estimation and control of systems with unknown covariance and multiplicative noise the problem of estimation and control for systems with multiplicative noise and unknown second order statistics is considered conditions are found for the existence of a solution based on game theoretic ideas the conditions for the existence of a saddle point for the time invariant filtering problem are necessary and sufficient whereas for all other cases only necessary the central idea of the solution is to convert the stochastic problem to a deterministic optimal control problem whose minimax point is sought with respect to the control filter and unknown statistics parameters the results that are derived show that the problem of estimation for systems with unknown covariances depends on the costate matrix which in turn is a function of the performance measure thus the filter loses one of its best known properties that of independence of the performance functional this property holds not only for the classical kalman filter but also for multiplicative systems

cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

diagnostic measures for the cox regression model with missing covariates this paper investigates diagnostic measures for assessing the influence of observations and model misspecification in the presence of missing covariate data for the cox regression model our diagnostics include case deletion measures conditional martingale residuals and score residuals the q distance is proposed to examine the effects of deleting individual observations on the estimates of finite dimensional and infinite dimensional parameters conditional martingale residuals are used to construct goodness of fit statistics for testing possible misspecification of the model assumptions a resampling method is developed to approximate the p values of the goodness of fit statistics simulation studies are conducted to evaluate our methods and a real data set is analyzed to illustrate their use

checking the marginal cox model for correlated failure time data summary correlated failure time data arise frequently in scientific investigations because there exists natural or artificial clustering of study subjects such that failure times within the same cluster are correlated it is convenient and useful to perform regression analysis by formulating the marginal distributions of the correlated failure times with the cox proportional hazards model in this paper we develop a class of graphical and numerical techniques for checking the adequacy of the marginal cox model the proposed methods are derived from cumulative sums of martingale based residuals over the failure time and or covariates the distributions of these stochastic processes under the assumed model can be approximated through simulating certain zero mean gaussian processes each observed residual pattern can then be compared objectively with a number of realisations from the approximating process an illustration with the diabetic retinopathy study is provided

a bayesian semiparametric survival model with longitudinal markers we consider inference for data from a clinical trial of treatments for metastatic prostate cancer patients joined the trial with diverse prior treatment histories the resulting heterogeneous patient population gives rise to challenging statistical inference problems when trying to predict time to progression on different treatment arms inference is further complicated by the need to include a longitudinal marker as a covariate to address these challenges we develop a semiparametric model for joint inference of longitudinal data and an event time the proposed approach includes the possibility of cure for some patients the event time distribution is based on a nonparametric polya tree prior for the longitudinal data we assume a mixed effects model incorporating a regression on covariates in a nonparametric event time model in general and for a polya tree model in particular is a challenging problem we exploit the fact that the covariate itself is a random variable we achieve an implementation of the desired regression by factoring the joint model for the event time and the longitudinal outcome into a marginal model for the event time and a regression of the longitudinal outcomes on the event time i e we implicitly model the desired regression by modeling the reverse conditional distribution

properties of hazard based residuals and implications in model diagnostics abstract model diagnostic procedures in failure time models using hazard based residuals rely on the assumption that the residual vector closely resembles a random sample from a unit exponential distribution when the model holds this article formally investigates the validity of this critical assumption by deriving and examining the properties of parametrically semiparametrically and nonparametrically estimated residuals for complete and right censored data the joint distribution of the residual vector is characterized and the behavior of some tests for exponentiality when applied to the residuals is examined analytically and through monte carlo methods findings reveal that the critical assumption of approximate unit exponentiality of the residual vector may not be viable and consequently the model diagnostic procedures considered which revolve on checking the approximate unit exponentiality of the residual vector specifically hazard plotting and the use of spacings and total time on test stati

assessing time by covariate interactions in proportional hazards regression models using cubic spline functions proportional hazards or cox regression is a popular method for modelling the effects of prognostic factors on survival use of cubic spline functions to model time by covariate interactions in cox regression allows investigation of the shape of a possible covariate time dependence without having to specify a specific functional form cubic spline functions allow one to graph such time by covariate interactions to test formally for the proportional hazards assumption and also to test for non linearity of the time by covariate interaction the functions can be fitted with existing software using relatively few parameters the regression coefficients are estimated using standard maximum likelihood methods

polynomial filtering for linear discrete time non gaussian systems in this work we propose a new filtering approach for linear discrete time non gaussian systems that generalizes a previous result concerning quadratic filtering a de santis a germani and m raimondi ieee trans automat control 40 1995 pp 1274 1278 a recursive nu th order polynomial estimate of finite memory delta is achieved by defining a suitable extended state which allows one to solve the filtering problem via the classical kalman linear scheme the resulting estimate will be the mean square optimal one among those estimators that take into account nu polynomials of the last delta observations numerical simulations show the effectiveness of the proposed method

a regression survival model for testing the proportional hazards hypothesis a semi parametric generalization of the proportional hazards regression model is defined whereby the hazard functions can cross for different values of the covariates in the two sample comparison it includes in particular the case of two weibull distributions differing in scale and shape parameters a global test of the proportional hazards assumption is proposed against such defined alternatives its power in the two sample case is compared to that of previously described tests by using simulation experiments survival data of patients with breast carcinoma including several prognostic factors are presented as an illustration

complex geometry and the asymptotics of harish chandra modules for real reductive lie groups i let g be a connected semisimple real matrix group it is now apparent that the representation theory of g is intimately connected with the complex geometry of the flag variety 9 by studying appropriate orbit structures on 9 we are naturally led to representation theory in the category of harish chandra modules x or the representation theory of category 9 the jacquet functor j f 9 has proved a useful tool in converting yfw problems into 9 problems which are often more tractable in this paper we advance the philosophy that the complex geometry of 9 associated to yf and 9 interacts in a natural way with the functor j leading to deep new information on the structure of jacquet modules this in turn gives new insight into the structure of certain nilpotent cohomology groups associated to harish chandra modules our techniques are based upon many of the ideas present in the proof of the kazhdan lusztig conjectures and bernstein s proof of the jantzen conjecture

model checking techniques based on cumulative residuals depuis longtemps on utilise les residus pour apprecier graphiquement et muneriquement l adequation de modeles de regression l analyse conventionnelle des residus fondee sur les graphes des residus bruts ou des courbes lissees est eminemment subjective tandis que la plupart des tests numeriques d ajustement informent peu sur la nature de l inadequation nous developpons des techniques d examen d un modele objectives et informatives qui reposent soit sur les sommes des residus cumulees selon certaines coordonnees comme une covariable on le predicteur lineaire soit sur des regroupements partiels sommes ou moyennes mobiles selon ces coordonnees pour une grande variete de modeles statistiques et de structures des observations dont les modeles lineaires generalises sur observations independantes ou correlees les distributions de ces processus stochastiques sous le modele postule peuvent etre approchees par des processus gaussiens centres faciles a simuler on pent done comparer graphiquement et numeriquement chaque processus observe a une serie de realisations du ment chaque processus observe a une serie de realisations du processus gaussien ces comparaisons permettent d evaluer objectivement dans quelle mesure une tendance observee sur un graphique serait attribuable a une mauvaise specification du modele plutot qu aux fluctuations d echantillonnage ces techniques sont particulierement utiles pour l examen de la fonction de lien ou de la forme fonctionnelle d une covariable plusieurs exemples medicaux illustrent cette approche

minimax linear observers and regulators for stochastic systems with uncertain second order statistics the problem of minimax design of linear observers and regulators for linear time varying multivariable stochastic systems with uncertain models of their second order statistics is treated in this paper general classes of allowable covariance matrices and means of the process and observation noises and of the random initial condition are considered a game formulation of the problem is adopted and it is shown that the optimal filter for the least favorable set of covariances is minimax robust for each of the filtering situations analyzed conditions satisfied by the saddle point solutions are given and their utility for finding the worst case covariances is illustrated by way of several examples of uncertainty classes of practical interest

on minimax robustness a general approach and applications the minimax approach to the design of systems that are robust with respect to modeling uncertainties is studied using a game theoretic formulation in which the performance functional and the sets of modeling uncertainties and admissible design policies are arbitrary the existence and characterization of minimax robust solutions that form saddle points are discussed through various methods that take into account several common features of the games encountered in applications in particular it is shown that if the performance functional and the uncertainty set are convex then a certain type of regularity condition on the functional is sufficient to ensure that the optimal strategy for a least favorable element of the uncertainty set is minimax robust the efficacy of the methods proposed for a general game is tested in the problems of matched filtering wiener filtering quadratic detection and output energy filtering in which uncertainties in their respective signal and noise models are assumed to exist these problems are analyzed in a common hilbert space framework and they serve to point out the advantages and limitations of the proposed techniques

goodness of fit test for proportional subdistribution hazards model this paper concerns using modified weighted schoenfeld residuals to test the proportionality of subdistribution hazards for the fine gray model similar to the tests proposed by grambsch and therneau for independently censored data we develop a score test for the time varying coefficients based on the modified schoenfeld residuals derived assuming a certain form of non proportionality the methods perform well in simulations and a real data analysis of breast cancer data where the treatment effect exhibits non proportional hazards copyright 2013 john wiley sons ltd

approximate case influence for the proportional hazards regression model with censored data a method is presented for approximating the influence of individual cases upon regression coefficient estimates obtained from the cox proportional hazards model observations can thus be identified which may greatly influence statistical inferences regarding the effects of prognostic factors upon survival time an example from a cancer clinical trial is given

minimax quadratic filtering of uncertain linear stochastic systems with partial fourth order information a minimax filtering problem for a class of uncertain linear stochastic systems is studied uncertainties involving fourth order moments of the noise distribution and of the initial state are considered under the hypothesis that the second order statistics are known hence the linear filter is available the quadratic minimax filter is found moreover it is shown that the minimax filter gives a worst case error variance even less than the exact error variance of the linear filter numerical simulations show that this improvement is meaningful also in cases of great uncertainty regarding the higher order statistics

information and asymptotic efficiency in parametric nonparametric models asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems for regular estimates and asymptotic minimax bounds the methods used involve i the notion of a hellinger differentiable root density where part of the differentiation is with respect to the nonparametric part of the model to obtain appropriate scores and ii calculation of the effective score for the real or vector finite dimensional parameter of interest as that component of the score function orthogonal to all nuisance parameter scores perhaps infinite dimensional the resulting asymptotic information for estimation of the parametric component of the model is just 4 times the squared l 2 norm of the effective score a corollary of these results is a simple necessary condition for adaptive estimation adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model examples considered include the one sample location model with and without symmetry mixture models the two sample shift model and cox s proportional hazards model

local central limit theorems the high order correlations of rejective sampling and logistic likelihood asymptotics let i 1 i n be independent but not necessarily identically distributed bernoulli random variables and let x n n j 1 i j for v in a bounded region a local central limit theorem expansion of p x n ex n v is developed to any given degree by conditioning this expansion provides information on the high order correlation structure of dependent weighted sampling schemes of a population e a special case of which is simple random sampling where a set d e is sampled with probability proportional to a d x a where x a are positive weights associated with individuals a e these results are used to determine the asymptotic information and demonstrate the consistency and asymptotic normality of the conditional and unconditional logistic likelihood estimator for unmatched case control study designs in which sets of controls of the same size are sampled with equal probability

optimal filtering in stochastic discrete time systems with unknown inputs in this note we derive a recursive filtering algorithm for the linear discrete time dynamic system with indeterminate stochastic inputs the algorithm is based on the minimax optimal method of parameter estimation in the linear regression model with parameters of two different types unknown and stochastic with partially known characteristics

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

checking the cox model with cumulative sums of martingale based residuals summary this paper presents a new class of graphical and numerical methods for checking the adequacy of the cox regression model the procedures are derived from cumulative sums of martingale based residuals over follow up time and or covariate values the distributions of these stochastic processes under the assumed model can be approximated by zero mean gaussian processes each observed process can then be compared both visually and analytically with a number of simulated realizations from the approximate null distribution these comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are special attention is given to checking the functional form of a covariate the form of the link function and the validity of the proportional hazards assumption an omnibus test consistent against any model misspecification is also studied the proposed techniques are illustrated with two real data sets

on residual plots for relative risk regression summary martingale based residuals proposed for proportional hazards regression by barlow prentice 1988 are considered plots of such residuals against time are shown to exhibit systematic patterns which are not a priori predictable even when the model is appropriate for the plots to be fully useful therefore account needs to be taken of the expected pattern under the model a simple amendment is suggested

influence functions for proportional hazards regression summary influence functions for the regression parameters in the proportional hazards model are presented it is suggested that empirical influence functions computed for each observation and each covariate can be useful in an informal way to identify influential observations this is illustrated on the stanford heart transplant data and two other examples

a diagnostic for cox regression and general conditional likelihoods abstract diagnostics for changes in the maximum likelihood estimate of due to deletion of single observations in linear and logistic regression models are equivalent to diagnostics obtained from fitting augmented regression models for the cox proportional hazards model and in conditional models for arbitrarily matched or stratified data diagnostics derived from augmented regression models are again useful and easy to compute these diagnostics are applicable to different formulations for the multiplicative covariable effects over time or over strata an exact or approximate treatment of tied failure times or multiple case matching and time dependent covariates

semiparametric inference in the proportional odds regression model abstract for fitting the proportional odds regression model with right censored survival times we introduce some weighted empirical odds functions these functions are solutions of some self consistency equations and have a nice martingale representation from these functions several classes of new regression estimators such as the pseudo maximum likelihood estimator martingale residual based estimators and minimum distance estimators are derived these estimators have desirable properties such as easy computation asymptotic normality via a martingale analysis and reliable asymptotic covariance estimation in closed form extensive numerical studies show that the minimum l 2 distance estimators have very good finite sample behaviors compared to existing methods results of some simulation studies and applications to a real dataset are given the weighted odds function based approach also provides inference on the baseline odds function and some measures for lack of fit analysis

a simple estimator for a shared frailty regression model summary we propose a simple estimation procedure for a proportional hazards frailty regression model for clustered survival data in which the dependence is generated by a positive stable distribution inferences for the frailty parameter can be obtained by using output from cox regression analyses the computational burden is substantially less than that of the other approaches to estimation the large sample behaviour of the estimator is studied and simulations show that the approximations are appropriate for use with realistic sample sizes the methods are motivated by studies of familial associations in the natural history of diseases their practical utility is illustrated with sib pair data from beaver dam wisconsin

regression splines in the cox model with application to covariate effects in liver disease abstract the cox proportional hazards model restricts the log hazard ratio to be linear in the covariates a smooth nonlinear covariate effect may go undetected in this model but can be well approximated by a spline function a survival model based on data from a clinical trial of primary biliary cirrhosis is developed using regression splines and the resulting log hazard ratio estimates are compared with those from nonparametric methods we remove the linear restriction on the log hazard ratio by transforming a continuous covariate into a vector of fixed knot basis splines b splines b splines are known to produce better conditioned systems of equations than the truncated power basis when used as interpolants and show similar behavior when fitting proportional hazards models we describe the procedures for and the issues arising in the estimation and the testing of the b spline coefficients although inference is not well developed for some nonparametric methods that estimate covariate effects the

degenerate whittaker functionals for real reductive groups in this paper we establish a connection between the associated variety of a representation and the existence of certain it degenerate whittaker functionals for both smooth and k finite vectors for all quasi split real reductive groups thereby generalizing results of kostant matumoto and others

smooth goodness of fit tests for composite hypothesis in hazard based models consider a counting process n t t j with compensator process a t t j where a t 0 t y s 0 s ds y t t j is an observable predictable process and 0 is an unknown hazard rate function a general procedure for extending neyman s smooth goodness of fit test for the composite null hypothesis h 0 0 c 0 r q is proposed and developed the extension is obtained by embedding c in the class a k whose members are of the form 0 exp t x r k where 1 k t is a vector of observable random processes satisfying certain regularity conditions the tests are based on quadratic forms of the statistic 0 t s dm s where m t n t 0 t y s 0 s ds and is a restricted maximum likelihood estimator of asymptotic properties of the test statistics are obtained under a sequence of local alternatives and the asymptotic local powers of the tests are examined the effect of estimating by is ascertained and the problem of choosing the process is discussed the procedure is illustrated by developing tests for testing that 0 belongs to i the class of constant hazard rates and ii the class of weibull hazard rates with particular emphasis on the random censorship model simulation results concerning the achieved levels and powers of the tests are presented and the procedures are applied to three data sets that have been considered in the literature

asymptotic distribution theory for cox type regression models with general relative risk form the theory and application of the cox 1972 failure time regression model has almost without exception assumed an exponential form for the dependence of the hazard function on regression variables other regression forms may be more natural or descriptive in some applications for example a linear relative risk regression model provides a convenient framework for studying epidemiologic risk factor interactions this note uses the counting process formulation of andersen and gill 1982 to develop asymptotic distribution theory for a class of intensity function regression models in which the usual exponential regression form is relaxed to an arbitrary non negative twice differentiable form some stability and regularity conditions beyond those of andersen and gill are required to show the consistency of the observed information matrix which in general need not be positive semidefinite

nonparametric estimation for a scale change with censored observations abstract nonparametric point and interval estimators of the ratio of two scale parameters are given for arbitrarily right censored data based on the idea of hodges and lehmann 1963 these estimators are defined in terms of rank test statistics for testing the equality of two survival distributions the asymptotic properties and efficiencies of estimators corresponding to the tests studied by gill 1980 are investigated

survival analysis with quantile regression models quantile regression offers great flexibility in assessing covariate effects on event times thereby attracting considerable interests in its applications in survival analysis but currently available methods often require stringent assumptions or complex algorithms in this article we develop a new quantile regression approach for survival data subject to conditionally independent censoring the proposed martingale based estimating equations naturally lead to a simple algorithm that involves minimizations only of l1 type convex functions we establish uniform consistency and weak convergence of the resultant estimators we develop inferences accordingly including hypothesis testing second stage inference and model diagnostics we evaluate the finite sample performance of the proposed methods through extensive simulation studies an analysis of a recent dialysis study illustrates the practical utility of our proposals

robust variance estimation for rate ratio parameter estimates from individually matched case control data the asymptotic variance and robust variance estimators of rate ratios estimated using conditional logistic regression from individually matched case control data are derived when the presumed proportional hazards model ismisspecified the robust variance estimators are easily computed using schoenfeld residuals generated from standard partial likelihood estimation software for failure time data simulation studies indicate that the robust variance estimators perform well for typical sizes and that the rare disease version should be adequate for all practical purposes it was also found that model misspecification must be quite extreme before the model based i e inverse information variance is significantly biased and that the robust variance estimators are somewhat more variable than the model based we conclude that the model based variance estimator can be used when model misspecification is not severe the robust estimator should be used when the presumed model clearly fits the data poorly

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

cure rate quantile regression for censored data with a survival fraction censored quantile regression offers a valuable complement to the traditional cox proportional hazards model for survival analysis survival times tend to be right skewed particularly when there exists a substantial fraction of long term survivors who are either cured or immune to the event of interest for survival data with a cure possibility we propose cure rate quantile regression under the common censoring scheme that survival times and censoring times are conditionally independent given the covariates in a mixture formulation we apply censored quantile regression to model the survival times of susceptible subjects and logistic regression to model the indicators of whether patients are susceptible we develop two estimation methods using martingale based equations one approach fully uses all regression quantiles by iterating estimation between the cure rate and quantile regression parameters and the other separates the two via a nonparametric kernel smoothing estimator we establish the uniform

understanding cox s regression model a martingale approach abstract an informal discussion is given of how martingale techniques can be used to extend cox s regression model and to derive its large sample properties

on prequential model assessment in life history analysis summary we extend the prequential approach introduced by a p dawid to continuous time marked point processes a prequential forecasting system is expressed by means of the stochastic intensities of the point process the exact independence and 1 exponentiality of the derived compensators are used to assess the predictive performance of the model and the inferential procedure

minimax control of linear stochastic systems with noise uncertainty the problem of linear quadratic gaussian control of multivariable linear stochastic systems with uncertain second order statistical properties is considered uncertainty is modeled by allowing process and observation noise spectral density matrices to vary arbitrarily within given classes and a minimax control formulation is applied to the quadratic objective functional general theorems proving the existence and characterization of saddle point solutions to this problem are presented and the relationship of these results to earlier results on minimax state estimation are discussed to illustrate the analytical results the specific example of regulating a double integrator plant is treated in detail

using cumulative sums of martingale residuals for model checking in nested case control studies summary standard use of cox regression requires collection of covariate information for all individuals in a cohort even when only a small fraction of them experiences the event of interest fail this may be very expensive for large cohorts further in biomarker studies it will imply a waste of valuable biological material that one may want to save for future studies a nested case control study offers a useful alternative for this design covariate information is only needed for the failing individuals cases and a sample of controls selected from the cases at risk sets methods based on martingale residuals are useful for checking the fit of cox s regression model for cohort data but similar methods have so far not been developed for nested case control data in this article it is described how one may define martingale residuals for nested case control data and it is shown how plots and tests based on cumulative sums of martingale residuals may be used to check model fit the plots and tests may be obtained using available software

smooth goodness of fit tests for the baseline hazard in cox s proportional hazards model abstract for the cox proportional hazards model the goodness of fit problem of testing whether the baseline hazard rate function equals some specified hazard rate function 0 in the presence of incomplete data is considered extending neyman s idea the goodness of fit tests are score tests obtained by embedding in a larger family of hazard rate functions developed through smooth and possibly random transformations of 0 but whereas their smooth alternatives were formulated through the density function my formulation is through the hazard rate function which is more natural in failure time models where model specification is now done through hazard rates furthermore my smooth alternatives are allowed to be data driven or dynamic through this machinery classes of omnibus as well as directional goodness of fit tests are obtained moreover this machinery provides a theoretically sound method for combining varied directional goodness of fit tests into omnibus tests some of the resul

the asymptotic joint distribution of the efficient scores test for the proportional hazards model calculated over time the log rank test derived by mantel 1966 and peto 1972 is a special case of the efficient scores test for the proportional hazards model it is used frequently in the analysis of survival duration in clinical trials where patients entering the study are assigned at random to different treatments and are followed until they either fail or the study is terminated most survival studies are designed so that the log rank test is calculated at the termination of the study however it is not unusual for the investigator to monitor the experiment periodically it would be very useful to know the joint distribution of the test statistic at different points in time in order to adjust for the effect of repeated significance testing although the joint distribution has not been derived previously sequential procedures for the analysis of survival data in clinical trials have been given by several authors including armitage 1975 pp 140 6 jones whitehead 1979 and nagelkerke hart 1980 in this paper we establish the asymptotic joint distribution of the more general efficient scores test for the proportional hazards model calculated at different points in time and indicate how this can be used for repeated significance testing

robust filtering and prediction for linear systems with uncertain dynamics a game theoretic approach we examine the existence and behavior of game theoretic solutions for robust linear filters and predictors our basic uncertainty class includes m th order time varying discrete time systems with uncertain dynamics uncertain initial state covariance and uncertain nonstationary input and observation noise covariance our results include recursive kalman filter predictor realizations for the resulting robust procedures our approach is based on saddle point theory we emphasize the notion of a least favorable prior distribution for the uncertain parameter values to obtain a worst case design technique in this paper we highlight the role such distributions with finite support play in these decision models in particular we demonstrate that in these decision models the least favorable prior distribution is always discrete

whittaker functions for pj principal series representations of sp 3 r abstract in this paper we give explicit formulas for the secondary and the primary whittaker functions for p j principal series representations of sp 3 r

residuals for proportional hazards models with interval censored survival data we develop diagnostic tools for use with proportional hazards models for interval censored survival data we propose counterparts t o the cox snell lagakos or martingale deviance and schoenfeld residuals many of the properties of these residuals carry over to the interval censored case in particular the interval censored versions of the lagakos and schoenfeld residuals may be derived as components of suitable score statistics the lagakos residuals may be used to check regression relationships while the schoenfeld residuals can help to detect nonproportional hazards in semiparametric models the methods apply to parametric models and to the semiparametric model with discrete observation times

sequential analysis of the proportional hazards model abstract for the proportional hazards model of survival analysis an appropriate large sample theory is developed for cases of staggered entry and sequential analysis the principal techniques involve an approximation of the score process by a suitable martingale and a random rescaling of time based on the observed fisher information as a result we show that the maximum partial likelihood estimator behaves asymptotically like brownian motion author

generalized martingale residual processes for goodness of fit inference in cox s type regression models in the paper a general class of stochastic processes based on the sums of weighted martingale transform residuals for goodness of fit inference in general cox s type regression models is studied their form makes the inference robust to covariate outliers a weak convergence result for such processes is obtained giving the possibility of establishing the randomness of their graphs together with the construction of the formal x 2 type goodness of fit tests by using the khmaladze innovation approach a modified version of the initial class of processes is also defined weak convergence results for the processes are derived this leads to the main application which concerns the formal construction of the kolmogorov smirnov and cramer von mises type goodness of fit tests this is done within the general situation considered

residuals for relative risk regression summary several possible definitions of residuals are given for relative risk regression with time varying covariates each such residual has a representation as an estimator of a stochastic integral with respect to the martingale arising from a subject s failure time counting process previously proposed residuals for individual study subjects and for specific time points are shown to be special cases of this definition as are previously derived regression diagnostics an illustration and various generalizations are also given able methods are required to detect various departures from modelling assumptions suitably defined residuals may play an important role in such identification however the nonparametric aspect of the model the possibility that modelled regression variables may be varying with follow up time and most importantly the usual presence of right censorship implies that specialized residual definitions are required the class of residuals considered here is most easily formulated using counting process notation for the failure time data let ni t yi t and zi t represent respectively for the ith subject the values of counting censoring and covariate processes at follow up time t i 1 n while ni u yi u zi u 0c u t specifies the corresponding histories for the ith subject prior to time t thus in a typical univariate failure time application ni with right continuous sample paths will take value zero prior to the time of failure on the ith subject and value one thereafter while yi with left continuous sample paths will take value one at times at which the ith subject is at risk for an observed failure and value zero otherwise the counting process ni can be uniquely decomposed so that for all t i

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

bootstrap based methods for estimating standard errors in cox s regression analyses of clustered event times we propose two bootstrap based methods to correct the standard errors ses from cox s model for within cluster correlation of right censored event times the cluster bootstrap method resamples with replacement only the clusters whereas the two step bootstrap method resamples i the clusters and ii individuals within each selected cluster with replacement in simulations we evaluate both methods and compare them with the existing robust variance estimator and the shared gamma frailty model which are available in statistical software packages we simulate clustered event time data with latent cluster level random effects which are ignored in the conventional cox s model for cluster level covariates both proposed bootstrap methods yield accurate ses and type i error rates and acceptable coverage rates regardless of the true random effects distribution and avoid serious variance under estimation by conventional cox based standard errors however the two step bootstrap method over estimates the variance for individual level covariates we also apply the proposed bootstrap methods to obtain confidence bands around flexible estimates of time dependent effects in a real life analysis of cluster event times

global measures of local influence for proportional hazards regression models cox proportional hazards regression models have been extended to multiple failure times and other situations by using a robust covariance matrix instead of the usual inverse of the information matrix we modify an existing measure of individual influence by substituting the robust covariance matrix for the inverse of the information matrix this provides a scalar measure of influence with known mean and bounded range on the interval 0 1 the measure is applicable to marginal multiple failure time models two examples are presented

noncausal minimax linear state estimation for systems with uncertain second order statistics this note considers the problem of minimax state estimation of the states of a linear time invariant system which is driven by and observed in the presence of noise processes with uncertain second order statistics when the process noise and observations are scalars the problem is shown to be equivalent to a scalar minimax estimation problem the existence of a minimax solution is thereby established and the minimax filter is shown to be a linear transformation of the minimax filter for the scalar problem

use of the box cox transformation with binary response models abstract the power transformation is applied to the odds ratio to generalize the logistic model and to parameterize a certain type of lack of fit transformation of the design variable within the context of the dose response problem is also considered

martingale based residuals for survival models summary graphical methods based on the analysis of residuals are considered for the setting of the highly used cox 1972 regression model and for the andersen gill 1982 generalization of that model we start with a class of martingale based residuals as proposed by barlow prentice 1988 these residuals and or their transforms are useful for investigating the functional form of a covariate the proportional hazards assumption the leverage of each subject upon the estimates of 13 and the lack of model fit to a given subject 1 1 model consider a set of n independent subjects such that the counting process ni ni t t o for the ith subject in the set indicates the number of observed events experienced over time t the sample paths of the ni are step functions with jumps of size 1 and with ni 0 0 we assume that the intensity function for ni t is given by yi t da t zi t yi t ep z t dao t 1 where yi t is a 0 1 process indicating whether the ith subject is a risk at time t 13 is a vector of regression coefficients zi t is a p dimensional vector of covariate processes and ao is the baseline cumulative hazard function several familar survival models fit into this framework the andersen gill 1982 generalization of the cox 1972 model arises when ao t is completely unspecified the further restriction that yi t 1 until the first event or censoring and is 0 thereafter yields the cox model the parametric form ao t t yields a poisson model or an exponential if restricted to a single event per subject and ao t tp a weibull model our attention will focus primarily on the andersen gill and cox models however the

a linear regression model for the analysis of life times a linear model is suggested for the influence of covariates on the intensity function this approach is less vulnerable than the cox model to problems of inconsistency when covariates are deleted or the precision of covariate measurements is changed a method of non parametric estimation of regression functions is presented this results in plots that may give information on the change over time in the influence of covariates a test method and two goodness of fit plots are also given the approach is illustrated by simulation as well as by data from a clinical trial of treatment of carcinoma of the oropharynx

a note on regression analysis of ordinal data with variability of classification summary the effect of observation variability in the classification of an ordinal response is examined within the framework of the proportional hazards regression model for ordinal data a model is proposed which incorporates this type of variability and the behaviour of the proportional hazards model when this new model is appropriate is examined four data sets are used to illustrate the practical application of the model

robust variance estimation for the case cohort design large cohort studies of rare outcomes require extensive data collection often for many relatively uninformative subjects sampling schemes have been proposed that oversample certain groups for example the case cohort design of prentice 1986 biometrika 73 1 11 provides an efficient method of analysis of failure time data however the variance estimate must explicitly correct for correlated score contributions a simple robust variance estimator is proposed that allows for more complicated sampling mechanisms the variance estimate uses a jackknife estimate of the variance of the individual influence function and is shown to be equivalent to a robust variance estimator proposed by lin and wei 1989 journal of the american statistical association 84 1074 1078 for the standard cox model simulation results indicate excellent agreement with corrected asymptotic estimates and appropriate test size the technique is illustrated with data evaluating the efficacy of mammography screening in reducing breast cancer mortality

polynomial filtering of discrete time stochastic linear systems with multiplicative state noise the problem of finding an optimal polynomial state estimate for the class of stochastic linear models with a multiplicative state noise term is studied for such models a technique of state augmentation is used leading to the definition of a general polynomial filter the theory is developed for time varying systems with nonstationary and non gaussian noises moreover the steady state polynomial filter for stationary systems is also studied numerical simulations show the high performances of the proposed method with respect to the classical linear filtering techniques

a partly parametric additive risk model summary aalen s additive risk model allows the influence of each covariate to vary separately over time although allowing greater flexibility of temporal structure than a cox model aalen s model is more limited in the number of covariates it can handle we introduce a partly parametric version of aalen s model in which the influence of only a few covariates varies nonparametrically over time and that of the remaining covariates is constant efficient procedures for fitting this new model are developed and studied the approach is applied to data from the medical research council s myelomatosis trials it is the first step of a taylor series expansion of a general hazard function about the zero of the covariate vector however in estimating the unknown functions in such a general model there is a variance bias trade off that may be critical in small and medium samples also after fitting the model one does not have parameters or formulae that are easily reported we propose a model that takes the additive structure of aalen s model and imposes parametric constraints to obtain a semiparametric submodel which may be more appropriate in some applications the model will be illustrated with data from clinical trials on myelomatosis covariates include treatment sex and four age strata which will be treated parametrically together with serum levels of haemoglobin and f32 microglobulin whose effects will be investigated nonparametrically the additive form can be interpreted loosely in terms of unobserved competing risks since the hazard function for the minimum of independent random vari ables is the sum of the hazard functions for the individual variables microglobulin levels are related to kidney function and tumour mass whereas haemoglobin is unaffected by kidney function hence one might anticipate that the hazard function associated with each covariate represents a different cause of death

on the regression analysis of multivariate failure time data summary the paper is concerned with the analysis of regression effects when individual study subjects may experience multiple failures the proposed methods are most likely to be useful when there are a fairly large number of study subjects two general classes of regression models are considered in order to relate the hazard or intensity function to covariates and to preceding failure time history both models are of a stratified proportional hazards type one model includes baseline intensity functions that are arbitrary as a function of time from the beginning of study while the other includes baseline intensity functions that are arbitrary as a function of time from the study subject s immediately preceding failure partial likelihood functions are derived for the regression coefficients generalizations and illustrations are given

sample size power calculation for stratified case cohort design the case cohort cc study design usually has been used for risk factor assessment in epidemiologic studies or disease prevention trials for rare diseases the sample size power calculation for a stratified cc scc design has not been addressed before this article derives such result based on a stratified test statistic simulation studies show that the proposed test for the scc design utilizing small sub cohort sampling fractions is valid and efficient for situations where the disease rate is low furthermore optimization of sampling in the scc design is discussed and compared with proportional and balanced sampling techniques an epidemiological study is provided to illustrate the sample size calculation under the scc design

generalized gelfand graev representations of exceptional simple algebraic groups over a finite field i introduction let ffi be an adjoint simple algebraic group of type e n 6 7 8 f 4 or g2 over an algebraically closed field k of characteristic p 0 we assume that 5 has a fixed fq rational structure for a finite subfield fq of k the purpose of this paper is to determine explicitly the values of the irreducible characters of g ffi fq at the unipotent elements when p is good 38 i 4 3 in this part i we shall do this only for large p and the full result will be proved in the forthcoming part ii after submitting the first version of the present paper the author was informed by lusztig that he solved analogous problems for classical groups using his theory of character sheaves and that he can also recover the result of this paper mentioned above our method depends upon 1 a parametrization of the irreducible characters of g and the determination of their multiplicities in deligne lusztig virtual representations lusztig 25 2 the computation of the green functions of g in large characteristic springer 36 7 1 shoji 33 beynon and spaltenstein 3 4 3 the determination of the values of generalized gelfand graev characters 20 of g in good characteristic see section 3 by 1 and 2 the uniform parts of the irreducible characters are already known explicitly in large characteristic this implies that in large characteristic our problem for groups of type e 6 o r e 7 has already been solved it should be mentioned here that the whole character table of the g2 group is also known by chang and ree 8 hence in the g2 case this paper gives just another approach to a part of the result of chang and ree if g is of type g2 f4 or e 8 the restriction of an irreducible character r of g to the set of unipotent elements can be written as a sum of its uniform part with a function of the form

on fitting cox s regression model with time dependent coefficients summary this paper examines goodness of fit testing in the cox regression model with timevarying regression coefficients arjas 1988 approach is used to define test statistics of the kolmogorov smirnov and cramer von mises types their asymptotic limits are shown to be well known functionals of standard brownian motion leading to the construction of formal goodness of fit tests some numerical studies are also included to illustrate the performance of the tests for moderate sample sizes

martingale difference residuals as a diagnostic tool for the cox model the proportional hazards model makes two major assumptions the hazard ratio is constant over time and the relationship between the hazard and continuous covariates is log linear methods exist for checking and relaxing each of these assumptions but in both cases the methods rely on the other assumption being true problems can occur if neither of the assumptions is appropriate or even if only one of the assumptions is appropriate but it is not known which we propose a new kind of residual for checking the two assumptions simultaneously the smoothed residuals provide a flexible estimate of the hazard ratio which may deviate from the standard proportional hazards model by having a time dependent hazard ratio transformed covariates or both the methods are illustrated using data from the medical research council s myeloma trials

weight filtrations for induced representations of real reductive lie groups soit g un groupe matriciel reel semi simple connexe et p man la decomposition de langlands d un sous groupe parabolique de g etant donnee une representation admissible de m et un caractere e de a on peut former la representation induite normalisee ip e de g on decrit une g filtration de ip e avec des sous quotients semi simples en termes d une m filtration connue a priori de a sous quotients semi simples dans le cas de caractere infinitesimal entier

specification diagnostics for duration models a martingale approach abstract this paper develops specification diagnostics for discrete time duration models which are based on the martingale theory of stochastic processes the methods can be applied to parametric grouped data duration models with time varying covariates right censoring and which account for unobserved heterogeneity a residual process is defined such that under the correct model specification the process is approximately a martingale difference sequence chi square tests based on these residuals are derived results from monte carlo simulations are presented

tests for variation over groups in survival data abstract when survival data can be divided into natural groups such as in a clinical trial with patients from a number of different institutions the question arises whether effects are different for the different groups for situations where one is concerned with a random pattern of variation over groups this article proposes testing the null hypothesis of no variation using statistics based on the sums of martingale residuals over subjects within groups these tests are closely related to the locally most powerful tests of liang for testing for variation across a large number of strata the tests are applied to data from a lung cancer trial conducted by the eastern cooperative oncology group

cox regression analysis of multivariate failure time data the marginal approach multivariate failure time data are commonly encountered in scientific investigations because each study subject may experience multiple events or because there exists clustering of subjects such that failure times within the same cluster are correlated in this paper i present a general methodology for analysing such data which is analogous to that of liang and zeger for longitudinal data analysis this approach formulates the marginal distributions of multivariate failure times with the familiar cox proportional hazards models while leaving the nature of dependence among related failure times completely unspecified the baseline hazard functions for the marginal models may be identical or different simple estimating equations for the regression parameters are developed which yield consistent and asymptotically normal estimators and robust variance covarinace estimators are constructed to account for the intra class correlation simulation results demonstrate that the large sample approximations are adequate for practical use and that ignoring the intra class correlation could yield rather misleading variance estimators the proposed methodology has been fully implemented in a simple computer program which also incorporates several alternative approaches detailed illustrations with data from four clinical or epidemiologic studies are provided