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

extensions of the i mmse relationship to gaussian channels with feedback and memory unveiling a fundamental link between information theory and estimation theory the i mmse relationship by guo et al together with its numerous extensions has great theoretical significance and various practical applications on the other hand its influences to date have been restricted to channels without feedback or memory due to the absence of its extensions to such channels in this paper we propose the extensions of the i mmse relationship to discrete time and continuous time gaussian channels with feedback and or memory our approach is based on a very simple observation which can be applied to other scenarios such as a simple and direct proof of the classical de bruijn s identity

strong convergence for an iterative method for the triple hierarchical constrained optimization problem abstract the variational inequality problem for a monotone operator over the fixed point set of a nonexpansive mapping is connected with many signal processing problems and such problems have hierarchical structure for example the convex optimization problem over the solution set of the variational inequality problem over the fixed point set has triple hierarchical structure in this paper we present an iterative algorithm for this problem the strong convergence for the proposed algorithm to the solution is guaranteed under some assumptions

three step iterative algorithm for a system of set valued variational inclusions with h η monotone operators abstract in this paper we introduce and study a new system of set valued variational inclusions with h monotone operators we prove the convergence of a new three step iterative algorithm for this system of set valued variational inclusions the results in this paper extend and improve some known results from the literature

on a class of nonconvex equilibrium problems in this paper we introduce a new class of equilibrium problems known as nonconvex equilibrium problems we suggest some algorithms for solving nonconvex equilibrium problems by using the auxiliary principle technique the convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomononicity we also introduce the concept of well posedness for the nonconvex equilibrium problems as special cases we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems our results can be considered a significant improvement of the previously known results

on a predictor corrector method for solving invex equilibrium problems abstract in this paper we use the auxiliary principle technique in conjunction with the bregman function to suggest and analyze a predictor corrector method for solving invex equilibrium problems we also study the convergence criteria of this new method under some mild conditions we obtain as special cases new and known methods for solving variational like inequalities and equilibrium problems

invex equilibrium problems in this paper we introduce a new class of equilibrium problems known as invex equilibrium problems in the setting of invexity this class of equilibrium problems includes equilibrium problems variational inequalities and variational like inequalities as special cases we use the auxiliary principle technique to suggest and analyze some iterative schemes for solving invex equilibrium problems and study the convergence criteria of these methods under some mild conditions we also consider the concept of well posedness for invex equilibrium problems our results represent significant and important refinements of the previously known results

a projection algorithm for general variational inequalities with perturbed constraint sets we consider the problem of general variational inequalities gvi with nonmonotone operator in a finite dimensional space we propose a method to solve gvi that at each iteration considers only one projection on an easy approximation of the constraint set which is important from a practical point of view we analyse the convergence of the algorithm under a weak cocoercivity condition using variational metric analysis computational experience is reported and comparative analysis with other two algorithms is also given for the monotone case

adjusted score tests of homogeneity for poisson processes abstract tests of homogeneity are being increasingly used for the analysis of event time data but relatively little attention has been paid to their distributional properties in settings with small to moderate sample sizes here we consider tests of homogeneity for recurrent event data in which the null model is a poisson process and the alternative is a mixed poisson process we examine score and adjusted score statistics in the context of parametric and semiparametric regression models where the adjustments correct for the bias induced by substituting the maximum likelihood estimates of the parameters into the test statistic we report the results of simulation studies suggesting that the adjusted score statistics are superior in terms of size and power we also find that the adjusted score test in the semiparametric regression model does not perform particularly well in small samples but the adjusted score test based on a piecewise exponential model has satisfactory performance the latter thus prov

variance component testing in generalised linear models with random effects summary there is considerable interest in testing for overdispersion correlation and heterogeneity across groups in biomedical studies in this paper we cast the problem in the framework of generalised linear models with random effects we propose a global score test for the null hypothesis that all the variance components are zero this test is a locally asymptotically most stringent test and is robust in the special sense that the test does not require specifying the joint distribution of the random effects we also propose individual score tests and their approximations for testing the variance components separately both tests can be easily implemented using existing statistical software we illustrate these tests with an application to the study of heterogeneity of mating success across males and females in an experiment on salamander matings and evaluate their performance through simulation

inertial extragradient algorithms for strongly pseudomonotone variational inequalities abstract the purpose of this paper is to study and analyze two different kinds of inertial type iterative methods for solving variational inequality problems involving strongly pseudomonotone and lipschitz continuous operators in hilbert spaces the projection method is used to design the algorithms which can be computed more easily the construction of solution approximations and the proof of convergence of the algorithms are performed without prior knowledge of the modulus of strong pseudomonotonicity and the lipschitz constant of cost operator instead of that the algorithms use variable stepsize sequences which are diminishing and non summable the numerical behaviors of the proposed algorithms on a test problem are illustrated and compared with several previously known algorithms

the forward backward forward method from continuous and discrete perspective for pseudo monotone variational inequalities in hilbert spaces abstract tseng s forward backward forward algorithm is a valuable alternative for korpelevich s extragradient method when solving variational inequalities over a convex and closed set governed by monotone and lipschitz continuous operators as it requires in every step only one projection operation however it is well known that korpelevich s method converges and can therefore be used also for solving variational inequalities governed by pseudo monotone and lipschitz continuous operators in this paper we first associate to a pseudo monotone variational inequality a forward backward forward dynamical system and carry out an asymptotic analysis for the generated trajectories the explicit time discretization of this system results into tseng s forward backward forward algorithm with relaxation parameters which we prove to converge also when it is applied to pseudo monotone variational inequalities in addition we show that linear convergence is guaranteed under strong pseudo monotonicity numerical experiments are carried out for pseudo monotone variational inequalities over polyhedral sets and fractional programming problems

generalized mixed quasi variational like inequalities in this paper we use the auxiliary principle technique to suggest and analyze some iterative methods for solving generalized mixed quasi variational like inequalities we show that the convergence of the proposed method requires either pseudomonotonicity or partially relaxed strongly monotonicity under some mild condition as special cases one can obtain a number of new and known algorithms for solving variational inequalities and complementarity problems our results represent significant and important refinements of the previously known results

a new system of generalized mixed quasi variational inclusions with h η monotone operators abstract in this paper we introduce and study a new system of generalized mixed quasi variational inclusions with h monotone operators we prove the convergence of a new iterative algorithm for this system of generalized mixed quasi variational inclusions the results in this paper extend and improve some known results in the literature

testing the correlation for clustered categorical and censored discrete time to event data when covariates are measured without with errors summary in the analysis of clustered categorical data it is of common interest to test for the correlation within clusters and the heterogeneity across different clusters we address this problem by proposing a class of score tests for the null hypothesis that the variance components are zero in random effects models for clustered nominal and ordinal categorical responses we extend the results to accommodate clustered censored discrete time to event data we next consider such tests in the situation where covariates are measured with errors we propose using the simex method to construct the score tests for the null hypothesis that the variance components are zero key advantages of the proposed score tests are that they can be easily implemented by fitting standard polytomous regression models and discrete failure time models and that they are robust in the sense that no assumptions need to be made regarding the distributions of the random effects and the unobserved covariates the asymptotic properties of the proposed tests are studied we illustrate these tests by analyzing two data sets and evaluate their performance with simulations

on iterative methods for equilibrium problems abstract in this paper we introduce a hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of finitely many nonexpansive mappings we prove that the approximate solution converges strongly to a solution of a class of variational inequalities under some mild conditions which is the optimality condition for some minimization problem we also give some comments on the results of plubtieng and punpaeng s plubtieng r punpaeng a general iterative method for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 336 2007 455 469 results obtained in this paper may be viewed as an improvement and refinement of the previously known results in this area

an outer approximation algorithm for solving general convex programs a new outer approximation algorithm is proposed for solving general convex programs a remarkable advantage of the algorithm over existing outer approximation methods is that the approximation of the constraint set is not cumulative that is the algorithm solves at each iteration a quadratic program whose constraints depend only on the current estimate of an optimal solution convergence of the algorithm is proved and possible applications are discussed

co coercivity and its role in the convergence of iterative schemes for solving variational inequalities this paper stresses the role of co coercivity and related notions in the convergence of iterative schemes for solving monotone but not necessarily strongly or even strictly monotone variational inequalities the analysis will be conducted within the conceptual framework of the auxiliary problem principle

a new alternating direction method for co coercive variational inequality problems this paper presents a new alternating direction method for solving co coercive variational inequality problems where the feasible set is the intersection of a simple set and polyhedron defined by a system of linear equations the proposed method can be viewed as a combination of han and lo s alternating direction method d r han h k lo a new alternating direction method for a class of nonlinear variational inequality problems journal of optimization theory and applications 112 3 2002 549 560 for such class of variational inequality problems and li liao and yuan s modified descent method for co coercive variational inequality problems m li l z liao x m yuan a modified descent projection method for co coercive variational inequalities european journal of operational research 189 2 2008 310 323 thus it possesses the advantages of both han and lo s alternating direction method which solves a series of small scale easier problems to solve the original variational inequality problem and li liao and yuan s modified descent method which is simple provided that the feasible set is simple we test the new method and compare it with han and lo s method and li liao and yuan s modified descent method and the numerical results show that our new method is suitable for such class of variational inequality problems

an explicit projection method for a system of nonlinear variational inequalities with different cocoercive mappings in this paper we introduce and consider a new system of variational inequalities involving two different operators using the projection technique we suggest and analyze new explicit iterative methods for this system of variational inequalities we also study the convergence analysis of the new iterative method under certain mild conditions since this new system includes the system of variational inequalities involving the single operator variational inequalities and related optimization problems as special cases results obtained in this paper continue to hold for these problems our results can be viewed as a refinement and improvement of the previously known results for variational inequalities

hemiequilibrium like problems abstract in this paper we introduce and consider a new class of equilibrium problems known as hemiequilibrium like problems this new class includes hemiequilibrium equilibrium like problems and several classes of variational inequalities as special cases a number of iterative methods for solving hemiequilibrium like problems are suggested and analyzed by using the auxiliary principle technique we also study the convergence analysis of these iterative methods under some mild conditions the results obtained in this paper can be considered as a novel application of the auxiliary principle technique

estimation of the survival function for gray s piecewise constant time varying coefficients model gray s extension of cox s proportional hazards ph model for right censored survival data allows for a departure from the ph assumption via introduction of time varying regression coefficients tvc using penalized splines gray s work focused on estimation inference and residual analyses but no estimator for the survival function has been proposed we derive a survival function estimator for one important member of the class of tvc models a piecewise constant time varying coefficients pc tvc model we also derive an estimate for the confidence limits of the survival function accuracy in estimating underlying survival times and survival quantiles is assessed for both cox s and gray s pc tvc model using a simulation study featuring scenarios violating the ph assumption finally an example of the estimated survival functions and the corresponding confidence limits derived from cox s ph and gray s pc tvc model respectively is presented for a liver transplant data set in the second part of the thesis we examine the effect of model misspecification for two classes of regression models for right censored survival data additive and multiplicative models for the conditional hazard rate a particular attention is given to data exhibiting time varying regression coefficients the class of multiplicative models is represented by cox ph model and gray s tvc model respectively and for additive models we use aalen s linear model both gray s tvc model and aalen s linear model incorporate time varying coefficients a simulation study is performed to cross analyze survival data which follows either a multiplicative or an additive model for the conditional hazard rate the effect of misspecifying the true model for the conditional hazard rate is assessed by looking at the power of the individual models to detect an existing effect bias and mean square error observed for each conditional model based estimator of survival we also show that aalen s model formulae is a first order taylor series approximation of that of gray s model which explains the comparably higher flexibility on part of the aalen s model as compared to the cox ph when the gray s tvc model for the data is misspecified

second order asymptotics of mutual information a formula for the second order expansion of the input output mutual information of multidimensional channels as the signal to noise ratio snr goes to zero is obtained while the additive noise is assumed to be gaussian we deal with very general classes of input and channel distributions as special cases these channel models include fading channels channels with random parameters and channels with almost gaussian noise when the channel is unknown at the receiver the second term in the asymptotic expansion depends not only on the covariance matrix of the input signal but also on the fourth mixed moments of its components the study of the second order asymptotics of mutual information finds application in the analysis of the bandwidth power tradeoff achieved by various signaling strategies in the wideband regime

some characterizations of strongly preinvex functions in this paper a new class of generalized convex function is introduced which is called the strongly preinvex function we study some properties of strongly preinvex function in particular we establish the equivalence among the strongly preinvex functions strongly invex functions and strongly monotonicity under some suitable conditions as special cases one can obtain several new and previously known results for preinvex invex functions

weak sharp solutions of variational inequalities xuexiang huang pointed out that the statement on line 5 of page 184 of p marcotte and d zhu siam j optim 9 1999 pp 179 189 is invalid and supplied an alternative proof of the inequality that appears on line 10 here is a simple proof of this result due to an anonymous referee the substitution starts at line 4 and ends at line 11

estimation in gaussian noise properties of the minimum mean square error consider the minimum mean square error mmse of estimating an arbitrary random variable from its observation contaminated by gaussian noise the mmse can be regarded as a function of the signal to noise ratio snr as well as a functional of the input distribution of the random variable to be estimated it is shown that the mmse is concave in the input distribution at any given snr for a given input distribution the mmse is found to be infinitely differentiable at all positive snr and in fact a real analytic function in snr under mild conditions the key to these regularity results is that the posterior distribution conditioned on the observation through gaussian channels always decays at least as quickly as some gaussian density furthermore simple expressions for the first three derivatives of the mmse with respect to the snr are obtained it is also shown that as functions of the snr the curves for the mmse of a gaussian input and that of a non gaussian input cross at most once over all snrs these properties lead to simple proofs of the facts that gaussian inputs achieve both the secrecy capacity of scalar gaussian wiretap channels and the capacity of scalar gaussian broadcast channels as well as a simple proof of the entropy power inequality in the special case where one of the variables is gaussian

heat flow derivatives and minimum mean square error in gaussian noise we connect recent developments on gaussian noise estimation and the minimum mean square error to earlier results on entropy and fisher information heat flow expansion in particular the derivatives of the minimum mean square error with respect to the noise parameter are related to the heat flow derivatives of the fisher information and a special lie algebra structure on iterated gradients the results lead in particular to a partial answer to the minimum mean square error conjecture

hemivariational like inequalities in this paper we introduce and consider a new class of variational inequalities known as the hemivariational like inequalities it is shown that the hemivariational like inequalities include hemivariational inequalities variational like inequalities and the classical variational inequalities as special cases the auxiliary principle is used to suggest and analyze some iterative methods for solving hemivariational like inequalities under mild conditions the results obtained in this paper can be considered as a novel application of the auxiliary principle technique

the extragradient method for finding a common solution of a finite family of variational inequalities and a finite family of fixed point problems in the presence of computational errors abstract in a hilbert space we study the convergence of the subgradient method to a common solution of a finite family of variational inequalities and of a finite family of fixed point problems under the presence of computational errors most results known in the literature establish the convergence of algorithms when computational errors are summable in the present paper the convergence of the subgradient method is established for nonsummable computational errors we show that the subgradient method generates a good approximate solution if the sequence of computational errors is bounded from above by a constant

a note on permutation tests for variance components in multilevel generalized linear mixed models in many applications of generalized linear mixed models to multilevel data it is of interest to test whether a random effects variance component is zero it is well known that the usual asymptotic chi square distribution of the likelihood ratio and score statistics under the null does not necessarily hold in this note we propose a permutation test based on randomly permuting the indices associated with a given level of the model that has the correct type i error rate under the null results from a simulation study suggest that it is more powerful than tests based on mixtures of chi square distributions the proposed test is illustrated using data on the familial aggregation of sleep disturbance

on nonconvex equilibrium problems in this paper we introduce a new class of equilibrium problems known as mixed quasi nonconvex equilibrium problems we suggest some iterative schemes for solving nonconvex equilibrium problems by using the auxiliary principle technique the convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomonotonicity as special cases we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems

generalized entropy power inequalities and monotonicity properties of information new families of fisher information and entropy power inequalities for sums of independent random variables are presented these inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables for an arbitrary collection of subsets as a consequence a simple proof of the monotonicity of information in central limit theorems is obtained both in the setting of independent and identically distributed i i d summands as well as in the more general setting of independent summands with variance standardized sums

functional inference in frailty measurement error models for clustered survival data using the simex approach we consider frailty models for clustered survival data in the presence of measurement errors in covariates we first show that when the measurement error is accounted for in a full likelihood analysis but the distribution of the unobserved covariate is misspecified the maximum likelihood estimators are asymptotically biased especially for the variance component whose bias can be substantial we then discuss making inference using functional estimation via the simex method where no distribution of the unobserved error prone covariate is assumed the simex method is easy to implement by repeatedly fitting standard frailty models we study the asymptotic properties of the simex estimates and show that they are consistent and asymptotically normal in simulation studies we compare the simex method and the likelihood method in terms of efficiency and robustness we also propose a simex score test for the variance component to test for the within cluster correlation and evaluate its performance through simu

selecting factors predictive of heterogeneity in multivariate event time data in multivariate survival analysis investigators are often interested in testing for heterogeneity among clusters both overall and within specific classes we represent different hypotheses about the heterogeneity structure using a sequence of gamma frailty models ranging from a null model with no random effects to a full model having random effects for each class following a bayesian approach we define prior distributions for the frailty variances consisting of mixtures of point masses at zero and inverse gamma densities since frailties with zero variance effectively drop out of the model this prior allocates probability to each model in the sequence including the overall null hypothesis of homogeneity using a counting process formulation the conditional posterior distributions of the frailties and proportional hazards regression coefficients have simple forms posterior computation proceeds via a data augmentation gibbs sampling algorithm a single run of which can be used to obtain model averaged estimates of the population parameters and posterior model probabilities for testing hypotheses about the heterogeneity structure the methods are illustrated using data from a lung cancer trial

a frailty model for informative censoring to account for the correlation between failure and censoring we propose a new frailty model for clustered data in this model the risk to be censored is affected by the risk of failure this model allows flexibility in the direction and degree of dependence between failure and censoring it includes the traditional frailty model as a special case it allows censoring by some causes to be analyzed as informative while treating censoring by other causes as noninformative it can also analyze data for competing risks to fit the model the em algorithm is used with markov chain monte carlo simulations in the e steps simulation studies and analysis of data for kidney disease patients are provided consequences of incorrectly assuming noninformative censoring are investigated

further applications of a splitting algorithm to decomposition in variational inequalities and convex programming a classical method for solving the variational inequality problem is the projection algorithm we show that existing convergence results for this algorithm follow from one given by gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators moreover we extend the projection algorithm to solveany monotone affine variational inequality problem when applied to linear complementarity problems we obtain a matrix splitting algorithm that is simple and for linear quadratic programs massively parallelizable unlike existing matrix splitting algorithms this algorithm converges under no additional assumption on the problem when applied to generalized linear quadratic programs we obtain a decomposition method that unlike existing decomposition methods can simultaneously dualize the linear constraints and diagonalize the cost function this method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets

the power of testing a semi parametric shared gamma frailty parameter in failure time data frailty is usually modeled as an unobserved random variable acting multiplicatively on the baseline hazard function where a shared unobserved quantity in the intensity induces a positive correlation among the observed failure times using the asymptotic properties of the nonparametric maximum likelihood estimator for the gamma frailty model we derive a power function for testing the shared frailty parameter and evaluate the effect of number of groups and number of individuals per groups on the power of the test by simulation studies the results show that choosing between 8 and 25 groups with sample sizes between 200 and 500 individuals will be enough to get a high power

a locally most powerful test for homogeneity with many strata summary a locally most powerful test is proposed for homogeneity of nuisance parameters from many strata in the mixed model setting the test is based on the difference between the square of the score and the observed information about the nuisance parameters all evaluated at the null hypothesis several examples including constant normal mean with possibly different variances and matched case control studies are presented in detail the relative efficiencies of the unstratified estimator of parameter of interest versus the stratified one are calculated two data sets on matched case control studies are used for illustration

perturbed projections and subgradient projections for the multiple sets split feasibility problem we study the multiple sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space by casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case we further derive and analyze a perturbed projection method for the multiple sets split feasibility problem and additionally furnish alternative proofs to two known results

weak co coercivity and its applications in several algorithms for solving variational inequalities abstract in this paper we investigate several algorithms for solving variational inequalities vis and the zero inclusion problem respectively it is shown that invested algorithms are applicable in broader area in addition for the purpose of practical implementation we further modify one of the algorithms discussed by adopting an armijo like search

mutual information and minimum mean square error in gaussian channels this paper deals with arbitrarily distributed finite power input signals observed through an additive gaussian noise channel it shows a new formula that connects the input output mutual information and the minimum mean square error mmse achievable by optimal estimation of the input given the output that is the derivative of the mutual information nats with respect to the signal to noise ratio snr is equal to half the mmse regardless of the input statistics this relationship holds for both scalar and vector signals as well as for discrete time and continuous time noncausal mmse estimation this fundamental information theoretic result has an unexpected consequence in continuous time nonlinear estimation for any input signal with finite power the causal filtering mmse achieved at snr is equal to the average value of the noncausal smoothing mmse achieved with a channel whose snr is chosen uniformly distributed between 0 and snr

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

the relaxed inexact projection methods for the split feasibility problem abstract in this paper we present several relaxed inexact projection methods for the split feasibility problem sfp each iteration of the first proposed algorithm consists of a projection onto a halfspace containing the given closed convex set the algorithm can be implemented easily and its global convergence to the solution can be established under suitable conditions moreover we present some modifications of the relaxed inexact projection method with constant stepsize by adopting armijo like search we furthermore present a variable step relaxed inexact projection method which does not require the computation of the matrix inverses and the largest eigenvalue of the matrix ata and the objective function can decrease sufficiently at each iteration we show convergence of these modified algorithms under mild conditions finally we perform some numerical experiments which show the behavior of the algorithms proposed

three measures of explained variation for correlated survival data under the proportional hazards mixed effects model measures of explained variation are useful in scientific research as they quantify the amount of variation in an outcome variable of interest that is explained by one or more other variables we develop such measures for correlated survival data under the proportional hazards mixed effects model because different approaches have been studied in the literature outside the classical linear regression model we investigate three measures r 2 rres2 and 2 that quantify three different population coefficients we show that although the three population measures are not the same they reflect similar amounts of variation explained by the predictors among the three measures we show that r 2 which is the simplest to compute is also consistent for the first population measure under the usual asymptotic scenario when the number of clusters tends to infinity the other two measures on the other hand all require that in addition the cluster sizes be large we study the properties of the measures both analytically and through simulation studies we illustrate their different usage on a multi center clinical trial and a recurrent events data set copyright 2016 john wiley sons ltd

solution of shannon s problem on the monotonicity of entropy it is shown that if x1 x2 are independent and identically distributed square integrable random variables then the entropy of the normalized sum ent x1 xn over n is an increasing function of n this resolves an old problem which goes back to 6 7 5 the result also has a version for non identically distributed random variables or random vectors

a note on solodov and tseng s methods for maximal monotone mappings this paper considers the problem of finding a zero of the sum of a single valued lipschitz continuous mapping a and a maximal monotone mapping b in a closed convex set c we first give some projection type methods and extend a modified projection method proposed by solodov and tseng for the special case of b n c to this problem then we give a refinement of tseng s method that replaces p c by p c k finally convergence of these methods is established

testing goodness of fit in regression a general approach for specified alternatives when fitting generalized linear models or the cox proportional hazards model it is important to have tools to test for lack of fit because lack of fit comes in all shapes and sizes distinguishing among different types of lack of fit is of practical importance we argue that an adequate diagnosis of lack of fit requires a specified alternative model such specification identifies the type of lack of fit the test is directed against so that if we reject the null hypothesis we know the direction of the departure from the model the goodness of fit approach of this paper allows to treat different types of lack of fit within a unified general framework and to consider many existing tests as special cases connections with penalized likelihood and random effects are discussed and the application of the proposed approach is illustrated with medical examples tailored functions for goodness of fit testing have been implemented in the r package globaltest copyright 2012 john wiley sons ltd

some new projection methods for variational inequalities in this paper we propose some new double projection methods for solving variational inequalities by using the wiener hopf equations technique it is shown that these methods converge linearly under mild conditions and include some existing projection methods as special cases some examples are given to illustrate the efficiency of the proposed methods

iterative algorithms for a new system of nonlinear variational inclusions with a η accretive mappings in banach spaces in this paper we introduce and study a new system of nonlinear variational inclusions with a h accretive mappings in banach spaces by using the resolvent operator associated with a h accretive mappings we construct some new iterative algorithms for approximating the solution of this system of variational inclusions we also prove the existence of solutions and the convergence of the sequences generated by the algorithm in banach spaces the results presented in this paper extend and improve some known results in the literature

conditional akaike information under generalized linear and proportional hazards mixed models we study model selection for clustered data when the focus is on cluster specific inference such data are often modelled using random effects and conditional akaike information was proposed in vaida blanchard 2005 and used to derive an information criterion under linear mixed models here we extend the approach to generalized linear and proportional hazards mixed models outside the normal linear mixed models exact calculations are not available and we resort to asymptotic approximations in the presence of nuisance parameters a profile conditional akaike information is proposed bootstrap methods are considered for their potential advantage in finite samples simulations show that the performance of the bootstrap and the analytic criteria are comparable with bootstrap demonstrating some advantages for larger cluster sizes the proposed criteria are applied to two cancer datasets to select models when the cluster specific inference is of interest

monotonic decrease of the non gaussianness of the sum of independent random variables a simple proof artstein ball barthe and naor have recently shown that the non gaussianness divergence with respect to a gaussian random variable with identical first and second moments of the sum of independent and identically distributed i i d random variables is monotonically nonincreasing we give a simplified proof using the relationship between non gaussianness and minimum mean square error mmse in gaussian channels as artstein we also deal with the more general setting of nonidentically distributed random variables

role of twice frechet differentiable mappings in general auxiliary problem principle abstract the approximation solvability of the following class of nonlinear variational inequality problems nvip based on the general auxiliary problem principle lapp is discussed find an element k such that x x x f x f x o for all x k where t k e is a nonlinear mapping from a nonempty closed convex subset k of a reflexive banach space e into its dual e and f k r is a continuous convex functional on k

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

on variable step relaxed projection algorithm for variational inequalities projection algorithms are practically useful for solving variational inequalities vi however some among them require the knowledge related to vi in advance such as lipschitz constant usually it is impossible in practice this paper studies the variable step basic projection algorithm and its relaxed version under weakly co coercive condition the algorithms discussed need not know constant function associated with the co coercivity or weak co coercivity and the step size is varied from one iteration to the next under certain conditions the convergence of the variable step basic projection algorithm is established for the practical consideration we also give the relaxed version of this algorithm in which the projection onto a closed convex set is replaced by another projection at each iteration and latter is easy to calculate the convergence of relaxed scheme is also obtained under certain assumptions finally we apply these two algorithms to the split feasibility problem sfp

regularized auxiliary problem principle for variational inequalities abstract this paper is devoted to the unification of the auxiliary problem principle and the principle of iterative regularization for the approximate solution of variational inequalities this unification permits the extension of the principle of iterative regularization to real reflexive banach spaces on the basis of this unification an algorithm is proposed and the strong convergence of the proposed algorithm is depicted

permutation tests for centre effect on survival endpoints with application in an acute myeloid leukaemia multicentre study when analysing multicentre data it may be of interest to test whether the distribution of the endpoint varies among centres in a mixed effect model testing for such a centre effect consists in testing to zero a random centre effect variance component it has been shown that the usual asymptotic 2 distribution of the likelihood ratio and score statistics under the null does not necessarily hold in the case of censored data mixed effects cox models have been used to account for random effects but few works have concentrated on testing to zero the variance component of the random effects we propose a permutation test using random permutation of the cluster indices to test for a centre effect in multilevel censored data results from a simulation study indicate that the permutation tests have correct type i error rates contrary to standard likelihood ratio tests and are more powerful the proposed tests are illustrated using data of a multicentre clinical trial of induction therapy in acute myeloid leukaemia patients copyright 2014 john wiley sons ltd

a parallel splitting method for coupled monotone inclusions a parallel splitting method is proposed for solving systems of coupled monotone inclusions in hilbert spaces and its convergence is established under the assumption that solutions exist unlike existing alternating algorithms which are limited to two variables and linear coupling our parallel method can handle an arbitrary number of variables as well as nonlinear coupling schemes the breadth and flexibility of the proposed framework is illustrated through applications in the areas of evolution inclusions variational problems best approximation and network flows

solving variational inequality and fixed point problems by line searches and potential optimization we introduce a general adaptive line search framework for solving fixed point and variational inequality problems our goals are to develop iterative schemes that i compute solutions when the underlying map satisfies properties weaker than contractiveness for example weaker forms of nonexpansiveness ii are more efficient than the classical methods even when the underlying map is contractive and iii unify and extend several convergence results from the fixed point and variational inequality literatures to achieve these goals we introduce and study joint compatibility conditions imposed upon the underlying map and the iterative step sizes at each iteration and consider line searches that optimize certain potential functions as a special case we introduce a modified steepest descent method for solving systems of equations that does not require a previous condition from the literature the square of the jacobian matrix is positive definite since the line searches we propose might be difficult to perform exactly we also consider inexact line searches

merit functions and error bounds for generalized variational inequalities we consider the generalized variational inequality and construct certain merit functions associated with this problem in particular those merit functions are everywhere nonnegative and their zero sets are precisely solutions of the variational inequality we further use those functions to obtain error bounds i e upper estimates for the distance to solutions of the problem

a relaxed projection method for variational inequalities this paper presents a modification of the projection methods for solving variational inequality problems each iteration of the proposed algorithm consists of projection onto a halfspace containing the given closed convex set rather than the latter set itself the algorithm can thus be implemented very easily and its global convergence to the solution can be established under suitable conditions

a free analogue of shannon s problem on monotonicity of entropy abstract we prove a free probability analog of a result of s artstein k ball f barthe a naor solution of shannon s problem on monotonicity of entropy j amer math soc 17 2004 975 982 in particular we prove that if x 1 x 2 are freely independent identically distributed random variables then the function n x 1 x n n is monotone increasing for all n

alternating projection proximal methods for convex programming and variational inequalities we consider a mixed problem composed in part of finding a zero of a maximal monotone operator and in part of solving a monotone variational inequality problem we propose a solution method for this problem that alternates between a proximal step for the maximal monotone operator part and a projection type step for the monotone variational inequality part and analyze its convergence and rate of convergence this method extends a decomposition method of chen and teboulle math programming 64 1994 pp 81 101 for convex programming and yields as a by product new decomposition methods

monotonicity of fixed point and normal mappings associated with variational inequality and its application we prove sufficient conditions for the monotonicity and the strong monotonicity of fixed point and normal maps associated with variational inequality problems over a general closed convex set sufficient conditions for the strong monotonicity of their perturbed versions are also shown these results include some well known in the literature as particular instances inspired by these results we propose a modified solodov and svaiter iterative algorithm for the variational inequality problem whose fixed point map or normal map is monotone

proportional hazards model with random effects we propose a general proportional hazards model with random effects for handling clustered survival data this generalizes the usual frailty model by allowing a multivariate random effect with arbitrary design matrix in the log relative risk in a way similar to the modelling of random effects in linear generalized linear and non linear mixed models the distribution of the random effects is generally assumed to be multivariate normal but other preferably symmetrical distributions are also possible maximum likelihood estimates of the regression parameters the variance components and the baseline hazard function are obtained via the em algorithm the e step of the algorithm involves computation of the conditional expectations of functions of the random effects for which we use markov chain monte carlo mcmc methods approximate variances of the estimates are computed by louis formula and posterior expectations and variances of the individual random effects can be obtained as a by product of the estimation the inference procedure is exemplified on two data sets

inverse variational inequalities with projection based solution methods abstract an inverse variational inequality is defined as to find a vector u r n such that f u v f u t u 0 v if an inverse function u f 1 x exists the above inverse variational inequality could be transformed as a regular variational inequality however in reality it is not uncommon that the inverse function of f 1 x does not have explicit form although its functional values can be observed existing line search algorithms cannot be applied directly to solve such inverse variational inequalities in this paper we propose two projection based methods using the co coercivity of mapping f a self adaptive strategy is developed to determine the step sizes efficiently when the co coercivity modulus is unknown the convergence of the proposed methods is proved rigorously

a bundle method for solving equilibrium problems we present a bundle method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle first we consider a general algorithm that we prove to be convergent then we explain how to make this algorithm implementable the strategy is to approximate the nonsmooth convex functions by piecewise linear convex functions in such a way that the subproblems are easy to solve and the convergence is preserved in particular we introduce a stopping criterion which is satisfied after finitely many iterations and which gives rise to stationary points finally we apply our implementable algorithm for solving the particular case of singlevalued and multivalued variational inequalities and we find again the results obtained recently by salmon et al 18

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

an interior point method with bregman functions for the variational inequality problem with paramonotone operators we present an algorithm for the variational inequality problem on convex sets with nonempty interior the use of bregman functions whose zone is the convex set allows for the generation of a sequence contained in the interior without taking explicitly into account the constraints which define the convex set we establish full convergence to a solution with minimal conditions upon the monotone operatorf weaker than strong monotonicity or lipschitz continuity for instance and including cases where the solution needs not be unique we apply our algorithm to several relevant classes of convex sets including orthants boxes polyhedra and balls for which bregman functions are presented which give rise to explicit iteration formulae up to the determination of two scalar stepsizes which can be found through finite search procedures 1998 the mathematical programming society inc published by elsevier science b v