maximum partial likelihood estimation is used where the maximization is taken over a suitably chosen finite dimensional estimation space whose dimension increases with the sample size and which is constructed from linear spaces of functions of one covariate and their tensor products the l 2 rate of convergence for the estimate and its anova components is obtained an adaptive numerical implementation is discussed whose performance is compared to full likelihood hazard regression both with and without the restriction to proportional hazards

linear maps preserving products of positive or hermitian matrices abstract in this note we consider linear maps on the complex n n matrix algebra m n preserving products of either two positive matrices a positive matrix and a hermitian matrix or two hermitian matrices we give characterizations of all those injections

an improved mathematical model for the stability of cylinder rows subject to cross flow abstract linearized quasi static fluid force coefficient data obtained from wind tunnel tests are used in an analysis of the fluidelastic stability of a double row of flexible circular cylinders subject to a cross flow although the analysis is quasi static frequency dependent terms are obtained in the aerodynamic stiffness and damping matrices the origin of these terms is twofold firstly because of the time lag between flow leaving an upstream row and arriving at a downstream row which becomes important at low values of the non dimensional flow velocity u fd secondly because of retardation of the flow approaching the cylinder which is particularly important when small displacements of the cylinder result in large changes in the fluid force coefficients this analysis is used to investigate the effects of a number of parameters on the critical flow velocity and the theoretical results are compared with those available in the literature in general agreement between theory and experiment is reasonably good indicating the validity of this analysis

similarity preserving linear maps on upper triangular matrix algebras in this note we consider similarity preserving linear maps on the algebra of all n n complex upper triangular matrices tn we give characterizations of similarity invariant subspaces in tn and similarity preserving linear injections on tn furthermore we considered linear injections on tn preserving similarity in mn as well

local likelihood and local partial likelihood in hazard regression in survival analysis the relationship between a survival time and a covariate is conveniently modeled with the proportional hazards regression model this model usually assumes that the covariate has a log linear effect on the hazard function in this paper we consider the proportional hazards regression model with a nonparametric risk effect we discuss estimation of the risk function and its derivatives in two cases when the baseline hazard function is parametrized and when it is not parametrized in the case of a parametric baseline hazard function inference is based on a local version of the likelihood function while in the case of a nonparametric baseline hazard we use a local version of the partial likelihood this results in maximum local likelihood estimators and maximum local partial likelihood estimators respectively we establish the asymptotic normality of the estimators it turns out that both methods have the same asymptotic bias and variance in a common situation even though the local likelihood method uses information about the baseline hazard function

similarity preserving linear maps on matrices abstract we determine all similarity preserving linear maps on the space of n x n complex matrices and all unitary equivalence preserving linear maps on the space of n x n hermitian matrices sub majorization preserving linear maps on hermitian matrices are also determined

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

a non linear investigation of single degree of freedom instability in cylinder arrays subject to cross flow abstract a quasi steady non linear analysis of single degree of freedom motion for a single flexible cylinder in an array of rigid cylinders subject to a fluid cross flow is presented then by using the so called first approximation of krylov and bogoliubov the stability and existence of limit cycle amplitudes are investigated it is shown that in general the stability behaviour predicted by the non linear analysis is in agreement with that obtained from the previously developed linearized analysis the analysis predicts stable limit cycle amplitudes of vibration which increase in magnitude as the velocity is increased also the rate of increase of cylinder vibrational amplitude with velocity is greater for low values of m d 2 than for higher values

spectrum preserving linear maps abstract let x and y be banach spaces we show that a spectrum preserving surjective linear map from b x to b y is either of the form t ata 1 for an isomorphism a of x onto y or the form t bt b 1 for an isomorphism b of x onto y

graphical methods for censored data abstract various methods of graphically portraying censored data are discussed these include smoothing and model checking many existing graphical methods may be expressed as functionals of the empirical distribution function when the data are subject to censoring the kaplan meier estimator of the distribution may replace the empirical distribution in these methods the plots and problems that arise from such a substitution will be discussed

joint estimation of time dependent and non linear effects of continuous covariates on survival in order to yield more flexible models the cox regression model lambda t x lambda 0 t exp betax has been generalized using different non parametric model estimation techniques one generalization is the relaxation of log linearity in x lambda t x lambda 0 t exp r x another is the relaxation of the proportional hazards assumption lambda t x lambda 0 t exp beta t x these generalizations are typically considered independently of each other we propose the product model lambda t x lambda 0 t exp beta t r x which allows for joint estimation of both effects and investigate its properties the functions describing the time dependent beta t and non linear r x effects are modelled simultaneously using regression splines and estimated by maximum partial likelihood likelihood ratio tests are proposed to compare alternative models simulations indicate that both the recovery of the shapes of the two functions and the size of the tests are reasonably accurate provided they are based on the correct model by contrast type i error rates may be highly inflated and the estimates considerably biased if the model is misspecified applications in cancer epidemiology illustrate how the product model may yield new insights about the role of prognostic factors

partially linear hazard regression for multivariate survival data this article studies estimation of partially linear hazard regression models for multivariate survival data a profile pseudo partial likelihood estimation method is proposed under the marginal hazard model framework the estimation on the parameters for the linear part is accomplished by maximization of a pseudo partial likelihood profiled over the nonparametric part this enables us to obtain consistent estimators of the parametric component asymptotic normality is obtained for the estimates of both the linear and nonlinear parts the new technical challenge is that the nonparametric component is indirectly estimated through its integrated derivative function from a local polynomial fit an algorithm of fast implementation of our proposed method is presented consistent standard error estimates using sandwich type ideas are also developed which facilitates inferences for the model it is shown that the nonparametric component can be estimated as well as if the parametric components were known and the

a practical approach to parameter identification for a lightly damped weakly nonlinear system fluidelastic systems are often lightly damped and exhibit weak nonlinear damping the weak linear and nonlinear damping forces have a significant effect on the long term behaviour of the system however parameter identification methods tend to concentrate on identifying the strongest forces in this paper the applicability of two identification methods to a single degree of freedom fluidelastic system is investigated using experimental data the system is weakly nonlinear under flow conditions and is prone to fluidelastic instability i e self excited limit cycle behaviour the crucial effect of the weak damping forces on the system stability and post stable behaviour has been demonstrated the nonlinearity detection and identification has been done using an analytic representation of the displacement signal freevib method and subsequently a nonlinear decrement method was used for comparison the identified models were used to predict the global behaviour of the system in the form of limit cycle amplitudes and this has been used as an indication of accuracy it was found that the nonlinear decrement method yields superior predictions but it suffers from the limitation that the functional form of the system must be known a priori therefore it is concluded that employing both methods together provides a more powerful approach for parameter estimation in a lightly damped system where weak nonlinear damping plays an important role

cross flow induced chaotic vibrations of heat exchanger tubes impacting on loose supports abstract the chaotic dynamics of heat exchanger tubes impacting on the generally loose baffle plates is studied using an analytic model that involves delay differential equations the critical flow velocity for the local instability of the flexible cylinder near the static equilibrium position is obtained by assuming a harmonic solution in the discretized linearized model and solving the resulting algebraic equations numerical solutions show that with increasing flow beyond the critical the amplitude of motion grows until impacting with the loose support occurs more complex motions then arise leading to chaos for a sufficiently high flow velocity a lyapunov exponent technique is developed for delay differential equations and is applied to the present model showing definitely and unequivocally for the first time for this system that chaotic motions do occur to understand the system behaviour better a one degree of freedom forced negatively damped impact oscillator is studied the response is quite complex and results in chaos in certain parameter regions the analysis is performed by finding periodic solutions and determining their stability and bifurcations with the poincare map technique

polynomial spline estimation and inference of proportional hazards regression models with flexible relative risk form the cox proportional hazards model usually assumes an exponential form for the dependence of the hazard function on covariate variables however in practice this assumption may be violated and other relative risk forms may be more appropriate in this article we consider the proportional hazards model with an unknown relative risk form issues in model interpretation are addressed we propose a method to estimate the relative risk form and the regression parameters simultaneously by first approximating the logarithm of the relative risk form by a spline and then employing the maximum partial likelihood estimation an iterative alternating optimization procedure is developed for efficient implementation statistical inference of the regression coefficients and of the relative risk form based on parametric asymptotic theory is discussed the proposed methods are illustrated using simulation and an application to the veteran s administration lung cancer data

asymptotic similarity preserving linear maps on b h it is proved that a bounded linear surjective map on b h which is asymptotic similarity preserving is nonzero scalar multiple of either an automorphism or an anti automorphism

adaptive regression splines in the cox model we develop a method for constructing adaptive regression spline models for the exploration of survival data the method combines cox s 1972 journal of the royal statistical society series b 34 187 200 regression model with a weighted least squares version of the multivariate adaptive regressi on spline mars technique of friedman 1991 annals of statistics 19 1 141 to adaptively select the knots and covariates the new technique can automatically fit models with terms that represent nonlinear effects and interactions among covariates applications based on simulated data and data from a clinical trial for myeloma are presented results from the myeloma application identified several important prognostic variables including a possible nonmonotone relationship with survival in one laboratory variable results are compared to those from the adaptive hazard regression hare method of kooperberg stone and truong 1995 journal of the american statistical association 90 78 94

some general techniques on linear preserver problems abstract several general techniques on linear preserver problems are described the first one is based on a transfer principle in model theoretic algebra that allows one to extend linear preserver results on complex matrices to matrices over other algebraically closed fields of characteristic 0 the second one concerns the use of some simple geometric technique to reduce linear preserver problems to standard types so that known results can be applied the third one is about solving linear preserver problems on more general operator algebras by reducing the problems to idempotent preservers numerous examples will be given to demonstrate the proposed techniques