symmetry of the poincare map and its influence on bifurcations in a vibro impact system abstract symmetric period n 2 motion of a three degree of freedom 3 dof vibro impact system with symmetric rigid constraints is considered the poincare map of the system is established and the symmetric fixed point of the poincare map corresponds to the associated symmetric period n 2 motion it is shown that the poincare map exhibits some symmetry property and can be expressed as the second iteration of another unsymmetric implicit map the symmetry of the poincare map influence bifurcation behaviors in vibro impact system significantly and suppresses not only period doubling bifurcation but also hopf flip bifurcation and pitchfork flip bifurcation based on the second iteration of another unsymmetric implicit map the normal forms in the case of hopf hopf bifurcation and hopf bifurcation satisfying 1 2 resonant conditions are obtained by numerical simulation general hopf bifurcation hopf hopf bifurcation and hopf bifurcation satisfying 1 2 resonant conditions of the symmetric period n 2 motion are represented however period doubling bifurcation hopf flip and pitchfork flip bifurcation have not been obtained which reflects upon the effect of the symmetry property on possible bifurcations it is interesting that the system can exhibit both the characteristic of 1 2 resonance and that of torus t2 under some parameter combination and 2 t1 torus is also obtained

almost periodic solution for a model of tumor growth a mathematical model of tumor growth governed by diffusion equation is studied where the source of mitotic inhibitor is almost periodic and time dependent within the tissue existence and uniqueness of an almost periodic solution for this model have been proved

a bernstein von mises theorem for smooth functionals in semiparametric models a bernstein von mises theorem is derived for general semiparametric functionals the result is applied to a variety of semiparametric problems in i i d and non i i d situations in particular new tools are developed to handle semiparametric bias in particular for nonlinear functionals and in cases where regularity is possibly low examples include the squared l 2 norm in gaussian white noise nonlinear functionals in density estimation as well as functionals in autoregressive models for density estimation a systematic study of bvm results for two important classes of priors is provided namely random histograms and gaussian process priors

codimension 2 hopf bifurcation of a two degree of freedom vibro impact system abstract codimension 2 hopf bifurcation problem of a two degree of freedom system vibrating against a rigid surface is investigated in this paper the four dimensional poincare map of the vibro impact system is reduced to a two dimensional normal form by virtue of a center manifold reduction and a normal form technique then the theory of hopf bifurcation of maps in r 2 is applied to conclude the existence of codimension 2 hopf bifurcation of the vibro impact system the quasi periodic response of the system by theoretical analysis is well supported by numerical simulations it is shown that there exists codimension 2 hopf bifurcation in multi degree of freedom vibro impact systems the codimension 2 tori doubling phenomenon and the routes of quasi periodic impacts to chaos are reported briefly

symmetry and bifurcations of a two degree of freedom vibro impact system abstract a two degree of freedom system with impact is considered the symmetry of the system and its poincare map is described the symmetric period n 2 motion corresponding to the symmetric fixed point of the poincare map is obtained if the jacobian matrix of the poincare map at the fixed point has a real eigenvalue crossing the unit circle at 1 the symmetric fixed point will bifurcate into two antisymmetric fixed points which have the same stability via pitchfork bifurcation the numerical simulation shows that the symmetric fixed points may have pitchfork bifurcations and hopf bifurcations while the control parameter changes continuously the two antisymmetric fixed points will give birth to two synchronous bifurcation sequences

rates of convergence for minimum contrast estimators we shall present here a general study of minimum contrast estimators in a nonparametric setting although our results are also valid in the classical parametric case for independent observations these estimators include many of the most popular estimators in various situations such as maximum likelihood estimators least squares and other estimators of the regression function estimators for mixture models or deconvolution the main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters optimal rates depending on entropy conditions are already known at least for some of the models involved and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large but under some circumstances large entropies or changes in the entropy structure due to local perturbations the resulting the rates are only suboptimal counterexamples are constructed which show that the phenomenon is real for non parametric maximum likelihood or regression this proves that under purely metric assumptions our theorem is optimal and that minimum contrast estimators happen to be suboptimal

on asymptotic normality of limiting density functions with bayesian implications walker 1969 presented a straightforward approach to asymptotic normality of posterior distributions based on independent and identically distributed i i d observations dawid 1970 extended this work by providing a weaker set of conditions to cover some distributions whose ranges depend on the parameters of interest more recently heyde and johnstone 1979 simplified walker s conditions for stochastic processes in this paper we introduce three basic normality conditions for limiting density functions in the bayesian context our approach is data determined and is nonprobabilistic in nature nevertheless the results thus obtained can be easily converted to probabilistic ones when compared with those given by other investigators it is shown that our conditions as a whole have a greater generality and flexibility than those previously assumed either by dawid or heyde and johnstone in particular for the i i d case the commonly assumed wald type uniform boundedness condition for tail behaviours is shown to be unnecessary for asymptotic normality section 3 3 the advantage of our approach is further exemplified by a simple proof of the asymptotic normality of distributions conjugate to the exponential family

a periodically forced piecewise linear oscillator a single degree of freedom non linear oscillator is considered the non linearity is in the restoring force and is piecewise linear with a single change in slope such oscillators provide models for mechanical systems in which components make intermittent contact a limiting case in which one slope approaches infinity an impact oscillator is also considered harmonic subharmonic and chaotic motions are found to exist and the bifurcations leading to them are analyzed

the profile sampler we consider frequentist inference for the parametric component separately from the nuisance parameter in semiparametric models based on sampling from the posterior of the profile likelihood we prove that this procedure gives a first order correct approximation to the maximum likelihood estimator and consistent estimation of the efficient fisher information for without computing derivatives or using complicated numerical approximations an exact bayesian interpretation is established under a certain data dependent prior the sampler is useful in particular when the nuisance parameter is not estimable at the rate where neither bootstrap validity nor general automatic variance estimation has been theoretically justified even when the nuisance parameter is consistent and the bootstrap is known to be valid the proposed markov chain monte carlo procedure can yield computational savings because maximization of the likelihood is not required the theory is verified for three examples the methods are

the semiparametric bernstein von mises theorem in a smooth semiparametric estimation problem the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior it is shown that under certain straightforward and interpretable conditions the assertion of le cam s acclaimed but strictly parametric bernstein von mises theorem univ california publ statist 1 1953 277 329 holds in the semiparametric situation as well as a consequence bayesian point estimators achieve efficiency for example in the sense of hajek s convolution theorem z wahrsch verw gebiete 14 1970 323 330 the model is required to satisfy differentiability and metric entropy conditions while the nuisance prior must assign nonzero mass to certain kullback leibler neighborhoods ghosal ghosh and van der vaart ann statist 28 2000 500 531 in addition the marginal posterior is required to converge at parametric rate which appears to be the most stringent condition in examples the results are applied to estimation of the linear coefficient in partial linear regression with a gaussian prior on a smoothness class for the nuisance

dirichlet process mixtures of generalized linear models we propose dirichlet process mixtures of generalized linear models dp glm a new class of methods for nonparametric regression given a data set of input response pairs the dp glm produces a global model of the joint distribution through a mixture of local generalized linear models dp glms allow both continuous and categorical inputs and can model the same class of responses that can be modeled with a generalized linear model we study the properties of the dp glm and show why it provides better predictions and density estimates than existing dirichlet process mixture regression models we give conditions for weak consistency of the joint distribution and pointwise consistency of the regression estimate

on the computational complexity of mcmc based estimators in large samples in this paper we examine the implications of the statistical large sample theory for the computational complexity of bayesian and quasi bayesian estimation carried out using metropolis random walks our anal ysis is motivated by the laplace bernstein von mises central limit theo rem which states that in large samples the posterior or quasi posterior approaches a normal density using this observation we establish polyno mial bounds on the computational complexity of general metropolis ran dom walks methods in large samples our analysis covers cases where the underlying log likelihood or extremum criterion function is possibly non concave discontinuous and of increasing dimension however the central limit theorem restricts the deviations from continuity and log concavity of the log likelihood or extremum criterion function in a very specific manner under minimal assumptions for the central limit theorem framework to hold we show that the metropolis algorithm is theoretically efficient even for the canonical gaussian walk which is studied in detail specifically we show that the running time of the algorithm in large samples is bounded in probability by a polynomial in the parameter dimension d and in partic ular is of stochastic order d2 in the leading cases after the burn in period we then give an application to exponential and curved exponential families of increasing dimension

the bernstein von mises theorem for the proportional hazard model we study large sample properties of bayesian analysis of the proportional hazard model with neutral to the right process priors on the baseline hazard function we show that the posterior distribution of the baseline cumulative hazard function and regression coefficients centered at the maximum likelihood estimator is jointly asymptotically equivalent to the sampling distribution of the maximum likelihood estimator

dissipative quasi geostrophic motion under temporally almost periodic forcing the full nonlinear dissipative quasi geostrophic model is shown to have a unique temporally almost periodic solution when the wind forcing is temporally almost periodic under suitable constraints on the spatial square integral of the wind forcing and the parameter ekman number viscosity and the domain size the proof involves the pullback attractor for the associated nonautonomous dynamical system

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

quasi genericity of bifurcations to high dimensional invariant tori for maps we consider a family of maps in a banach spacee near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle the other part of the spectrum being strictly inside the unit disc we focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantt 1 circles into a family of invariantt 2 tori we show that this problem needs a 3 dimensional parameter unfolding and that for the complete maps bifurcation occurs at points where is the rotation number on the non normally hyperbolict 1 circle ande 2i are the eigenvalues of the constant matrix conjugated to the non contracting part of the linearization on the normal fiber bundle overt 1 making some non resonance and diophantine assumptions on leading to a positive measure cantor set int 2 we show that in paraboloidal regions of the 3 dim parameter space we have clean bifurcations as for the truncated normal form the complement of these regions forms a set of bubbles such as the ones obtained by chenciner in chen for a codimension 2 problem for maps in 2 the main tool here is a generalization for a matrix function ont 1 close to a constant of the quasi conjugacy to a constant modulo a minimum of additional parameters moved quasi conjugacy for the infinite dimensional case we use ac decoupling result on the angular dependent linear parts into a contraction still angular dependent and another part quasi conjugated to a constant matrix this type of analysis applies for a wide range of problems where truncated normal forms of the maps give bifurcations fromt n tot n 1 tori and this needs a n 1 dimensional parameter unfolding

on the bernstein von mises theorem with infinite dimensional parameters if there are many independent identically distributed observations governed by a smooth finite dimensional statistical model the bayes estimate and the maximum likelihood estimate will be close furthermore the posterior distribution of the parameter vector around the posterior mean will be close to the distribution of the maximum likelihood estimate around truth thus bayesian confidence sets have good frequentist coverage properties and conversely however even for the simplest infinite dimensional models such results do not hold the object here is to give some examples

a frequentist approach to bayesian asymptotics abstract ergodic theorem shows that ergodic averages of the posterior draws converge in probability to the posterior mean under the stationarity assumption the literature also shows that the posterior distribution is asymptotically normal when the sample size of the original data considered goes to infinity to the best of our knowledge there is little discussion on the large sample behaviour of the posterior mean in this paper we aim to fill this gap in particular we extend the posterior mean idea to the conditional mean case which is conditioning on a given vector of summary statistics of the original data we establish a new asymptotic theory for the conditional mean estimator for the case when both the sample size of the original data concerned and the number of markov chain monte carlo iterations go to infinity simulation studies show that this conditional mean estimator has very good finite sample performance in addition we employ the conditional mean estimator to estimate a garch 1 1 model for s p 500 stock returns and find that the conditional mean estimator performs better than quasi maximum likelihood estimation in terms of out of sample forecasting

a new criterion of period doubling bifurcation in maps and its application to an inertial impact shaker a new critical criterion of period doubling bifurcations is proposed for high dimensional maps without the dependence on eigenvalues as in the classical bifurcation criterion this criterion is composed of a series of algebraic conditions under which period doubling bifurcation occurs the proposed criterion is applied to the analysis of period doubling bifurcation in a two degree of freedom inertial shaker model it can be seen in this example that the proposed criterion is preferable to the classical bifurcation criterion in high dimensional maps

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

bayesian and non bayesian approaches to statistical inference and decision making bayesian and non bayesian approaches to statistical inference and decision making are discussed and compared a pragmatic criterion success in practice as well as logical consistency are emphasized in comparing alternative approaches attention is given to learning models probability concepts prior information axiom systems and selected technical aspects of bayesian and non bayesian approaches

asymptotic normality of semiparametric and nonparametric posterior distributions in this article it is shown that the marginal semiparametric and nonparametric posterior distributions for a parameter of interest behave like an ordinary parametric posterior distribution this in practice provides support of the utility of marginal semiparametric and nonparametric posterior distributions in particular the marginal semiparametric and nonparametric posterior distributions are asymptotically normal and centered at the corresponding maximum likelihood estimates mles or posterior means with covariance matrix the inverse of the fisher information additionally the semiparametric and nonparametric mles for the parameter of interest and the marginal posterior means are asymptotically normal and centered at the true parameter with the same covariance matrix the results are a semiparametric version and a nonparametric version of the parametric bayesian central limit theorem that establish a connection between the semiparametric and the nonparametric bayesian inference and their frequentis

hopf bifurcation of a two degree of freedom vibro impact system abstract the bifurcation problem of a two degree of freedom system vibrating against a rigid surface is studied in this paper it is shown that there exist hopf bifurcations in the vibro impact systems with two or more degrees of freedom under suitable system parameters in the paper a centre manifold theorem technique is applied to reduce the poincare map of the vibro impact system to a two dimensional one and then the theory of hopf bifurcation of maps in r 2 is applied to conclude the existence of hopf bifurcation of the vibro impact system the theoretical solutions are verified by numerical computations the quasi periodic response of the system represented by invariant circles in the projected poincare sections is obtained by numerical simulations and routes of quasi periodic impacts to chaos are stated briefly

a semiparametric bernstein von mises theorem for gaussian process priors this paper is a contribution to the bayesian theory of semiparametric estimation we are interested in the so called bernstein von mises theorem in a semiparametric framework where the unknown quantity is f with the parameter of interest and f an infinite dimensional nuisance parameter two theorems are established one in the case with no loss of information and one in the information loss case with gaussian process priors the general theory is applied to three specific models the estimation of the center of symmetry of a symmetric function in gaussian white noise a time discrete functional data analysis model and cox s proportional hazards model in all cases the range of application of the theorems is investigated by using a family of gaussian priors parametrized by a continuous parameter

an inconsistent maximum likelihood estimate abstract an example is given of a family of distributions on 1 1 with a continuous one dimensional parameterization that joins the triangular distribution when 0 to the uniform when 1 for which the maximum likelihood estimates exist and converge strongly to 1 as the sample size tends to infinity whatever be the true value of the parameter a modification that satisfies cramer s conditions is also given

bernstein von mises theorems for gaussian regression with increasing number of regressors this paper brings a contribution to the bayesian theory of nonparametric and semiparametric estimation we are interested in the asymptotic normality of the posterior distribution in gaussian linear regression models when the number of regressors increases with the sample size two kinds of bernstein von mises theorems are obtained in this framework nonparametric theorems for the parameter itself and semiparametric theorems for functionals of the parameter we apply them to the gaussian sequence model and to the regression of functions in sobolev and c classes in which we get the minimax convergence rates adaptivity is reached for the bayesian estimators of functionals in our applications

variable selection and function estimation in additive nonparametric regression using a data based prior abstract a hierarchical bayesian approach is proposed for variable selection and function estimation in additive nonparametric gaussian regression models and additive nonparametric binary regression models the prior for each component function is an integrated wiener process resulting in a posterior mean estimate that is a cubic smoothing spline each of the explanatory variables is allowed to be in or out of the model and the regression functions are estimated by model averaging to allow variable selection and model averaging data based priors are used for the smoothing parameter and the slope at 0 of each component function a two step markov chain monte carlo method is used to efficiently obtain the data based prior and to carry out variable selection and function estimation it is shown by simulation that significant improvements in the function estimators can be obtained over an approach that estimates all the unknown functions simultaneously the methodology is illustrated for a binary regressio

the effect of frequency and clearance variations on single degree of freedom impact oscillators abstract a simple piecewise linear forced oscillator with impacts is taken as a model for more complicated vibro impacting systems the presence of impacts makes the system discontinuous and non linear and to study it effectively the theory of dynamical systems is used by a combination of analytical and numerical techniques many phenomena are identified which are characteristic of non linear dynamical systems as the system parameters are varied in particular these include chaotic regimes and multiply coexisting stable states the domains of attraction of the periodic states are plotted and compared with the strange attractors of the chaotic states in addition phenomena are observed related to the discontinuity which have no counterpart in smooth dynamical systems chatter when infinitely many impacts occur in a finite time and grazing when a stable periodic orbit encounters a discontinuity and disappears catastrophically under a smooth change in the parameters

anti controlling quasi periodic impact motion of an inertial impact shaker system in this paper anti controlling quasi periodic impact motion of an inertial impact shaker system is addressed there exist two aspects of difficulty in the anti control design one is from the implicit poincare map of the system itself and the other from the limitation of the classical critical criterion of hopf bifurcation described by the properties of eigenvalues through using linear feedback control method in the original differential system and applying an explicit criterion of hopf bifurcation without using eigenvalues to the poincare map of the close loop system the two difficulties above can be overcome and the control design for creation of the quasi periodic impact motion at a specified system parameter location is achieved numerical simulation shows that the stable quasi periodic impact motion of the system is created at a desired parameter location by adjusting control parameter appropriately

probability inequalities for likelihood ratios and convergence rates of sieve mles let y 1 y n be independent identically distributed with density p 0 and let f be a space of densities we show that the supremum of the likelihood ratios i 1 n p y i p 0 y i where the supremum is over p f with p l 2 p 0 1 2 2 e is exponentially small with probability exponentially close to 1 the exponent is proportional to ne 2 the only condition required for this to hold is that e exceeds a value determined by the bracketing hellinger entropy of f a similar inequality also holds if we replace f by f n and p 0 by q n where q n is an approximation to p 0 in a suitable sense these results are applied to establish rates of convergence of sieve mles furthermore weak conditions are given under which the optimal rate e n defined by h e n f ne n 2 where h f is the hellinger entropy of f is nearly achievable by sieve estimators

on maximum likelihood estimation in infinite dimensional parameter spaces an approximate maximum likelihood estimate is known to be consistent under some compactness and integrability conditions in this paper we study its convergence rate and its asymptotic efficiency in estimating smooth functionals of the parameter we provide conditions under which the rate of convergence can be established this rate is essentially governed by the size of the space of score functions as measured by an entropy index we also show that for a large class of smooth functionals the plug in maximum likelihood estimate is asymptotically efficient that is it achieves the minimal fisher information bound the theory is illustrated by several nonparametric or semiparametric examples

higher order semiparametric frequentist inference with the profile sampler we consider higher order frequentist inference for the parametric component of a semiparametric model based on sampling from the posterior profile distribution the first order validity of this procedure established by lee kosorok and fine 2005 is extended to second order validity in the setting where the infinite dimensional nuisance parameter achieves the parametric rate specifically we obtain higher order estimates of the maximum profile likelihood estimator and of the efficient fisher information moreover we prove that an e frequentist confidence interval for the parametric component at level alpha can be estimated by the alpha level credible set from the profile sampler with an error of order op n 1 as far as we are aware these results are the first higher order frequentist results obtained for semiparametric estimation a fully bayesian interpretation is established under a certain data dependent prior the theory is verified for three specific examples 1 introduction the focus of this paper is on higher order frequentist inference for the parametric component of a semiparametric model in addition to the d dimensional euclidean parameter semiparametric models also have an infinite dimensional parameter which is sometimes called the nuisance parameter a classic example is the cox proportional hazards model for right censored survival data 7 where interest focuses on the log hazard ratios for the regression covariate vector z the integrated baseline hazard function is the infinite dimensional nuisance parameter the involvement of an infinite dimensional nuisance parameter in semiparametric models generally complicates maximum likelihood inference for in particular estimating the limiting variance of n n 0 where 0 is the true value of usually requires estimating an infinite dimensional operator of course this is not a problem with the cox model since the profile likelihood for does not involve the nuisance parameter but for most semiparametric models this simplification of the profile likelihood does not occur and supported in part by grant ca075142

distribution functions of means of a dirichlet process let be a random probability measure chosen by a dirichlet process on r b with parameter and such that x dx turns out to be a finite random variable the main concern of this paper is the statement of a suitable expression for the distribution function of that random variable such an expression is deduced through an extension of a procedure based on the use of generalized stieltjes transforms

quasi likelihood estimation in semiparametric models abstract suppose the expected value of a response variable y may be written h x t where x and t are covariates each of which may be vector valued is an unknown parameter vector is an unknown smooth function and h is a known function in this article we outline a method for estimating the parameter of this type of semiparametric model using a quasi likelihood function algorithms for computing the estimates are given and the asymptotic distribution theory for the estimators is developed the generalization of this approach to the case in which y is a multivariate response is also considered the methodology is illustrated on two data sets and the results of a small monte carlo study are presented

bernstein von mises theorem for linear functionals of the density in this paper we study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semi parametric version of the bernstein von mises theorem the special case of the cumulative distributive function evaluated at a specific point is widely considered in particular we show that for infinite dimensional exponential families under quite general assumptions the asymptotic posterior distribution of the functional can be either gaussian or a mixture of gaussian distributions with different centering points this illustrates the positive but also the negative phenomena that can occur in the study of bernstein von mises results

bayesian regression with heteroscedastic error density and parametric mean function abstract in this paper we consider bayesian estimation of restricted conditional moment models with the linear regression as a particular example a common practice in the bayesian literature for linear regression and other semi parametric models is to use flexible families of distributions for the errors and to assume that the errors are independent from covariates however a model with flexible covariate dependent error distributions should be preferred for the following reason assuming that the error distribution is independent of predictors might lead to inconsistent estimation of the parameters of interest when errors and covariates are dependent to address this issue we develop a bayesian regression model with a parametric mean function defined by a conditional moment condition and flexible predictor dependent error densities sufficient conditions to achieve posterior consistency of the regression parameters and conditional error densities are provided in experiments the proposed method compares favorably with classical and alternative bayesian estimation methods for the estimation of the regression coefficients and conditional densities

profile likelihood and conditionally parametric models in this paper we outline a general approach to estimating the parametric component of a semiparametric model for the case of a scalar parametric component the method is based on the idea of first estimating a one dimensional subproblem of the original problem that is least favorable in the sense of stein the likelihood function for the scalar parameter along this estimated subproblem may be viewed as a generalization of the profile likelihood for the problem the scalar parameter is then estimated by maximizing this generalized profile likelihood this method of estimation is applied to a particular class of semiparametric models where it is shown that the resulting estimator is asymptotically efficient

laplace expansion for posterior densities of nonlinear functions of parameters summary an asymptotic expansion for marginal posterior densities of nonlinear functions of parameters is derived the density in question is proportional to an integral over a lower dimensional manifold with the integrand including an appropriate change in volume term laplace s method is applied to obtain the expansion avoiding the difficult step of the explicit parameterization of the manifold we carry out the expansion via an implicit and local parameterization and the derivatives required can be evaluated recursively an example indicates a considerable improvement by including the term of order o n 1

bayesian regression with nonparametric heteroskedasticity abstract this paper studies large sample properties of a semiparametric bayesian approach to inference in a linear regression model the approach is to model the distribution of the regression error term by a normal distribution with the variance that is a flexible function of covariates the main result of the paper is a semiparametric bernstein von mises theorem under misspecification even when the distribution of the regression error term is not normal the posterior distribution of the properly recentered and rescaled regression coefficients converges to a normal distribution with the zero mean and the variance equal to the semiparametric efficiency bound

an analysis of bayesian inference for nonparametric regression abstract the observation model y sub i beta i n epsilon sub i 1 or n is considered where the epsilon s are i i d mean zero and variance sigma sq and beta is an unknown smooth function a gaussian prior distribution is specified by assuming beta is the solution of a high order stochastic differential equation the estimation error delta beta beta average is analyzed where beta average is the posterior expectation of beta asymptotic posterior and sampling distributional approximations are given for abs val del square when abs val square is one of a family of norms natural to the problem it is shown that the frequentist coverage probability of a variety of 1 alpha posterior probability regions tends to be larger than 1 alpha but will be infinitely often less than any epsilon 0 as n approaches infinity with prior probability 1 a related continuous time signal estimation problem is also studied keywords bayesian inference nonparametric regression confidence regions signal extraction smoothing splices

general frequentist properties of the posterior profile distribution in this paper inference for the parametric component of a semiparametric model based on sampling from the posterior profile distribution is thoroughly investigated from the frequentist viewpoint the higher order validity of the profile sampler obtained in cheng and kosorok ann statist 36 2008 is extended to semiparametric models in which the infinite dimensional nuisance parameter may not have a root n convergence rate this is a nontrivial extension because it requires a delicate analysis of the entropy of the semiparametric models involved we find that the accuracy of inferences based on the profile sampler improves as the convergence rate of the nuisance parameter increases simulation studies are used to verify this theoretical result we also establish that an exact frequentist confidence interval obtained by inverting the profile log likelihood ratio can be estimated with higher order accuracy by the credible set of the same type obtained from the posterior profile distribution our theory is verified for several specific examples

non periodic motion caused by grazing incidence in an impact oscillator the motion of a single degree of freedom periodically forced oscillator subjected to a rigid amplitude constraint is considered using analytical methods the singularities caused by grazing impact are studied it is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter a special type of bifurcation occurs the motion after the bifurcation may be non periodic and a criterion for this based on orientation and eigenvalues is given

interaction of hopf and period doubling bifurcations of a vibro impact system abstract an inertial shaker as a vibratory system with impact is considered by means of differential equations periodicity and matching conditions the theoretical solution of periodic n 1 impacting motion can be obtained and the poincare map is established dynamics of the system are studied with special attention to interaction of hopf and period doubling bifurcations corresponding to a codimension 2 one when a pair of complex conjugate eigenvalues crosses the unit circle and the other eigenvalue crosses 1 simultaneously for the jacobi matrix the four dimensional map can be reduced to a three dimensional normal form by the center manifold theorem and the theory of normal forms the two parameter unfoldings of local dynamical behavior are put forward and the singularity is investigated it is proved that there exist curve doubling bifurcation a torus doubling bifurcation hopf bifurcation of 2 2 fixed points as well as period doubling bifurcation and hopf bifurcation of 1 1 fixed points near the critical point numerical results indicate that the vibro impact system presents complicated and interesting curve doubling bifurcation and hopf bifurcation as the two controlling parameters vary

on methods of sieves and penalization we develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates mle s in parametric semiparametric and nonparametric models we find that the asymptotic behavior of substitution estimates for estimating smooth functionals are essentially governed by two indices the degree of smoothness of the functional and the local size of the underlying parameter space we show that when the local size of the parameter space is not very large the substitution standard nonsieve substitution sieve and substitution penalized mle s are asymptotically efficient in the fisher sense under certain stochastic equicontinuity conditions of the loglikelihood moreover when the convergence rate of the estimate is slow the degree of smoothness of the functional needs to compensate for the slowness of the rate in order to achieve efficiency when the size of the parameter space is very large the standard and penalized maximum likelihood procedures may be inefficient whereas the method of sieves may be able to overcome this difficulty this phenomenon is particularly manifested when the functional of interest is very smooth especially in the semiparametric case

bifurcations and chaos in a system with impacts abstract a vibro impact system is considered a body bounces on a flat horizontal surface of the vibro bench hopf bifurcations of the vibro impact system in two kinds of strong resonance cases 03 1 and 04 1 are investigated a poincare map of the system is established the period 1 single impact motion of the system and its stability are studied by analytical methods and hopf bifurcation values and intersecting conditions of the system in strong resonance cases are determined a center manifold theorem technique is applied to reduce the poincare map to a two dimensional one which is put into normal form by theory of normal forms by the theory of hopf and subharmonic bifurcations of fixed points in r 2 strong resonance dynamic behavior of the vibro impact system near points of resonance is analyzed in the resonance case of 03 1 the system generally exhibits unstable period 3 three impact motion in the resonance case of 04 1 the system can exhibit stable period 4 four impact motion and quasi periodic motion the theoretical analyses are verified by numerical solutions routes to chaos in two kinds of strong resonance cases considered are obtained by numerical simulations quasi periodic impacts in non resonance and weak resonance cases doubling periodic bifurcations and chaotic motions are also stated briefly by numerical analyses

almost periodic passive tracer dispersion the authors investigate the impact of external sources on the pattern formation of concentration profiles of passive tracers in a two dimensional shear flow by using the pullback attractor technique for the associated nonautonomous dynamical system it is shown that a unique temporally almost periodic concentration profile exists for the temporally almost periodic external source

bifurcations and chaos in autonomous self excited oscillators with impact damping dynamics and bifurcations of a class of single degree of freedom self excited oscillators with an impact damper have been studied the basic motivation is to determine a persistent birfurcation structure of co dimension one which is independent of the exact functional form of the self excitation mechanism in the process an analytical method for studying the qualitative dynamical features is presented a new numerical scheme for analyzing the eigenproperties of a solution is developed a detailed numerical study revealing the superstructure of the bifurcation phenomena has been performed

convergence rates of posterior distributions for non i i d observations we consider the asymptotic behavior of posterior distributions and bayes estimators based on observations which are required to be neither independent nor identically distributed we give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion we then specialize our results to independent nonidentically distributed observations markov processes stationary gaussian time series and the white noise model we apply our general results to several examples of infinite dimensional statistical models including nonparametric regression with normal errors binary regression poisson regression an interval censoring model whittle estimation of the spectral density of a time series and a nonlinear autoregressive model