a large sample study of cox s regression model strong consistency and asymptotic normality are established for the maximum partial likelihood estimate of the regression parameter in cox s regression model estimates are also derived for the underlying cumulative hazard function and survival distribution we establish the asymptotic normality of these estimates and calculate the limiting variances

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

julia sets and mandelbrot sets in noor orbit in recent literature researchers have generated julia sets and mandelbrot sets in mann and ishikawa orbits that are examples of two step and three step feedback processes respectively this paper presents further generalization of julia and mandelbrot sets for complex valued polynomials such as quadratic cubic and higher degree polynomials using a noor orbit which is a four step iterative procedure the graphical images of julia and mandelbrot sets have been visualized and certain patterns in mandelbrot sets have been recognized it is fascinating to see that a few mandelbrot sets are akin to a butterfly or a coupled urn or a coupled trident

estimation and variable selection for generalized additive partial linear models we study generalized additive partial linear models proposing the use of polynomial spline smoothing for estimation of nonparametric functions and deriving quasi likelihood based estimators for the linear parameters we establish asymptotic normality for the estimators of the parametric components the procedure avoids solving large systems of equations as in kernel based procedures and thus results in gains in computational simplicity we further develop a class of variable selection procedures for the linear parameters by employing a nonconcave penalized quasi likelihood which is shown to have an asymptotic oracle property monte carlo simulations and an empirical example are presented for illustration

statistical estimation in varying coefficient models varying coefficient models are a useful extension of classical linear models they arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age the appeal of these models is that the coef cient functions can easily be estimated via a simple local regression this yields a simple one step estimation procedure we show that such a one step method cannot be optimal when different coefficient functions admit different degrees of smoothness this drawback can be repaired by using our proposed two step estimation procedure the asymptotic mean squared error for the two step procedure is obtained and is shown to achieve the optimal rate of convergence a few simulation studies show that the gain by the two step procedure can be quite substantial the methodology is illustrated by an application to an environmental data set

efficient estimation of the partly linear additive cox model the partly linear additive cox model is an extension of the linear cox model and allows flexible modeling of covariate effects semiparametrically we study asymptotic properties of the maximum partial likelihood estimator of this model with right censored data using polynomial splines we show that with a range of choices of the smoothing parameter the number of spline basis functions required for estimation of the nonparametric components the estimator of the finite dimensional regression parameter is root n consistent asymptotically normal and achieves the semiparametric information bound rates of convergence for the estimators of the nonparametric components are obtained they are comparable to the rates in nonparametric regression implementation of the estimation approach can be done easily and is illustrated by using a simulated example

estimation of the mean function with panel count data using monotone polynomial splines we study nonparametric likelihood based estimators of the mean function of counting processes with panel count data using monotone polynomial splines the generalized rosen algorithm proposed by zhang jamshidian 2004 is used to compute the estimators we show that the proposed spline likelihood based estimators are consistent and that their rate of convergence can be faster than n 1 3 simulation studies with moderate samples show that the estimators have smaller variances and mean squared errors than their alternatives proposed by wellner zhang 2000 a real example from a bladder tumour clinical trial is used to illustrate this method

efficient estimation and inferences for varying coefficient models abstract this article deals with statistical inferences based on the varying coefficient models proposed by hastie and tibshirani local polynomial regression techniques are used to estimate coefficient functions and the asymptotic normality of the resulting estimators is established the standard error formulas for estimated coefficients are derived and are empirically tested a goodness of fit test technique based on a nonparametric maximum likelihood ratio type of test is also proposed to detect whether certain coefficient functions in a varying coefficient model are constant or whether any covariates are statistically significant in the model the null distribution of the test is estimated by a conditional bootstrap method our estimation techniques involve solving hundreds of local likelihood equations to reduce the computational burden a one step newton raphson estimator is proposed and implemented the resulting one step procedure is shown to save computational cost on an order of tens with no

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

nonparametric regression using local kernel estimating equations for correlated failure time data we study nonparametric regression for correlated failure time data kernel estimating equations are used to estimate nonparametric covariate effects independent and weighted kernel estimating equations are studied the derivative of the nonparametric function is first estimated and the nonparametric function is then estimated by integrating the derivative estimator we show that the nonparametric kernel estimator is consistent for any arbitrary working correlation matrix and that its asymptotic variance is minimized by assuming working independence we evaluate the performance of the proposed kernel estimator using simulation studies and apply the proposed method to the western kenya parasitaemia data

a joint model of recurrent events and a terminal event with a nonparametric covariate function we extend the shared frailty model of recurrent events and a dependent terminal event to allow for a nonparametric covariate function we include a gaussian random effect frailty in the intensity functions of both the recurrent and terminal events to capture correlation between the two processes we employ the penalized cubic spline method to describe the nonparametric covariate function in the recurrent events model we use laplace approximation to evaluate the marginal penalized partial likelihood without a closed form we also propose the variance estimates for regression coefficients numerical analysis results show that the proposed estimates perform well for both the nonparametric and parametric components we apply this method to analyze the hospitalization rate of patients with heart failure in the presence of death

semiparametric estimation methods for panel count data using monotone b splines we study semiparametric likelihood based methods for panel count data with proportional mean model e t z 0 t exp 0tz where z is a vector of covariates and 0 t is the baseline mean function we propose to estimate 0 t and 0 jointly with 0 t approximated by monotone b splines and to compute the estimators using generalized rosen algorithm proposed by jamshidian 2004 we show that the proposed spline based likelihood estimators of 0 t are consistent with a possibly better than n1 3 convergence rate if 0 t is sufficiently smooth the normality of the estimators of 0 is also established comparisons between the proposed estimators and their alternatives studied in wellner and zhang 2007 are made through simulations studies regarding their finite sample performance and computational complexity a real example from a bladder tumor clinical trial is used to illustrate the methods

information bounds for cox regression models with missing data we derive information bounds for the regression parameters in cox models when data are missing at random these calculations are of interest for understanding the behavior of efficient estimation in case cohort designs a type of two phase design often used in cohort studies the derivations make use of key lemmas appearing in robins rotnitzky and zhao j amer statist assoc 89 1994 846 866 and robins hsieh and newey j roy statist soc ser b 57 1995 409 424 but in a form suited for our purposes here we begin by summarizing the results of robins rotnitzky and zhao in a form that leads directly to the projection method which will be of use for our model of interest we then proceed to derive new information bounds for the regression parameters of the cox model with data missing at random mar in the final section we exemplify our calculations with several models of interest in cohort studies including an i i d version of the classical case cohort design of prentice biometrika 73 1986 1 11 and self and prentice ann statist 16 1988 64 81

generalized additive models some applications abstract generalized additive models have the form x fj x j where might be the regression function in a multiple regression or the logistic transformation of the posterior probability pr y 1 x in a logistic regression in fact these models generalize the whole family of generalized linear models x x where x g x is some transformation of the regression function we use the local scoring algorithm to estimate the functions fj xj nonparametrically using a scatterplot smoother as a building block we demonstrate the models in two different analyses a nonparametric analysis of covariance and a logistic regression the procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in a standard linear analysis a variety of inferential tools have been developed to aid the analyst in assessing the relevance and significance of the estimated functions these include confidence curves degrees of freedom estimates and approximat

estimating multiplicative and additive hazard functions by kernel methods we propose new procedures for estimating the component functions in both additive and multiplicativen onparametricm arker dependenht azard models we work with a full counting process framework that allows for left truncation and right censoring and time varying covariates our procedures are based on kernel hazard estimation as developed by nielsen and linton and on the idea of marginal integration we provide a central limit theorem for the marginal integration estimator we then define estimators based on finite step backfitting in both additive and multiplicative cases and prove that these estimators are asymptotically normal and have smaller variance than the marginal integration method

on estimation of partially linear transformation models we study a general class of partially linear transformation models which extend linear transformation models by incorporating nonlinear covariate effects in survival data analysis a new martingale based estimating equation approach consisting of both global and kernel weighted local estimation equations is developed for estimating the parametric and nonparametric covariate effects in a unified manner we show that with a proper choice of the kernel bandwidth parameter one can obtain the consistent and asymptotically normal parameter estimates for the linear effects asymptotic properties of the estimated nonlinear effects are established as well we further suggest a simple resampling method to estimate the asymptotic variance of the linear estimates and show its effectiveness to facilitate the implementation of the new procedure an iterative algorithm is developed numerical examples are given to illustrate the finite sample performance of the procedure supplementary materials are available online

residual based tree structured survival analysis extensions of various non parametric regression techniques for example additive models trees mars have been devised for right censored survival data these approaches directly handle the difficulties posed by censoring however it is possible to bypass these difficulties by utilizing standard non parametric regression procedures applied with say martingale residuals as outcome analytic correspondences between the direct and residual based approaches have been established for additive models while more qualitative comparisons have been provided for mars here we develop such correspondences for tree structured regression in particular we provide an analytic relationship between logrank and martingale residual sum of squares split functions that explains the widely observed similarity of the resultant trees further investigation is provided by simulation and an illustrative example using time to aids with data deriving from a western australian hiv cohort study

combining dependent tests with incomplete repeated measurements summary in comparing the effectiveness of two treatments repeated measurements of the same characteristic are often taken under two or more distinct conditions for each experimental subject a locally best linear combination of individual test statistics obtained under different conditions is proposed to test the equality of the two treatments the test procedure allows different patterns of missing observations for the two groups to be compared special cases such as combining wilcoxon tests t tests and 2 x 2 tables are discussed in detail with real examples

nonsmooth backfitting for the excess risk additive regression model with two survival time scales we consider an extension of aalen s additive regression model allowing covariates to have effects that vary on two different time scales the two time scales considered are equal up to a constant that varies for each individual such as for example follow up time and age in medical studies or calendar time and age in longitudinal studies the model has been introduced in scheike 2001 where it was solved via smoothing techniques we present a new backfitting 20 algorithm estimating the structured model without having to use smoothing estimators of the cumulative regression functions on the two time scales are suggested by solving local estimating equations jointly on the two time scales we provide large sample properties and simultaneous confidence bands the model is applied to data on myocardial infarction providing a separation of the two effects stemming from time since diagnosis and age

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

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

nonparametric estimation and tests of fit for dose response relations summary various parametric models have been used to describe the dependence of disease risks on continuous explanatory variables the fit of such models is generally tested either against some larger but still parametric class of alternatives or by making the covariates discrete in some arbitrary fashion this communication introduces a more general concept of residuals which may be useful for visualizing the fit of the data subject only to the constraint of monotonicity the maximum likelihood estimate mle of the dose response relation for a single covariate is shown to be a step function with discontinuities only at the covariate values of the failures the mle is found by a constrained newton raphson search and a procedure for reducing the dimensions of the search is suggested applications to the risk of lung cancer in relation to asbestos exposure are given

efficient estimation of semiparametric transformation models for counting processes summary a class of semiparametric transformation models is proposed to characterise the effects of possibly time varying covariates on the intensity functions of counting processes the class includes the proportional intensity model and linear transformation models as special cases nonparametric maximum likelihood estimators are developed for the regression parameters and cumulative intensity functions of these models based on censored data the estimators are shown to be consistent and asymptotically normal the limiting variances for the estimators of the regression parameters achieve the semi parametric efficient bounds and can be consistently estimated the limiting variances for the estimators of smooth functionals of the cumulative intensity function can also be consistently estimated simulation studies reveal that the proposed inference procedures perform well in practical settings two medical studies are provided

bounds for eigenforms on arithmetic hyperbolic 3 manifolds on a family of arithmetic hyperbolic 3 manifolds of square free level we prove an upper bound for the sup norm of hecke maas cusp forms with a power saving over the local geometric bound simultaneously in the laplacian eigenvalue and the volume by a novel combination of diophantine and geometric arguments in a noncommutative setting we obtain bounds as strong as the best corresponding results on arithmetic surfaces

penalized log likelihood estimation for partly linear transformation models with current status data we consider partly linear transformation models applied to current status data the unknown quantities are the transformation function a linear regression parameter and a nonparametric regression effect it is shown that the penalized mle for the regression parameter is asymptotically normal and efficient and converges at the parametric rate although the penalized mle for the transformation function and nonparametric regression effect are only n 1 3 consistent inference for the regression parameter based on a block jack knife is investigated we also study computational issues and demonstrate the proposed methodology with a simulation study the transformation models and partly linear regression terms coupled with new estimation and inference techniques provide flexible alternatives to the cox model for current status data analysis

exploring the nature of covariate effects in the proportional hazards model we discuss an exploratory technique for investigating the nature of covariate effects in cox s proportional hazards model this technique features an additive term sigma p1 fj chi ij in place of the usual linear term sigma p1 chi ij beta j where chi i1 chi i2 chi ip are covariate values for the ith individual the fj are unspecified smooth functions that are estimated using scatterplot smoothers these functions can be used for descriptive purposes or to suggest transformations of the covariates the estimation technique is a variation of the local scoring algorithm for generalized additive models hastie and tibshirani 1986 statistical science 1 297 318

estimation and testing in a two sample generalized odds rate model abstract the generalized odds ratio for a survival variable t is defined as t t c c 1 1 sc t sc t for c 0 and log s t for c 0 where s t pr t t this ratio coincides with the integrated hazard for c 0 and the odds ratio for c 1 when the distribution of t depends on a covariate vector we assume that conditionally on the covariates log t t c is linear in the covariates this model is a generalization of the proportional hazard model phm which has an interpretation both as a phm with random nuisance effects clayton and cuzick 1986 and as a proportional odds rate model with the odds rate defined from the response times of series systems harrington and fleming 1982 and bickel 1986a considered rank tests for this semiparametric model clayton and cuzick 1986 considered estimation we use the odds rate representation to define a class of estimates of the proportionality parameter in the two sample case we show that the estimates are consistent and asymptotica

on the sup norm of maass cusp forms of large level iii let f be a hecke maass cuspidal newform of square free level n and laplacian eigenvalue lambda it is shown that left f right infty ll lambda epsilon n frac 1 6 epsilon left f right 2 for any epsilon 0

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

empirical likelihood based inference for generalized additive partial linear models abstract empirical likelihood based inference for the parameters in generalized additive partial linear models gaplm is investigated with the use of the polynomial spline smoothing for estimation of nonparametric functions an estimated empirical likelihood ratio statistic based on the quasi likelihood equation is proposed we show that the resulting statistic is asymptotically standard chi squared distributed and the confidence regions for the parametric components are constructed some simulations are conducted to illustrate the proposed methods

varying coefficient transformation models with censored data summary a maximum likelihood method with spline smoothing is proposed for linear transformation models with varying coefficients the estimation and inference procedures are computationally easy under some regularity conditions the estimators are proved to be consistent and asymptotically normal a simulation study using the stanford transplant data is presented to show that the proposed method performs well with a finite sample and is easy to use in practice

sup norms of eigenfunctions in the level aspect for compact arithmetic surfaces let d be an indefinite quaternion division algebra over mathbb q we approach the problem of bounding the sup norms of automorphic forms phi on d times mathbb a that belong to irreducible automorphic representations and transform via characters of unit groups of orders of d we obtain a non trivial upper bound for vert phi vert infty in the level aspect that is valid for arbitrary orders this generalizes and strengthens previously known upper bounds for vert phi vert infty in the setting of newforms for eichler orders in the special case when the index of the order in a maximal order is a squarefull integer n our result specializes to vert phi vert infty ll pi infty epsilon n 1 3 epsilon vert phi vert 2 a key application of our result is to automorphic forms phi which correspond at the ramified primes to either minimal vectors in the sense of hu et al commun math helv to appear or p adic microlocal lifts in the sense of nelson in microlocal lifts and and quantum unique ergodicity on mathrm gl 2 mathbb q p algebra number theory 12 9 2033 2064 2018 for such forms our bound specializes to vert phi vert infty ll epsilon c frac 1 6 epsilon vert phi vert 2 where c is the conductor of the representation pi generated by phi this improves upon the previously known local bound vert phi vert infty ll lambda epsilon c frac 1 4 epsilon vert phi vert 2 in these cases

convergence rate of sieve estimates in this paper we develop a general theory for the convergence rate of sieve estimates maximum likelihood estimates mle s and related estimates obtained by optimizing certain empirical criteria in general parameter spaces in many cases especially when the parameter space is infinite dimensional maximization over the whole parameter space is undesirable in such cases one has to perform maximization over an approximating space sieve of the original parameter space and allow the size of the approximating space to grow as the sample size increases this method is called the method of sieves in the case of the maximum likelihood estimation an mle based on a sieve is called a sieve mle we found that the convergence rate of a sieve estimate is governed by a the local expected values variances and l 2 entropy of the criterion differences and b the approximation error of the sieve a robust nonparametric regression problem a mixture problem and a nonparametric regression problem are discussed as illustrations of the theory we also found that when the underlying space is too large the estimate based on optimizing over the whole parameter space may not achieve the best possible rates of convergence whereas the sieve estimate typically does not suffer from this difficulty

efficient estimation for the cox model with varying coefficients summary a proportional hazards model with varying coefficients allows one to examine the extent to which covariates interact nonlinearly with an exposure variable a global partial likelihood method in contrast with the local partial likelihood method of fan et al 2006 is proposed for estimation of varying coefficient functions the proposed estimators are proved to be consistent and asymptotically normal semiparametric efficiency of the estimators is demonstrated in terms of their linear functionals evidence in support of the superiority of the method is presented in numerical studies and real examples

relative risk regression models with inverse polynomials the proportional hazards model assumes that the log hazard ratio is a linear function of parameters in the current paper we model the log relative risk as an inverse polynomial which is particularly suitable for modeling bounded and asymmetric functions the parameters estimated by maximizing the partial likelihood are consistent and asymptotically normal the advantages of the inverse polynomial model over the ordinary polynomial model and the fractional polynomial model for fitting various asymmetric log relative risk functions are shown by simulation the utility of the method is further supported by analyzing two real data sets addressing the specific question of the location of the minimum risk threshold

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

semiparametric analysis of transformation models with censored data summary a unified estimation procedure is proposed for the analysis of censored data using linear transformation models which include the proportional hazards model and the proportional odds model as special cases this procedure is easily implemented numerically and its validity does not rely on the assumption of independence between the covariates and the censoring variable the estimator is the same as the cox partial likelihood estimator in the case of the proportional hazards model moreover the asymptotic variance of the proposed estimator has a closed form and its variance estimator is easily obtained by plug in rules the method is illustrated by simulation and is applied to the veterans administration lung cancer data

maximum likelihood estimation for the proportional odds model with random effects in this article we study the semiparametric proportional odds model with random effects for correlated right censored failure time data we establish that the maximum likelihood estimators for the parameters of this model are consistent and asymptotically gaussian furthermore the limiting variances achieve the semiparametric efficiency bounds and can be consistently estimated simulation studies show that the asymptotic approximations are accurate for practical sample sizes and that the efficiency gains of the proposed estimators over those of cai cheng and wei can be substantial a real example is provided to illustrate the proposed methods

smoothing spline based score tests for proportional hazards models we propose score type tests for the proportional hazards assumption and for covariate effects in the cox model using the natural smoothing spline representation of the corresponding nonparametric functions of time or covariate the tests are based on the penalized partial likelihood and are derived by viewing the inverse of the smoothing parameter as a variance component and testing an equivalent null hypothesis that the variance component is zero we show that the tests have a size close to the nominal level and good power against general alternatives and we apply them to data from a cancer clinical trial

on the symmetries of the julia sets for the process z zp c the self replicating properties of the julia sets jc p for the iterative processes z implies zp c are examined for integers p 1 it is shown that the corresponding mandelbrot sets m p contain p 1 fold symmetries which results in jc p having p fold symmetries

bounding sup norms of cusp forms of large level let f be an l2 normalized weight zero hecke maas cusp form of square free level n character and laplacian eigenvalue 1 4 it is shown that f n 1 37 from which the hybrid bound f 1 4 n for some 0 is derived the first bound holds also for f yk 2f where f is a holomorphic cusp form of weight k with the implied constant now depending on k

analysis of clustered interval censored data using a class of semiparametric partly linear frailty transformation models a flexible class of semiparametric partly linear frailty transformation models is considered for analyzing clustered interval censored data which arise naturally in complex diseases and dental research this class of models features two nonparametric components resulting in a nonparametric baseline survival function and a potential nonlinear effect of a continuous covariate the dependence among failure times within a cluster is induced by a shared unobserved frailty term a sieve maximum likelihood estimation method based on piecewise linear functions is proposed the proposed estimators of the regression dependence and transformation parameters are shown to be strongly consistent and asymptotically normal whereas the estimators of the two nonparametric functions are strongly consistent with optimal rates of convergence an extensive simulation study is conducted to study the finite sample performance of the proposed estimators we provide an application to a dental study for illustration

focused information criterion and model averaging for generalized additive partial linear models we study model selection and model averaging in generalized additive partial linear models gaplms polynomial spline is used to approximate nonparametric functions the corresponding estimators of the linear parameters are shown to be asymptotically normal we then develop a focused information criterion fic and a frequentist model average fma estimator on the basis of the quasi likelihood principle and examine theoretical properties of the fic and fma the major advantages of the proposed procedures over the existing ones are their computational expediency and theoretical reliability simulation experiments have provided evidence of the superiority of the proposed procedures the approach is further applied to a real world data example

a sieve m theorem for bundled parameters in semiparametric models with application to the efficient estimation in a linear model for censored data in many semiparametric models that are parameterized by two types of parameters a euclidean parameter of interest and an infinite dimensional nuisance parameter the two parameters are bundled together i e the nuisance parameter is an unknown function that contains the parameter of interest as part of its argument for example in a linear regression model for censored survival data the unspecified error distribution function involves the regression coefficients motivated by developing an efficient estimating method for the regression parameters we propose a general sieve m theorem for bundled parameters and apply the theorem to deriving the asymptotic theory for the sieve maximum likelihood estimation in the linear regression model for censored survival data the numerical implementation of the proposed estimating method can be achieved through the conventional gradient based search algorithms such as the newton raphson algorithm we show that the proposed estimator is consistent and asymptotically normal and achieves the semiparametric efficiency bound simulation studies demonstrate that the proposed method performs well in practical settings and yields more efficient estimates than existing estimating equation based methods illustration with a real data example is also provided

conditional transformation models the ultimate goal of regression analysis is to obtain information about the conditional distribution of a response given a set of explanatory variables this goal is however seldom achieved because most established regression models only estimate the conditional mean as a function of the explanatory variables and assume that higher moments are not aected by the regressors the underlying reason for such a restriction is the assumption of additivity of signal and noise we propose to relax this common assumption in the framework of transformation models the novel class of semiparametric regression models proposed herein allows transformation functions to depend on explanatory variables these transformation functions are estimated by regularised optimisation of scoring rules for probabilistic forecasts e g the continuous ranked probability score the corresponding estimated conditional distribution functions are consistent based on applications from dierent domains we show that these conditional transformation models are useful for describing possible heteroscedasticity comparing spatially varying distributions identifying extreme events deriving prediction intervals and selecting variables beyond mean regression eects an empirical investigation based on a heteroscedastic varying coecient simulation model demonstrates that semiparametric estimation of conditional distribution functions can be more benecial than kernel based non parametric approaches or parametric generalised additive models for location scale and shape

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

bayesian analysis of proportional hazards models built from monotone functions we consider the usual proportional hazards model in the case where the baseline hazard the covariate link and the covariate coefficients are all unknown both the baseline hazard and the covariate link are monotone functions and thus are characterized using a dense class of such functions which arises upon transformation as a mixture of beta distribution functions we take a bayesian approach for fitting such a model since interest focuses more upon the likelihood we consider vague prior specifications including jeffreys s prior computations are carried out using sampling based methods model criticism is also discussed finally a data set studying survival of a sample of lung cancer patients is analyzed

power transformed linear quantile regression with censored data we propose a class of power transformed linear quantile regression models for survival data subject to random censoring the estimation procedure follows two sequential steps first for a given transformation parameter we can easily obtain the estimates for the regression coefficients by minimizing a well defined convex objective function second we can estimate the transformation parameter based on a model discrepancy measure by constructing cumulative sum processes we show that both the regression and transformation parameter estimates are strongly consistent and asymptotically normal the variance covariance matrix depends on the unknown density function of the error term so we estimate the variance by the usual bootstrap approach we examine the performance of the proposed method for finite sample sizes through simulation studies and illustrate it with a real data example

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 generalized additive regression model for survival times we present a non parametric survival model withtwo time scales the time scales are equivalent up to a constant that varies over the subjects covariate effects are modelled linearly on eachtime scale by additive aalen models estimators of the cumulative intensities on the two time scales are suggested by solving approximate local maximum likelihood estimating equations the local estimating equations necessitate only the choice of one bandwidth the estimators are provided with large sample properties the model is applied to data on patients with myocardial infarction and used to describe the prognostic effect of covariates on the two time scales time since myocardial infarction and age 1 introduction in many bio medical applications in survival analysis it is of interest to study the effect of covariates on various time scales we consider the situation where multiple time scales are involved and focus on the specific situation withtwo time scales that are equivalent up to a constant for eachindividual suchas for example follow up time and age a class of models where it is relevant to consider multiple time scales is the the three state model known as the illness death model or the disability model where the additional time scale may be duration in the illness state of the model see keiding 1991 for a general discussion of these models oakes 1995 discussed how multiple time scales may be combined into a single scale previous work has considered semi parametric survival models where one time scale is modelled parametrically and the other time scale is considered non parametric an example of this type of analysis with three time scales applied to diabetes patients can be found in ramlau hansen et al 1987 in th is paper we present a non parametric regression approachwithtwo time scales where each time scale contribute additively to the mortality the effect of covariates are modelled by additive aalen models on eachtime scale aalen 1980 1989 1993 mckeague 1988 huffer and mckeague 1991 this allows covariates to have effects that vary on two different time scales in a motivating example we consider patients that experience myocardial infarction and aim at predicting the intensity considering the two time scales age and time since myocardial infarction as an example one of the covariates describes the heart function and is allowed to have an effect that varies

local partial likelihood estimation for lifetime data this paper considers a proportional hazards model which allows one to examine the extent to which covariates interact nonlinearly with an exposure variable for analysis of lifetime data a local partial likelihood technique is proposed to estimate nonlinear interactions asymptotic normality of the proposed estimator is established the baseline hazard function the bias and the variance of the local likelihood estimator are consistently estimated in addition a one step local partial likelihood estimator is presented to facilitate the computation of the proposed procedure and is demonstrated to be as efficient as the fully iterated local partial likelihood estimator furthermore a penalized local likelihood estimator is proposed to select important risk variables in the model numerical examples are used to illustrate the effectiveness of the proposed procedures

partially linear structure selection in cox models with varying coefficients summary to explore the nonlinear interactions between covariates and an index variable partially linear proportional hazards models have been proposed for censored survival data however specification of the partially linear structure was usually carried out in an ad hoc manner by first fitting a full varying coefficient model and visually examining the resulting fit to identify the linear part in this article we consider the problem of coefficient estimation and constant coefficient identification based on a double shrinkage approach variable selection is also considered in a coherent estimation framework resulting in a double penalization procedure under the mild assumptions we establish asymptotic properties for the procedure such as consistency sparesistency constansistency and asymptotic normality we evaluate the performance of the proposed method by numerical simulations and demonstrate its application using a breast cancer data set

sieve maximum likelihood estimator for semiparametric regression models with current status data in a randomized controlled clinical trial study where the response variable of interest is the time to occurrence of a certain event it is often too expensive or even impossible to observe the exact time however the current status of the subject at a random time of inspection is much more natural feasible and practical in terms of cost effectiveness this article considers a semiparametric regression model that consists of parametric and nonparametric regression components a sieve maximum likelihood estimator mle is proposed to estimate the regression parameter allowing exploration of the nonlinear relationship between a certain covariate and the response function asymptotic properties of the proposed sieve mles are discussed under some mild conditions the estimators are shown to be strongly consistent moreover the estimators of the unknown parameters are asymptotically efficient and normally distributed and the estimator of the nonparametric function has an optimal convergence rate simula

information bounds for the conditional hazard ratio in a nested family of regression models information bounds are obtained for a parametric treatment effect in a nested family of models for the conditional distribution of survival given confounding variables the family extends from the fully parametric weibull model to a completely nonparametric model for right censored data and includes the cox model and the stratified version of it it is shown that in certain special cases there is no loss of information about the treatment effect when going from a fully parametric to a nonparametric specification of the conditional hazard function

a semi parametric threshold regression analysis of sexually transmitted infections in adolescent women time to event analysis of sexually transmitted infection data is often complicated by the existence of nonproportional hazards and nonlinear independent variable effects methods without the proportional hazards assumption such as threshold regression models have been successfully used in many applications this paper seeks to extend the existing threshold regression models to accommodate the nonlinear independent variable effects specifically we incorporated penalized and regression splines to the threshold regression models for added modeling flexibility cross validation methods were used for the selection of the number of knots and for the determination of smoothing parameters variance estimates were proposed for inference purposes simulation results showed that the proposed methods were able to achieve nonparametric function and parametric coefficient estimates that are close to their true values simulation also demonstrated satisfactory performance of variance estimates using the proposed methods we analyzed time from sexual debut to the first infection with chlamydia trachomatis infection in a group of young women analysis shows that the lifetime number of sexual partners has a nonlinear effect on the risk of c trachomatis infection and the infection risks were differential by ethnicity and age of sexual debut

generalized julia sets from a non analytic complex mapping abstract the method constructing the julia sets from a simple non analytic complex mapping developed by michelitsch and rossler was expanded according to the complex mapping expanded by the author a series of the generalized julia sets for real index number were constructed using the experimental mathematics method combining the theory of analytic function of one complex variable with computer aided drawing the fractal features and evolutions of the generalized julia sets are studied the results show 1 the geometry structure of the generalized julia sets depends on the parameters r and c 2 the generalized julia sets have symmetry and fractal feature 3 the generalized julia sets for decimal index number have discontinuity and collapse and their evolutions depend on the choice of the principal range of the phase angle

smoothed cox regression nonparametric regression was shown by beran and mckeague and utikal to provide a flexible method for analysis of censored failure times and more general counting processes models in the presence of covariates we discuss application of kernel smoothing towards estimation in a generalized cox regression model with baseline intensity dependent on a covariate under regularity conditions we show that estimates of the regression parameters are asymptotically normal at rate root n and we also discuss estimation of the baseline cumulative hazard function and related parameters

testing goodness of fit for proportional hazards model with censored observations abstract a numerical omnibus test of fit for the two sample proportional hazards model is proposed for randomly censored observations the test is derived from cox s partial likelihood and does not need a dummy time dependent covariate or any partition of the time axis consistency of the test is established examples are provided for illustration

semiparametric frailty models for clustered failure time data summary we consider frailty models with additive semiparametric covariate effects for clustered failure time data we propose a doubly penalized partial likelihood dppl procedure to estimate the nonparametric functions using smoothing splines we show that the dppl estimators could be obtained from fitting an augmented working frailty model with parametric covariate effects whereas the nonparametric functions being estimated as linear combinations of fixed and random effects and the smoothing parameters being estimated as extra variance components this approach allows us to conveniently estimate all model components within a unified frailty model framework we evaluate the finite sample performance of the proposed method via a simulation study and apply the method to analyze data from a study of sexually transmitted infections sti

analysis of transformation models with censored data summary in this paper we consider a class of semi parametric transformation models under which an unknown transformation of the survival time is linearly related to the covariates with various completely specified error distributions this class of regression models includes the proportional hazards and proportional odds models inference procedures derived from a class of generalised estimating equations are proposed to examine the covariate effects with censored observations numerical studies are conducted to investigate the properties of our proposals for practical sample sizes these transformation models coupled with the new simple inference procedures provide many useful alternatives to the cox regression model in survival analysis

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

smooth backfitting of proportional hazards with multiplicative components smooth backfitting has proven to have a number of theoretical and practical advantages in structured regression by projecting the data down onto the structured space of interest smooth backfitting

the dimensionality reduction principle for generalized additive models soit x y une paire de variables aleatoires telles que x x 1 x j s etende sur c 0 1 j la distribution conditionnelle de y etant donne x x est supposee appartenir a une famille exponentielle appropriee ayant pour parametre r f x denote la dependance de vis a vis de x et f l approximation additive de f ayant la log vraisemblance esperee maximum possible