one dimensional space discrete transport subject to levy perturbations in this paper we study a one dimensional space discrete transport equation subject to additive levy forcing the explicit form of the solutions allows their analytic study in particular we discuss the invariance of the covariance structure of the stationary distribution for levy perturbations with finite second moment the situation of more general levy perturbations lacking the second moment is considered as well we moreover show that some of the properties of the solutions are pertinent to a discrete system and are not reproduced by its continuous analogue

rank based inference for the accelerated failure time model summary a broad class of rank based monotone estimating functions is developed for the semiparametric accelerated failure time model with censored observations the corresponding estimators can be obtained via linear programming and are shown to be consistent and asymptotically normal the limiting covariance matrices can be estimated by a resampling technique which does not involve nonparametric density estimation or numerical derivatives the new estimators represent consistent roots of the non monotone estimating equations based on the familiar weighted log rank statistics simulation studies demonstrate that the proposed methods perform well in practical settings two real examples are provided

a simple resampling method by perturbing the minimand suppose that under a semiparametric setting an estimator of a vector of parameters of interest is obtained by optimising an objective function which has a u process structure the covariance matrix of the estimator is generally a function of the underlying density function which may be difficult to estimate well by conventional methods in this paper we present a simple resampling method by perturbing the objective function repeatedly inferences of the parameters can then be made based on a large collection of the resulting optimisers we illustrate our proposal by three examples with a heteroscedastic regression model

weighted rank regression for clustered data analysis we consider ranked based regression models for clustered data analysis a weighted wilcoxon rank method is proposed to take account of within cluster correlations and varying cluster sizes the asymptotic normality of the resulting estimators is established a method to estimate covariance of the estimators is also given which can bypass estimation of the density function simulation studies are carried out to compare different estimators for a number of scenarios on the correlation structure presence absence of outliers and different correlation values the proposed methods appear to perform well in particular the one incorporating the correlation in the weighting achieves the highest efficiency and robustness against misspecification of correlation structure and outliers a real example is provided for illustration

bias analysis and simex approach in generalized linear mixed measurement error models abstract we consider generalized linear mixed models glmms for clustered data when one of the predictors is measured with error when the measurement error is additive and normally distributed and the error prone predictor is itself normally distributed we show that the observed data also follow a glmm but with a different fixed effects structure from the original model a different and more complex random effects structure and restrictions on the parameters this characterization enables us to compute the biases that result in common glmms when one ignores measurement error for instance in one common situation the biases in parameter estimates become larger as the number of observations within a cluster increases both for regression coefficients and for variance components parameter estimation is described using the simex method a relatively new functional method that makes no assumptions about the structure of the unobservable predictors simulations and an example illustrate the results

estimation in covariance components models abstract estimation techniques for linear covariance components models are developed and illustrated with special emphasis on explaining computational processes the estimation of fixed and random effects when the variances and covariances are known is presented in bayesian terms point estimates of the unknown variances and covariances are computed using the em algorithm for maximum likelihood estimation from incomplete data the techniques are illustrated with data on law schools field mice and professional football teams

simple systems with anomalous dissipation and energy cascade we analyze a class of dynamical systems of the type dot a n t c n 1 a n 1 t c n a n 1 t f n t n epsilon mathbb n a 0 0 where fn t is a forcing term with f n t not 0 only for n le n star infty and the coupling coefficients cn satisfy a condition ensuring the formal conservation of energy frac12 sum n a n t 2 despite being formally conservative we show that these dynamical systems support dissipative solutions suitably defined and as a result may admit unique statistical steady states when the forcing term fn t is nonzero this claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients cn the mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n this is responsible for solutions with interesting energy spectra namely mathbb e a n 2 scales as n alpha as n here the exponents depend on the coupling coefficients cn and mathbb e denotes expectation with respect to the equilibrium measure this is reminiscent of the conjectured properties of the solutions of the navier stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence hence these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence similar in that respect to the random passive scalar or random burgers equation but even simpler and fully solvable

repeated probit regression when covariates are measured with error this paper develops a model for repeated binary regression when a covariate is measured with error the model allows for estimating the effect of the true value of the covariate on a repeated binary response the choice of a probit link for the effect of the error free covariate coupled with normal measurement error for the error free covariate results in a probit model after integrating over the measurement error distribution we propose a two stage estimation procedure where in the first stage a linear mixed model is used to fit the repeated covariate in the second stage a model for the correlated binary responses conditional on the linear mixed model estimates is fit to the repeated binary data using generalized estimating equations the approach is demonstrated using nutrient safety data from the diet intervention of school age children disc study

anomalous dissipation in a stochastically forced infinite dimensional system of coupled oscillators we study a system of stochastically forced infinite dimensional coupled harmonic oscillators although this system formally conserves energy and is not explicitly dissipative we show that it has a nontrivial invariant probability measure this phenomenon which has no finite dimensional equivalent is due to the appearance of some anomalous dissipation mechanism which transports energy to infinity this prevents the energy from building up locally and allows the system to converge to the invariant measure the invariant measure is constructed explicitly and some of its properties are analyzed

a monte carlo implementation of the em algorithm and the poor man s data augmentation algorithms abstract the first part of this article presents the monte carlo implementation of the e step of the em algorithm given the current guess to the maximizer of the posterior distribution latent data patterns are generated from the conditional predictive distribution the expected value of the augmented log posterior is then updated as a mixture of augmented log posteriors mixed over the generated latent data patterns multiple imputations in the m step of the algorithm this mixture is maximized to obtain the update to the maximizer of the observed posterior the gradient and hessian of the observed log posterior are also expressed as mixtures mixed over the multiple imputations the relation between the monte carlo em mcem algorithm and the data augmentation algorithm is noted two modifications to the mcem algorithm the poor man s data augmentation algorithms which allow for the calculation of the entire posterior are then presented these approximations serve as diagnostics for the validity o

rank based estimating equations with general weight for accelerated failure time models an induced smoothing approach the induced smoothing technique overcomes the difficulties caused by the non smoothness in rank based estimating functions for accelerated failure time models but it is only natural when the estimating function has gehan s weight for a general weight the induced smoothing method does not provide smooth estimating functions that can be easily evaluated we propose an iterative induced smoothing procedure for general weights with the estimator from gehan s weight initial value the resulting estimators have the same asymptotic properties as those from the non smooth estimating equations with the same weight their variances are estimated with an efficient resampling approach that avoids solving estimating equations repeatedly the methodology is generalized to incorporate an additional weight to accommodate missing data and various sampling schemes in a numerical study the proposed estimators were obtained much faster without losing accuracy in comparison to those from non smooth estimating equations and the variance estimators provided good approximation of the variation in estimation the methodology was applied to two real datasets the first one from an adolescent depression study and the second one from a cancer study with missing covariates by design the implementation is available in an r package aftgee

induced smoothing for rank regression with censored survival times adaptions of weighted rank regression to the accelerated failure time model for censored survival data have been successful in yielding asymptotically normal estimates and flexible weighting schemes to increase statistical efficiencies however for only one simple weighting scheme gehan or wilcoxon weights are estimating equations guaranteed to be monotone in parameter components and even in this case are step functions requiring the equivalent of linear programming for computation the lack of smoothness makes standard error or covariance matrix estimation even more difficult an induced smoothing technique overcame these difficulties in various problems involving monotone but pure jump estimating equations including conventional rank regression the present paper applies induced smoothing to the gehan wilcoxon weighted rank regression for the accelerated failure time model for the more difficult case of survival time data subject to censoring where the inapplicability of permutation arguments necessitates a new method of estimating null variance of estimating functions smooth monotone parameter estimation and rapid reliable standard error or covariance matrix estimation is obtained

mixed effects models with censored data with application to hiv rna levels summary mixed effects models are often used for estimating fixed effects and variance components in longitudinal studies of continuous data when the outcome being modelled is a laboratory measurement however it may be subject to lower and upper detection limits i e censoring in this paper the usual em estimation procedure for mixed effects models is modified to account for left and or right censoring

censored observations repeated measures and mixed effects models an approach using the em algorithm and normal errors summary maximum likelihood estimation of a vector regression parameter and variance com ponents is considered for the mixed effects model when observations are right censored a general scheme of estimation is given using the em algorithm and detailed results found for the model with between and within block variation this model is applied to the logarithms of survival times from a repeated measures design in this paper we consider the estimation of the parameters when some of the observa tions are right censored this can occur with repeated measurements on subjects where the response is a waiting time and the variability in log times is modelled by a normal distribution alternative models for this situation are given by clayton cuzick 1985 and their methods involve partially parametric techniques using ranks and are based on models involving extreme value log gamma and logistic distributions we are concerned with use of the em algorithm dempster laird rubin 1977 to estimate the parameters of the mixed model for uncensored data hartley rao 1967 have considered this solution the analysis leads to a method which is straightforward to implement for the repeated measures model dempster et al 1984 give an account of the method with uncensored observations in 2 we consider a general implementation of the em algorithm for the pure random effects model with censored data and then in 3 we give explicit results for the simple random effects model section 4 uses these results for the analysis of repeated measures and 5 discusses some miscellaneous points arising finally in 6 we give an example involving skin graft data where there has been extensive analysis using matched pairs techniques

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

quantile association regression models it is often important to study the association between two continuous variables in this work we propose a novel regression framework for assessing conditional associations on quantiles we develop general methodology which permits covariate effects on both the marginal quantile models for the two variables and their quantile associations the proposed quantile copula models have straightforward interpretation facilitating a comprehensive view of association structure which is much richer than that based on standard product moment and rank correlations we show that the resulting estimators are uniformly consistent and weakly convergent as a process of the quantile index simple variance estimators are presented which perform well in numerical studies extensive simulations and a real data example demonstrate the practical utility of the methodology supplementary materials for this article are available online

generalized linear mixed model for binary outcomes when covariates are subject to measurement errors and detection limits longitudinal measurement of biomarkers is important in determining risk factors for binary endpoints such as infection or disease however biomarkers are subject to measurement error and some are also subject to left censoring due to a lower limit of detection statistical methods to address these issues are few we herein propose a generalized linear mixed model and estimate the model parameters using the monte carlo newton raphson mcnr method inferences regarding the parameters are made by applying louis s method and the delta method simulation studies were conducted to compare the proposed mcnr method with existing methods including the maximum likelihood ml method and the ad hoc approach of replacing the left censored values with half of the detection limit hdl the results showed that the performance of the mcnr method is superior to ml and hdl with respect to the empirical standard error as well as the coverage probability for the 95 confidence interval the hdl method uses an incorrect imputation method and the computation is constrained by the number of quadrature points while the ml method also suffers from the constrain for the number of quadrature points the mcnr method does not have this limitation and approximates the likelihood function better than the other methods the improvement of the mcnr method is further illustrated with real world data from a longitudinal study of local cervicovaginal hiv viral load and its effects on oncogenic hpv detection in hiv positive women

concordance assisted learning for estimating optimal individualized treatment regimes in this article we propose a new concordance assisted learning for estimating optimal individualized treatment regimes we first introduce a type of concordance function for prescribing treatment and propose a robust rank regression method for estimating the concordance function we then find treatment regimes up to a threshold to maximize the concordance function named prescriptive index finally within the class of treatment regimes that maximize the concordance function we find the optimal threshold to maximize the value function we establish the convergence rate and asymptotic normality of the proposed estimator for parameters in the prescriptive index an induced smoothing method is developed to estimate the asymptotic variance of the proposed estimator we also establish the n1 3 consistency of the estimated optimal threshold and its limiting distribution in addition a doubly robust estimator of parameters in the prescriptive index is developed under a class of monotonic index models the practical use and effectiveness of the proposed methodology are demonstrated by simulation studies and an application to an aids data

a copula model for repeated measurements with non ignorable non monotone missing outcome a normal copula based selection model is proposed for continuous longitudinal data with a non ignorable non monotone missing data process the normal copula is used to combine the distribution of the outcome of interest and that of the missing data indicators given the covariates parameters in the model are estimated by a pseudo likelihood method we first use the gee with a logistic link to estimate the parameters associated with the marginal distribution of the missing data indicator given the covariates assuming that covariates are always observed then we estimate other parameters by inserting the estimates from the first step into the full likelihood function a simulation study is conducted to assess the robustness of the assumed model under different missing data processes the proposed method is then applied to one example from a community cohort study to demonstrate its capability to reduce bias

standard errors and covariance matrices for smoothed rank estimators a pseudo bayesian interpretation of standard errors yields a natural induced smoothing of statistical estimating functions when applied to rank estimation the lack of smoothness which prevents standard error estimation is remedied efficiency and robustness are preserved while the smoothed estimation has excellent computational properties in particular convergence of the iterative equation for standard error is fast and standard error calculation becomes asymptotically a one step procedure this property also extends to covariance matrix calculation for rank estimates in multi parameter problems examples and some simple explanations are given

induced smoothing for the semiparametric accelerated failure time model asymptotics and extensions to clustered data summary this paper extends the induced smoothing procedure of brown wang 2006 for the semiparametric accelerated failure time model to the case of clustered failure time data the resulting procedure permits fast and accurate computation of regression parameter estimates and standard errors using simple and widely available numerical methods such as the newton raphson algorithm the regression parameter estimates are shown to be strongly consistent and asymptotically normal in addition we prove that the asymptotic distribution of the smoothed estimator coincides with that obtained without the use of smoothing this establishes a key claim of brown wang 2006 for the case of independent failure time data and also extends such results to the case of clustered data simulation results show that these smoothed estimates perform as well as those obtained using the best available methods at a fraction of the computational cost

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

a smoothing expectation and substitution algorithm for the semiparametric accelerated failure time frailty model this paper proposes an estimation procedure for the semiparametric accelerated failure time frailty model that combines smoothing with an expectation and maximization like algorithm for estimating equations the resulting algorithm permits simultaneous estimation of the regression parameter the baseline cumulative hazard and the parameter indexing a general frailty distribution we develop novel moment based estimators for the frailty parameter including a generalized method of moments estimator standard error estimates for all parameters are easily obtained using a randomly weighted bootstrap procedure for the commonly used gamma frailty distribution the proposed algorithm is very easy to implement using widely available numerical methods simulation results demonstrate that the algorithm performs very well in this setting we re analyz several previously analyzed data sets for illustrative purposes copyright 2012 john wiley sons ltd

maximizing generalized linear mixed model likelihoods with an automated monte carlo em algorithm summary two new implementations of the em algorithm are proposed for maximum likelihood fitting of generalized linear mixed models both methods use random independent and identically distributed sampling to construct monte carlo approximations at the e step one approach involves generating random samples from the exact conditional distribution of the random effects given the data by rejection sampling using the marginal distribution as a candidate the second method uses a multivariate t importance sampling approximation in many applications the two methods are complementary rejection sampling is more efficient when sample sizes are small whereas importance sampling is better with larger sample sizes monte carlo approximation using random samples allows the monte carlo error at each iteration to be assessed by using standard central limit theory combined with taylor series methods specifically we construct a sandwich variance estimate for the maximizer at each approximate e step this suggests a rule for automatically increasing the monte carlo sample size after iterations in which the true em step is swamped by monte carlo error in contrast techniques for assessing monte carlo error have not been developed for use with alternative implementations of monte carlo em algorithms utilizing markov chain monte carlo e step approximations three different data sets including the infamous salamander data of mccullagh and nelder are used to illustrate the techniques and to compare them with the alternatives the results show that the methods proposed can be considerably more efficient than those based on markov chain monte carlo algorithms however the methods proposed may break down when the intractable integrals in the likelihood function are of high dimension

maximum likelihood algorithms for generalized linear mixed models abstract maximum likelihood algorithms are described for generalized linear mixed models i show how to construct a monte carlo version of the em algorithm propose a monte carlo newton raphson algorithm and evaluate and improve the use of importance sampling ideas calculation of the maximum likelihood estimates is feasible for a wide variety of problems where they were not previously i also use the newton raphson algorithm as a framework to compare maximum likelihood to the joint maximization or penalized quasi likelihood methods and explain why the latter can perform poorly

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

semiparametric accelerated failure time modeling for clustered failure times from stratified sampling clustered failure times often arise from studies with stratified sampling designs where it is desired to reduce both cost and sampling error semiparametric accelerated failure time aft models have not been used as frequently as cox relative risk models in such settings due to lack of efficient and reliable computing routines for inferences the challenge roots in the nonsmoothness of the rank based estimating functions and for clustered data the asymptotic properties of the estimator from the weighted version have not been available the recently proposed induced smoothing approach which provides fast and accurate rank based inferences for aft models is generalized to incorporate weights to accommodate stratified sampling designs the estimator from the induced smoothing weighted estimating equations are shown to be consistent and have the same asymptotic distribution as that from the nonsmooth version which has not been developed before the variance of the estimator is estimated by computational

monotone estimating equations for censored data the monotone class rank test based estimating equations for regression models with right censored data is considered we introduce an estimator which is a solution of a monotone estimating equation that is an extension of the gehan test the estimator is easy to derive sqrt n consistent and asymptotically normal under minimal conditions all monotone estimating equations are characterized and a simulation study which shows that our suggested procedure performs well is included

induced smoothing for rank based regression with recurrent gap time data various semiparametric regression models have recently been proposed for the analysis of gap times between consecutive recurrent events among them the semiparametric accelerated failure time aft model is especially appealing owing to its direct interpretation of covariate effects on the gap times in general estimation of the semiparametric aft model is challenging because the rank based estimating function is a nonsmooth step function as a result solutions to the estimating equations do not necessarily exist moreover the popular resampling based variance estimation for the aft model requires solving rank based estimating equations repeatedly and hence can be computationally cumbersome and unstable in this paper we extend the induced smoothing approach to the aft model for recurrent gap time data our proposed smooth estimating function permits the application of standard numerical methods for both the regression coefficients estimation and the standard error estimation large sample properties and an asymptotic variance estimator are provided for the proposed method simulation studies show that the proposed method outperforms the existing nonsmooth rank based estimating function methods in both point estimation and variance estimation the proposed method is applied to the data analysis of repeated hospitalizations for patients in the danish psychiatric center register

estimation and inference on correlations between biomarkers with repeated measures and left censoring due to minimum detection levels statistical approaches for estimating and drawing inference on the correlation between two biomarkers that are repeatedly assessed over time and subject to left censoring because minimum detection levels are lacking we propose a linear mixed effects model and estimate the parameters with the monte carlo expectation maximization mcem method inferences regarding the model parameters and the correlation between the biomarkers are performed by applying louis s method and the delta method simulation studies were conducted to compare the proposed mcem method with existing methods including the maximum likelihood estimation method the multiple imputation method and two widely used ad hoc approaches replacing the censored values with the detection limit or with half of the detection limit the results show that the performance of the mcem with respect to relative bias and coverage probability for the 95 confidence interval is superior to the detection limit and half of the detection limit approaches and exceeds that of the multiple imputation method at medium to high levels of censoring and the standard error estimates from the mcem method are close to ideal the maximum likelihood estimation method can estimate the parameters accurately however a nonpositive definite information matrix can occur so that the variances are not estimable these five methods are illustrated with data from a longitudinal human immunodeficiency virus study to estimate and draw inference on the correlation between human immunodeficiency virus rna levels measured in plasma and in cervical secretions at multiple time points copyright 2012 john wiley sons ltd

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures