orbit automata as a new tool to attack the order problem in automaton groups abstract we introduce a new tool called the orbit automaton that describes the action of an automaton group g on the subtrees corresponding to the orbits of g on levels of the tree the connection between g and the groups generated by the orbit automata is used to find elements of infinite order in certain automaton groups for which other methods failed to work

orbit decidability and the conjugacy problem for some extensions of groups given a short exact sequence of groups with certain conditions 1 f g h 1 we prove that g has solvable conjugacy problem if and only if the corresponding action subgroup a 6 aut f is orbit decidable from this we deduce that the conjugacy problem is solvable among others for all groups of the form z2 fm f2 fm fn z and zn a fm with virtually solvable action group a 6 gln z also we give an easy way of constructing groups of the form z4 fn and f3 fn with unsolvable conjugacy problem on the way we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem as an application an alternative solution to the conjugacy problem in aut f2 is given

improving generalised estimating equations using quadratic inference functions summary generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified however under such misspecification the estimator of the regression parameter can be inefficient in this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter and that remains optimal even if the working correlation structure is misspecified the idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices a representation that is valid for the working correlations most commonly used both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations this approach also provides a chi squared inference function for testing nested models and a chi squared regression misspecification test furthermore the test statistic follows a chi squared distribution asymptotically whether or not the working correlation structure is correctly specified

the conjugacy problem in automaton groups is not solvable abstract free abelian by free self similar groups generated by finite self similar sets of tree automorphisms and having unsolvable conjugacy problem are constructed along the way finitely generated orbit undecidable free subgroups of gl d z for d 6 and aut f d for d 5 are constructed as well

power analyses for longitudinal study designs with missing data existing methods for power analysis for longitudinal study designs are limited in that they do not adequately address random missing data patterns although the pattern of missing data can be assessed during data analysis it is unknown during the design phase of a study the random nature of the missing data pattern adds another layer of complexity in addressing missing data for power analysis in this paper we model the occurrence of missing data with a two state first order markov process and integrate the modelling information into the power function to account for random missing data patterns the markov model is easily specified to accommodate different anticipated missing data processes we develop this approach for the two most popular longitudinal models the generalized estimating equations gee and the linear mixed effects model under the missing completely at random mcar assumption for gee we also limit our consideration to the working independence correlation model the proposed methodology is illustrated with numerous examples that are motivated by real study designs

existence uniqueness and comparisons for bsdes in general spaces we present a theory of backward stochastic differential equations in continuous time with an arbitrary filtered probability space no assumptions are made regarding the left continuity of the filtration of the predictable quadratic variations of martingales or of the measure integrating the driver we present conditions for existence and uniqueness of square integrable solutions using lipschitz continuity of the driver these conditions unite the requirements for existence in continuous and discrete time and allow discrete processes to be embedded with continuous ones we also present conditions for a comparison theorem and hence construct time consistent nonlinear expectations in these general spaces

tests of homogeneity for generalized linear models abstract we propose two tests for testing homogeneity among clustered data adjusting for the effects of covariates the first is a score test for a generalized linear model with random effect in which the distribution of the response variable given the random effect is entirely defined in contrast to the likelihood ratio test however the score test does not require estimation of the parameters of a mixed effects model nor specification of the mixing distribution the second test is proposed in the framework of the generalized estimating equation gee approach in deriving this test we need only the specification of the marginal expectation and variance of the response variable and the fourth moment for the overdispersion term whereas for deriving the score test for mixed effects models the entire conditional distribution must be specified we demonstrate that the two tests are identical when the covariance matrix assumed in the gee approach is that of the random effects model in both approaches

estimating a multivariate familial correlation using joint models for canonical correlations application to memory score analysis from familial hispanic alzheimer s disease study analysis of multiple traits can provide additional information beyond analysis of a single trait allowing better understanding of the underlying genetic mechanism of a common disease to accommodate multiple traits in familial correlation analysis adjusting for confounders we develop a regression model for canonical correlation parameters and propose joint modeling along with mean and scale parameters the proposed method is more powerful than the regression method modeling pairwise correlations because it captures familial aggregation manifested in multiple traits through maximum canonical correlation

mean field backward stochastic differential equations in general probability spaces abstract in this paper we deal with a class of mean field backward stochastic differential equations in continuous time with an arbitrary filtered probability space we prove the existence and uniqueness of a solution for those equations with strengthened lipschitz assumption a comparison theorem is also established

on linear and quadratic estimating functions summary in some cases maximum quasi likelihood estimation which is at the core of glim can fail to give reasonable results a more general class of estimating functions is investigated which avoids such failure and also is not restricted to the particular forms of mean and variance function of glim

estimating equations for parameters in means and covariances of multivariate discrete and continuous responses generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response these equations are to be solved jointly with score equations from a generalized linear model for mean parameters a class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion and proposals for the use of such equations are developed comments on the relative merits of the ad hoc and model based approaches to estimation are given and a regression illustration with a bivariate response is provided

incorporating marginal covariate information in a nonparametric regression model for a sample of r x c tables summary nonparametric regression models are proposed in the framework of ecological inference for exploratory modeling of disease prevalence rates adjusted for variables such as age ethnicity race and socio economic status ecological inference is needed when a response variable and covariate are not available at the subject level because only summary statistics are available for the reporting unit for example in the form of r x c tables in this article only the marginal counts are assumed available in the sample of r x c contingency tables for modeling the joint distribution of counts a general form for the ecological regression model is proposed whereby certain covariates are included as a varying coefficient regression model whereas others are included as a functional linear model the nonparametric regression curves are modeled as splines fit by penalized weighted least squares a data driven selection of the smoothing parameter is proposed using the pointwise maximum squared bias computed from averaging kernels explained by o sullivan 1986 statistical science 1 502 517 analytic expressions for bias and variance are provided that could be used to study the rates of convergence of the estimators instead this article focuses on demonstrating the utility of the estimators in a study of disparity in health outcomes by ethnicity race

macroscopic limits for stochastic partial differential equations of mckean vlasov type a class of quasilinear stochastic partial differential equations spdes driven by spatially correlated brownian noise is shown to become macroscopic i e deterministic as the length of the correlations tends to 0 the limit is the solution of a quasilinear partial differential equation the quasilinear spdes are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations sodes coupled though a mean field interaction and driven by correlated brownian noise the limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes we also compare our approach to sodes with the one introduced by kunita

a combined score test for binary and ordinal endpoints from clinical trials there is growing interest especially for trials in stroke in combining multiple endpoints in a single clinical evaluation of an experimental treatment the endpoints might be repeated evaluations of the same characteristic or alternative measures of progress on different scales often they will be binary or ordinal and those are the cases studied here in this paper we take a direct approach to combining the univariate score statistics for comparing treatments with respect to each endpoint the correlations between the score statistics are derived and used to allow a valid combined score test to be applied a sample size formula is deduced and application in sequential designs is discussed the method is compared with an alternative approach based on generalized estimating equations in an illustrative analysis and replicated simulations and the advantages and disadvantages of the two approaches are discussed

zero sum stochastic differential games and backward equations abstract in this paper we are concerned with the applications to stochastic zero sum differential games of the results on backward stochastic differential equations using these techniques the existence of a saddle point in the bounded case is obtained if the isaacs condition holds furthermore this technique is also a very simple approach for the existence of an optimal strategy in the case of controlled diffusions

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

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

computational methods for multiplicative intensity models using weighted gamma processes proportional hazards marked point processes and panel count data we develop computational procedures for a class of bayesian nonparametric and semiparametric multiplicative intensity models incorporating kernel mixtures of spatial weighted gamma measures a key feature of our approach is that explicit expressions for posterior distributions of these models share many common structural features with the posterior distributions of bayesian hierarchical models using the dirichlet process using this fact along with an approximation for the weighted gamma process we show that with some care one can adapt efficient algorithms used for the dirichlet process to this setting we discuss blocked gibbs sampling procedures and polya urn gibbs samplers we illustrate our methods with applications to proportional hazard models poisson spatial regression models recurrent events and panel count data

analyzing bivariate ordinal data using a global odds ratio abstract a moment method is proposed for regression analysis of bivariate ordered categorical data using the global odds ratio as the measure of association for modeling the margins this method utilizes the stochastic ordering implicit in the data this method allows for covariate effects in modeling the association between responses an application of the proposed model is illustrated using the ophthalmological data from the wisconsin epidemiologic study of diabetic retinopathy for identifying risk factors among younger onset diabetics this model is also extended to a longitudinal data setting with more than two repeated measures

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

a note on extended quasi likelihood abstract we study the method of extended quasi likelihood estimation and inference of a variance function recently proposed by nelder pregibon 1987 the estimates are inconsistent in general and the test levels can be biased but in many cases such as the exponential family the inconsistency and bias will not be a major concern extended quasi likelihood is compared with carroll ruppert s 1982 pseudo likelihood method which gives consistent estimates and when slightly modified asymptotically unbiased tests we quantify the showing in this instance that the two estimates are closely related and may be asymptotically equivalent in many important cases however in some cases outside the exponential family an asymptotic bias can persist heteroscedastic regression model

on the reliability and precision of within and between population estimates of relative rate parameters comparisons of individual and aggregate level analyses of data from a multigroup observational study are made using an exponential form relative rate model stratified analytic random effects and aggregate random effects analyses are studied estimating equations are developed to give a consistent estimation procedure across analyses and corresponding information matrices are compared simulation studies provide insight into the efficiency and bias of relative rate parameter estimates with respect to covariate dispersion confounding and covariate measurement error

mean field backward stochastic differential equations a limit approach mathematical mean field approaches play an important role in different fields of physics and chemistry but have found in recent works also their application in economics finance and game theory the objective of our paper is to investigate a special mean field problem in a purely stochastic approach for the solution y z of a mean field backward stochastic differential equation driven by a forward stochastic differential of mckean vlasov type with solution x we study a special approximation by the solution x n y n z n of some decoupled forward backward equation which coefficients are governed by n independent copies of x n y n z n we show that the convergence speed of this approximation is of order 1 root n moreover our special choice of the approximation allows to characterize the limit behavior of root n x n x y n y z n z we prove that this triplet converges in law to the solution of some forward backward stochastic differential equation of mean field type which is not only governed by a brownian motion but also by an independent gaussian field

the combination of ecological and case control data ecological studies in which data are available at the level of the group rather than at the level of the individual are susceptible to a range of biases due to their inability to characterize within group variability in exposures and confounders in order to overcome these biases we propose a hybrid design in which ecological data are supplemented with a sample of individual level case control data we develop the likelihood for this design and illustrate its benefits via simulation both in bias reduction when compared to an ecological study and in efficiency gains relative to a conventional case control study an interesting special case of the proposed design is the situation where ecological data are supplemented with case only data the design is illustrated using a dataset of county specific lung cancer mortality rates in the state of ohio from 1988

aggregate data studies of disease risk factors summary statistical methods are proposed for estimating relative rate parameters based on estimated disease rates and covariate data from random samples of individuals from each of several cohorts a random effects model is used to derive mean and variance models for estimated disease rates estimating equations for relative rate parameters are then developed by replacing cohort covariate averages by corresponding sample averages the asymptotic distribution of regression parameter estimates is derived and the asymptotic bias is shown to be small even if covariates are contaminated by classical random measurement errors provided the covariate sample size in each cohort is not small simulation studies motivated by international data on diet and breast cancer provide insights into the properties of the proposed estimators

nonlinear diffusion governed by mckean vlasov equation on hilbert space and optimal control this paper deals with optimal feedback control of measure valued solutions of nonlinear diffusion governed by mckean vlasov equations questions of existence uniqueness and regularity properties of measure valued solutions are addressed a class of feedback controls furnished with a weak topology is introduced and some important topological properties of the attainable set corresponding to these controls are presented we consider several typical control problems with objective functionals which are functions of measures and prove the existence of optimal controls

on the accommodation of disease rate correlations in aggregate data studies of disease risk factors prentice and sheppard 1995 biometrika 82 113 125 proposed a method for estimating relative risks associated with poorly measured exposures using disease rates from multiple populations and exposure and confounding factor data from sample surveys of persons in each population the method involved an assumption of independence of disease rates across populations conditional on exposures and confounding factors here this assumption is relaxed by allowing dependencies among the disease rates within a partition of the populations perhaps defined on the basis of geographic proximity or cultural similarity such dependencies could for example derive from unmeasured risk factors that are shared among neighboring populations dependencies within an element of the partition are modeled by allowing a common correlation between the disease rates of contiguous populations and by inducing correlations between the rates for noncontiguous populations using a multivariate lognormal assumption estimating equations are proposed for relative risk parameter estimation and robustness and efficiency properties are assessed through simulations the relaxed estimation procedure is shown to yield useful efficiency gains when moderate or strong disease rate dependencies are present especially when disease rate variances are large relative to binomial variance

freeness of automaton groups vs boundary dynamics abstract we prove that the boundary dynamics of the semi group generated by the enriched dual transducer characterizes the algebraic property of being free for an automaton group we specialize this result to the class of bireversible transducers and we show that the property of being not free is equivalent to the existence of a finite schreier graph in the boundary of the enriched dual pointed at some essentially non trivial point from these results we derive some consequences from the algebraic algorithmic and dynamical points of view

on the burnside problem for periodic groups narain gupta 1 and said sidki 2 1 university of manitoba department of mathematics winnipeg manitoba r3t 2n2 canada 2 university of brasilia department of mathematics brasilia d f brazil introduction the generalized burnside problem refers to the question are finitely generated periodic groups finite this was answered in the negative by golod 1 who proved that for each prime p there exists a finitely generated infinite p group golod s construction in not however direct and is based on his celebrated work with safarevi recently grigor uk 2 has given a direct and elegant construction of an infinite 2 group which is generated by three elements of order 2 in this paper we give for each odd prime p a direct construction of an infinite p group on two generators each of order p our group is a subgroup of the automorphism group of a regular tree of degree p and as might be expected it is residually finite and has infinite exponent preliminaries let p be an odd prime and let t 0 be the infinite regular tree of degree p with vertex 0 so that through each vertex u of t 0 there are p regular subtrees

bivariate modelling of clustered continuous and ordered categorical outcomes simultaneous observation of continuous and ordered categorical outcomes for each subject is common in biomedical research but multivariate analysis of the data is complicated by the multiple data types here we construct a model for the joint distribution of bivariate continuous and ordinal outcomes by applying the concept of latent variables to a multivariate normal distribution the approach is then extended to allow for clustering of the bivariate outcomes the model can be parameterized in a way that allows writing the joint distribution as a product of a standard random effects model for the continuous variable and a correlated cumulative probit model for the ordinal outcome this factorization suggests convenient parameter estimation using estimating equations foetal weight and malformation data from a developmental toxicity experiment illustrate the results

an automaton group with undecidable order and engel problems abstract for every turing machine we construct an automaton group that simulates it precisely starting from an initial configuration of the turing machine we explicitly construct an element of the group such that the turing machine stops if and only if this element is of finite order if the turing machine is universal the corresponding automaton group has an undecidable order problem this solves a problem raised by grigorchuk the above group also has an undecidable engel problem there is no algorithm that given g h in the group decides whether there exists an integer n such that the n iterated commutator g h h h is the identity or not this solves a problem raised by bartholdi

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

surprising effects of measurement error on an aggregate data estimator summary in a generalised linear model with a single normally distributed covariate for the most part the effect of normally distributed additive measurement error is attenuation i e asymptotic bias towards the null prentice sheppard 1995 suggested a marginalised random effects approach to combining the results of different studies on binary outcomes we show that in probit regression when the number of observations per study is large under the stated normality assumptions attenuation never occurs in fact the asymptotic bias is away from the null this appears to be the first known case under reasonable distributional assumptions that the effect of measurement error is reverse attenuation

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

gaussian estimation for correlated binomial data summary the mean variance structure of correlated binomial data is discussed and whittle s 1961 gaussian estimation is suggested as a useful general method in this context comparisons are made with other monrent methods in current use