asymptotic results for maximum likelihood estimators in joint analysis of repeated measurements and survival time maximum likelihood estimation has been extensively used in the joint analysis of repeated measurements and survival time however there is a lack of theoretical justification of the asymptotic properties for the maximum likelihood estimators this paper intends to fill this gap specifically we prove the consistency of the maximum likelihood estimators and derive their asymptotic distributions the maximum likelihood estimators are shown to be semiparametrically efficient

attractors of continuous difference equations abstract this paper aims at arranging the general ideas and results of the research on the nonlinear continuous difference equations and acquainting the reader with the properties of the solutions which seem unusual in the context of the theory of ordinary differential equations and have great importance in simulating sophisticated oscillatory processes such as self organization and deterministic chaos and just in those cases when it is difficult to apply ordinary differential equations

semiparametric efficient estimation for shared frailty models with doubly censored clustered data in this paper we investigate frailty models for clustered survival data that are subject to both left and right censoring termed doubly censored data this model extends current survival literature by broadening the application of frailty models from right censoring to a more complicated situation with additional left censoring our approach is motivated by a recent hepatitis b study where the sample consists of families we adopt a likelihood approach that aims at the nonparametric maximum likelihood estimators npmle a new algorithm is proposed which not only works well for clustered data but also improve over existing algorithm for independent and doubly censored data a special case when the frailty variable is a constant equal to one this special case is well known to be a computational challenge due to the left censoring feature of the data the new algorithm not only resolves this challenge but also accommodate the additional frailty variable effectively asymptotic properties of the npmle are established along with semi parametric efficiency of the npmle for the finite dimensional parameters the consistency of bootstrap estimators for the standard errors of the npmle is also discussed we conducted some simulations to illustrate the numerical performance and robustness of the proposed algorithm which is also applied to the hepatitis b data

aubry mather theory for conformally symplectic systems in this article we develop an analogue of aubry mather theory for a class of dissipative systems namely conformally symplectic systems and prove the existence of interesting invariant sets which in analogy to the conservative case will be called the aubry and the mather sets besides describing their structure and their dynamical significance we shall analyze their attracting repelling properties as well as their noteworthy role in driving the asymptotic dynamics of the system

the fibonacci unimodal map the fibonacci recurrence of the critical orbit appeared in the work of branner and hubbard on complex cubic polynomials bh 12 and in yoccoz s work yl y2 on quadratic ones as the worst pattern of recurrence on the other hand a real quadratic fibonacci map was suggested by hofbauer and keller hk as a possible candidate for a map having a wild attractor that is a set a which is the w limit set for lebesgue almost every orbit but is strictly smaller than the w limit set for a generic orbit the w limit set of the critical orbit in hk possesses all known topological properties of wild attractors compare bl2 in fact we will see below that the quadratic fibonacci map does not have a wild attractor however the corresponding question for a map with a degenerate critical point remains open actually the first indication of the fibonacci map appeared in the numerical work of tsuda t related to the belousova zhabotinskii reaction and also in numerical work of shibayama sh more precisely they studied the sequence of fibonacci bifurcations creating the fibonacci map this paper will study topological geometrical and measure theoretical properties of the real fibonacci map our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by sullivan s it turns out that the situation can be understood completely and is of quite regular nature in particular any fibonacci map with negative schwarzian and nondegenerate critical point has an absolutely continuous invariant measure so we deal with a regular type of chaotic dynamics it turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the feigenbaum map its hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter branner and hubbard introduce the concept of a tableau in order to describe recurrence of critical orbits their fibonacci tableau is a basic example which corresponds to one particularly close and regular pattern of recurrence if a complex quadratic map z 1 z2 c realizes this fibonacci tableau then the

maximum likelihood estimation in the proportional odds model abstract we consider maximum likelihood estimation of the parameters in the proportional odds model with right censored data the estimator of the regression coefficient is shown to be asymptotically normal with efficient variance the maximum likelihood estimator of the unknown monotonic transformation of the survival time converges uniformly at a parametric rate to the true transformation estimates for the standard errors of the estimated regression coefficients are obtained by differentiation of the profile likelihood and are shown to be consistent a likelihood ratio test for the regression coefficient is also considered

on the unfolding of a blowout bifurcation abstract suppose a chaotic attractor a in an invariant subspace loses stability on varying a parameter at the point of loss of stability the most positive lyapunov exponent of the natural measure on a crosses zero at what has been called a blowout bifurcation we introduce the notion of an essential basin of an attractor a this is the set of points x such that accumulation points of the sequence of measures 1 n n 1 k 0 f k x are supported on a we characterise supercritical and subcritical scenarios according to whether the lebesgue measure of the essential basin of a is positive or zero we study a drift diffusion model and a model class of piecewise linear mappings of the plane in the supercritical case we find examples where a lyapunov exponent of the branch of attractors may be positive hyperchaos or negative depending purely on the dynamics far from the invariant subspace for the mappings we find asymptotically linear scaling of lyapunov exponents average distance from the subspace and basin size on varying a parameter we conjecture that these are general characteristics of blowout bifurcations

slyrb measures natural invariant measures for chaotic systems in many applications it is useful to consider not only the set that constitutes an attractor but also if it exists the asymptotic distribution of a typical trajectory converging to the attractor indeed in the physics literature such a distribution is often assumed to exist when it exists it is called a natural invariant measure the results by lasota and yorke and by sinai ruelle and bowen represent two approaches both of which establish the existence of an invariant measure the goal of this paper is to relate the lasota yorke measure for chaotic attractors in one dimensional maps and the sinai ruelle bowen measure for chaotic attractors in higher dimensional dynamical systems we introduce the notion of slyrb measure we pronounce the term slyrb as a single word slurb the srb concept of measure can be motivated by asking how a trajectory from a typical initial point is distributed asymptotically similarly the slyrb concept of measure can be motivated by asking what the average distribution is for trajectories of a large collection of initial points in some region not necessarily restricted to a single basin the latter is analogous to ask where all the rain drops from a rain storm go and the former asks about where a single rain drop goes perhaps winding up distributed throughout a particular lake

ascent based monte carlo expectation maximization summary the expectation maximization em algorithm is a popular tool for maximizing likelihood functions in the presence of missing data unfortunately em often requires the evaluation of analytically intractable and high dimensional integrals the monte carlo em mcem algorithm is the natural extension of em that employs monte carlo methods to estimate the relevant integrals typically a very large monte carlo sample size is required to estimate these integrals within an acceptable tolerance when the algorithm is near convergence even if this sample size were known at the onset of implementation of mcem its use throughout all iterations is wasteful especially when accurate starting values are not available we propose a data driven strategy for controlling monte carlo resources in mcem the algorithm proposed improves on similar existing methods by recovering em s ascent i e likelihood increasing property with high probability being more robust to the effect of user defined inputs and handling classical monte carlo and markov chain monte carlo methods within a common framework because of the first of these properties we refer to the algorithm as ascent based mcem we apply ascent based mcem to a variety of examples including one where it is used to accelerate the convergence of deterministic em dramatically

star products and nonstandard quantization for klein gordon equation a study of star quantization of the klein gordon equation based on poincare group is presented several examples are discussed and the work is finished with some remarks on perturbation theory induced by a star product

ensemble estimation and variable selection with semiparametric regression models we consider scenarios in which the likelihood function for a semiparametric regression model factors into separate components with an efficient estimator of the regression parameter available for each component an optimal weighted combination of the component estimators named an ensemble estimator may be employed as an overall estimate of the regression parameter and may be fully efficient under uncorrelatedness conditions this approach is useful when the full likelihood function may be difficult to maximize but the components are easy to maximize it covers settings where the nuisance parameter may be estimated at different rates in the component likelihoods as a motivating example we consider proportional hazards regression with prospective doubly censored data in which the likelihood factors into a current status data likelihood and a left truncated right censored data likelihood variable selection is important in such regression modelling but the applicability of existing techniques is unclear in the ensemble approach we propose ensemble variable selection using the least squares approximation technique on the unpenalized ensemble estimator followed by ensemble re estimation under the selected model the resulting estimator has the oracle property such that the set of nonzero parameters is successfully recovered and the semiparametric efficiency bound is achieved for this parameter set simulations show that the proposed method performs well relative to alternative approaches analysis of an aids cohort study illustrates the practical utility of the method

strange attractors and asymptotic measures of discrete time dissipative systems we investigate asymptotic properties of certain discrete time dynamical systems in two and three dimensions with solenoidal attractor it is proved that the asymptotic measures relevant for the generalized version of the ergodic theorem all derive from one haar measure

star exponentials for any ordering of the elements of the inhomogeneous symplectic lie algebra we compute for any ordering the star exponentials of all polynomials of degree not greater than two on the 2l dimensional phase space of a quantum system with l degrees of freedom and we show in the particular case of the moyal star product that the weyl transform of the moyal star exponential of the one dimensional harmonic oscillator hamiltonian is the evolution operator of this quantum system

asymptotic theory for the frailty model the frailty model is a generalization of cox s proportional hazards model which includes a random effect nielsen gill andersen and sorensen 1992 proposed an em algorithm to estimate the cumulative baseline hazard and the variance of the random effect here the asymptotic distribution of the estimators is given along with a consistent estimator of the asymptotic variance

the minimal center of attraction of measurable systems and their discretizations a generalization of the birkhoff minimal center of attraction is introduced and investigated for dynamical systems generated by single and multi valued borel measurable mappings in particular it is shown to be equal to the minimal closed set containing the supports of the semi invariant measures of the system its approximation under the spatial discretization of the state space is also considered

sensitivity of dendrite maps abstract suppose that x is a dendrite and f x x is a sensitive continuous map we show that a x f contains a bilaterally transitive subsystem with nonempty interior b the system x f satisfies only one of the following two conditions b1 x f contains a topologically transitive subsystem with nonempty interior b2 there exists an f invariant nowhere dense closed subset a of x such that the attraction basin basin a f contains a residual subset b of an open set and the strong attraction basin sbasin a f is dense in b c if x is completely regular then x f contains a relatively strongly mixing subsystem with nonempty interior dense periodic points and positive topological entropy unlike for interval maps we construct a sensitive dendrite map with zero topological entropy

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

nonlinear dynamics of networks the groupoid formalism a formal theory of symmetries of networks of coupled dynamical systems stated in terms of the group of permutations of the nodes that preserve the network topology has existed for some time global network symmetries impose strong constraints on the corresponding dynamical systems which affect equilibria periodic states heteroclinic cycles and even chaotic states in particular the symmetries of the network can lead to synchrony phase relations resonances and synchronous or cycling chaos symmetry is a rather restrictive assumption and a general theory of networks should be more flexible a recent generalization of the group theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network the input sets now the symmetry group becomes a groupoid which is an algebraic structure that resembles a group except that the product of two elements may not be defined the groupoid formalism makes it possible to extend group theoretic methods to more general networks and in particular it leads to a complete classification of robust patterns of synchrony in terms of the combinatorial structure of the network many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology a network of dynamical systems is not just a dynamical system with a high dimensional phase space it is also equipped with a canonical set of observables the states of the individual nodes of the network moreover the form of the underlying ode is constrained by the network topology which variables occur in which component equations and how those equations relate to each other the result is a rich and new range of phenomena only a few of which are yet properly understood

asymptotic theory for the correlated gamma frailty model the frailty model is a generalization of cox s proportional hazard model where a shared unobserved quantity in the intensity induces a positive correlation among the survival times murphy showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator npmle for the shared gamma frailty model without covariates in this paper we extend this result to the correlated gamma frailty model and we allow for covariates we discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator which goes back to wald our proof of the consistency for the npmle is essentially the same as the classical proof for the maximum likelihood estimator a new central limit theorem for processes of bounded variation is given furthermore we prove that a consistent estimator for the asymptotic variance of the npmle is given by the inverse of a discrete observed information matrix

monte carlo em estimation for time series models involving counts abstract the observations in parameter driven models for time series of counts are generated from latent unobservable processes that characterize the correlation structure these models result in very complex likelihoods and even the em algorithm which is usually well suited for problems of this type involves high dimensional integration in this article we discuss a monte carlo em mcem algorithm that uses a markov chain sampling technique in the calculation of the expectation in the e step of the em algorithm we propose a stopping criterion for the algorithm and provide rules for selecting the appropriate monte carlo sample size we show that under suitable regularity conditions an mcem algorithm will with high probability get close to a maximizer of the likelihood of the observed data we also discuss the asymptotic efficiency of the procedure we illustrate our monte carlo estimation method on a time series involving small counts the polio incidence time series previously analyzed by zeger

transcritical riddling in a system of coupled maps the transition from fully synchronized behavior to two cluster dynamics is investigated for a system of n globally coupled chaotic oscillators by means of a model of two coupled logistic maps an uneven distribution of oscillators between the two clusters causes an asymmetry to arise in the coupling of the model system while the transverse period doubling bifurcation remains essentially unaffected by this asymmetry the transverse pitchfork bifurcation is turned into a saddle node bifurcation followed by a transcritical riddling bifurcation in which a periodic orbit embedded in the synchronized chaotic state loses its transverse stability we show that the transcritical riddling transition is always hard for this we study the sequence of bifurcations that the asynchronous point cycles produced in the saddle node bifurcation undergo and show how the manifolds of these cycles control the magnitude of asynchronous bursts in the case where the system involves two subpopulations of oscillators with a small mismatch of the parameters the transcritical riddling will be replaced by two subsequent saddle node bifurcations or the saddle cycle involved in the transverse destabilization of the synchronized chaotic state may smoothly shift away from the synchronization manifold in this way the transcritical riddling bifurcation is substituted by a symmetry breaking bifurcation which is accompanied by the destruction of a thin invariant region around the symmetrical chaotic state

on the strength of attractors in a high dimensional system milnor attractor network robust global attraction and noise induced selection abstract strength of attractor is studied by the return rate to itself after perturbations for a multi attractor state of a globally coupled map it is found that fragile milnor attractors have a large basin volume at the partially ordered phase such dominance of fragile attractors is understood by robustness of global attraction in the phase space change of the attractor strength and basin volume against the parameter and size are studied in the partially ordered phase the dynamics is often described as milnor attractor network which leads to a new interpretation of chaotic itinerancy noise induced selection of fragile attractors is found that has a sharp dependence on the noise amplitude relevance of the observed results to neural dynamics and cell differentiation is also discussed

cycling chaos its creation persistence and loss of stability in a model of nonlinear magnetoconvection we examine a model system where attractors may consist of a heteroclinic cycle between chaotic sets this cycling chaos manifests itself as trajectories that spend increasingly long periods lingering near chaotic invariant sets interspersed with short transitions between neighbourhoods of these sets such behaviour is robust to perturbations that preserve the symmetry of the system we examine bifurcations of this state we discuss a scenario where an attracting cycling chaotic state is created at a blowout bifurcation of a chaotic attractor in an invariant subspace this differs from the standard scenario for the blowout bifurcation in that in our case the blowout is neither subcritical nor supercritical the robust cycling chaotic state can be followed to a point where it loses stability at a resonance bifurcation and creates a series of large period attractors the model we consider is a ninth order truncated ordinary differential equation ode model of three dimensional incompressible convection in a plane layer of conducting fluid subjected to a vertical magnetic field and a vertical temperature gradient symmetries of the model lead to the existence of invariant subspaces for the dynamics in particular there are invariant subspaces that correspond to regimes of two dimensional flows with variation in the vertical but only one of the two horizontal directions stable two dimensional chaotic flow can go unstable to three dimensional flow via the cross roll instability we show how the bifurcations mentioned above can be located by examination of various transverse liapunov exponents we also consider a reduction of the ode to a map and demonstrate that the same behaviour can be found in the corresponding map this allows us to describe and predict a number of observed transitions in these models the dynamics we describe is new but nonetheless robust and so should occur in other applications

least uncertainty principle in deformation quantization deformation quantization generally produces families of cohomologically equivalent quantizations of a single physical system we conjecture that the physically meaningful ones i allow enough observable energy distributions i e ones for which no pure quantum state has negative probability and ii reduce the uncertainty in the probability distribution of the resulting quantum states for the simple harmonic oscillator this principle selects the classic groenewold moyal or weyl product on phase space while for the free particle in which there is no real quantization all cohomologically equivalent quantizations are equally good

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

quantum groups and deformation quantization explicit approaches and implicit aspects deformation quantization which gives a development of quantum mechanics independent of the operator algebra formulation and quantum groups which arose from the inverse scattering method and a study of yang baxter equations share a common idea abstracted earlier in algebraic deformation theory that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations i e quantizations in deformation quantization the algebraic object is the algebra of observables functions on symplectic phase space whose infinitesimal deformation is the poisson bracket and global deformation a star product in quantum groups it is a hopf algebra generally either of functions on a lie group or often its dual in the topological vector space sense as we briefly explain a completed universal enveloping algebra of a lie algebra with for infinitesimal a matrix satisfying the modified classical yang baxter equation mcybe frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract one success here comes from universal deformation formulas udfs expressions built from a lie algebra which deform any algebra on which the lie algebra operates as derivations the most famous of these is the moyal product a special case of a class in which the lie algebra is abelian another comes from recognition that the belavin drinfel d solutions to the mcybe are in fact infinitesimal deformations for which in the case of the special linear groups it is possible to give explicit formulas for the corresponding quantum yang baxter equations this review paper discusses necessarily in brief these and related topics including twisting as a form of udf and finding formulas for preferred deformations of hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation