a combinatorial perturbation method and arnold s whiskered tori in vortex dynamics abstract a combinatorial perturbation method for some n body problems is presented this method is used to reformulate a class of n body problems in the form of near integrable hamiltonian systems with many degrees of freedom the kam theory is worked out for n body problems defined on subgraphs g of the complete graph kn the main result in this paper is the existence of arnold s whiskered tori and diffusion for vortex lattice dynamics throughout the paper combinatorial and graph theoretic concepts play important roles in particular the kam nondegeneracy and melnikov transversability conditions are given in terms of binary trees t n one of the main tools here is a combinatorial algorithm for symplectic transformations to jacobi relative coordinates which is based on binary trees t n

computation of homoclinic solutions to periodic orbits in a reduced water wave problem abstract this paper concerns homoclinic solutions to periodic orbits in a fourth order hamiltonian system arising from a reduction of the classical water wave problem in the presence of surface tension these solutions correspond to travelling solitary waves which converge to non decaying ripples at infinity an analytical result of amick and toland 1992 showing the existence of such homoclinic orbits to small amplitude periodic orbits in a singular limit is extended numerically also a related result by amick and mcleod 1991 showing the non existence of homoclinic solutions to zero is motivated geometrically a general boundary value method is constructed for continuation of homoclinic orbits to periodic orbits in hamiltonian and reversible systems numerical results are presented using the path following software auto showing that the amick toland solutions persist well away from the singular limit and for large amplitude periodic orbits special account is taken of the phase shift between the two periodic solutions in the asymptotic limits furthermore new multi modal homoclinic solutions to periodic orbits are shown to exist under a transversality hypothesis which is verified a posteriori by explicit computation continuation of these new solutions reveals limit points with respect to the singular parameter

bayesian inference for small sample capture recapture data we consider data on the survival of a population of cephalorhynchus hectori hector s dolphins in a marine area of new zealand to estimate survival probabilities of animal populations a multiple capture recapture sampling scheme can be used in this paper we propose a practical methodology to derive approximations to posterior distributions based on laplace methods we show how to calculate bayes estimates and credible intervals in this setting

chaotic motion of the n vortex problem on a sphere ii saddle centers in three degree of freedom hamiltonians we study the motion of n point vortices with n on a sphere in the presence of fixed pole vortices which are governed by a hamiltonian dynamical system with n degrees of freedom special attention is paid to the evolution of their polygonal ring configuration called the n ring in which they are equally spaced along a line of latitude of the sphere when the number of the point vortices is n 5n or 6n with n the system is reduced to a two degree of freedom hamiltonian with some saddle center equilibria one of which corresponds to the unstable n ring using a melnikov type method applicable to two degree of freedom hamiltonian systems with saddle center equilibria and a numerical method to compute stable and unstable manifolds we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle centers and hence chaotic motions occur especially the evolution of the unstable n ring is shown to be chaotic

on coherence in parametric density estimation summary in parametric density estimation the existence of a prior distribution on the parameter dictates the use of the corresponding predictive density function as estimate this paper argues that in a decision theory approach to parameter density estimation any loss function introduced should lead to this predictive result for situations where prior information is available the kullbackleibler directed divergence has this coherence property whereas the corresponding symmetric divergence does not the role of a recently suggested sequential improvement criterion in density estimation is also discussed

empirical bayes methods for combining likelihoods comment suppose that several independent experiments are observed each one yielding a likelihood l k k for a real valued parameter of interest k for example k might be the log odds ratio for a 2 x 2 table relating to the kth population in a series of medical experiments this article concerns the following empirical bayes question how can we combine all of the likelihoods l k to get an interval estimate for any one of the k s say 1 the results are presented in the form of a realistic computational scheme that allows model building and model checking in the spirit of a regression analysis no special mathematical forms are required for he priors or the likelihoods this scheme is designed to take advantage of recent methods that produce approximate numerical likelihoods l k k even in very complicated situations with all nuisance parameters eliminated the empirical bayes likelihood theory is extended to situations where the k s have a regression structure as well as an empirical bayes relationship most of the discussion is presented in terms of a hierarchical bayes model and concerns how such a model can be implemented without requiring large amounts of bayesian input frequentist approaches such as bias correction and robustness play a central role in the methodology

nonintegrability of some hamiltonian systems scattering and analytic continuation we consider canonical two degrees of freedom analytic hamiltonian systems with hamiltonian functionh 1 2 p12 p22 u q1 q2 where u q1 q2 1 2 v2q12 2q22 o q12 q22 3 2 and q2 u q1 0 0 under some additional not so restrictive hypothesis we present explicit conditions for the exisstence of transversal homoclinic orbits to some periodic orbits of these systems we use a theorem of lerman 1991 and an analogy between one of its conditions with the usual one dimensional quantum scattering problem the study of the scattering equation leads us to an analytic continuation problem for the solutions of a linear second order differential equation we apply our results to some classical problems

horseshoes in two degree of freedom hamiltonian systems with saddle centers we consider a class of two degree of freedom hamiltonian systems which have saddle centers with homoclinic orbits and do not take the form of small perturbations of integrable systems using a melnikov type global perturbation technique we present a condition under which orbits transversely homoclinic to periodic orbits exist and horseshoes are created on surfaces of energy level near the saddle centers we apply the theory to systems with potentials and give concrete examples in particular we can detect the presence of horseshoes in a class of systems to which previous results did not apply

practical use of higher order asymptotics for multiparameter exponential families recently developed asymptotics based on saddlepoint methods provide important practical methods for multiparameter exponential families especially in generalized linear models the aim here is to clarify and explore these attention is restricted to tests and confidence intervals regarding a single parametric function which can be represented as a natural parameter of a full rank exponential family excellent approximations to exact conditional inferences are often available in terms of simple adjustments to the signed square root of the likelihood ratio statistic

bayesian information criteria and smoothing parameter selection in radial basis function networks summary by extending schwarz s 1978 basic idea we derive a bayesian information criterion which enables us to evaluate models estimated by the maximum penalised likelihood method or the method of regularisation the proposed criterion is applied to the choice of smoothing parameters and the number of basis functions in radial basis function net work models monte carlo experiments were conducted to examine the performance of the nonlinear modelling strategy of estimating the weight parameters by regularisation and then determining the adjusted parameters by the bayesian information criterion the simulation results show that our modelling procedure performs well in various situations

goodness of prediction fit for multivariate linear models abstract using both frequentist and bayesian techniques predicting densities are derived for future observations from a multivariate linear model with matrix normal error terms all the candidates belong to a general location scale family of predicting densities an analytic comparison is undertaken using a kullback leibler loss by citing an optimal member of a subclass including most of these predicting densities as members the subclass is based on an invariant student t random matrix and the optimal member is the bayesian predictive density corresponding to a jeffreys noninformative prior information based numerical comparisons illustrate the nature of the dominance

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

on the generalization of the likelihood function and the likelihood principle abstract the parametric likelihood and likelihood principle lp play a central role in parametric methodology and in the foundations of statistics the main purpose of this article is to extend the concepts of likelihood and lp to general inferential aims and models covering for example prediction and empirical bayes models the likelihood function is the joint distribution of the data and the unobserved variables of inferential interest considered as a function of the parameters and these inferential variables lp is based on this likelihood and the principles of sufficiency sp and conditionality cp are modified such that the equivalence sp and cp lp is valid generalizing birnbaum s theorem

nonlinear galerkin methods based on the concept of determining modes for the magnetohydrodynamic equations a new nonlinear galerkin method based on an approximate inertial manifold and on a finite number of determining modes has been implemented for the three dimensional magnetohydrodynamic equations it is found that these active modes which regulate the dynamics of the system are not necessarily those with the smallest wavenumbers but those containing most of the time averaged enstrophy it is demonstrated that owing to the reduced number of equations the modified nonlinear galerkin method which includes only these determining modes in a truncation improves the computational efficiency in comparison with the traditional galerkin method

bayesian inference in neural networks summary approximate marginal bayesian computation and inference are developed for neural network models the marginal considerations include determination of approximate bayes factors for model choice about the number of nonlinear sigmoid terms approximate predictive density computation for a future observable and determination of approximate bayes estimates for the nonlinear regression function standard conjugate analysis applied to the linear parameters leads to an explicit posterior on the nonlinear parameters further marginalisation is performed using laplace approximations the choice of prior and the use of an alternative sigmoid lead to posterior invariance in the nonlinear parameter which is discussed in connection with the lack of sigmoid identifiability a principal finding is that parsimonious model choice is best determined from the list of modal estimates used in the laplace approximation of the bayes factors for various numbers of sigmoids by comparison the values of the various bayes factors are of only secondary importance the proposed methods are illustrated in the context of two nonlinear datasets that involve respectively univariate and multivariate nonlinear regression models

a representation of the posterior mean for a location model summary an exact representation of the posterior mean is developed for a location model and the class of priors that are normal scale mixtures the result makes use of the conditional distribution of the maximum likelihood estimator and masreliez s theorem directions for future development are indicated an exact representation for the posterior mean e 01y is given where y is a 1 x n vector of observations from a location model f x 0 and 0 has a prior density p 0 that is a normal scale mixture let l 0 denote the likelihood function and let y 0 a where 0 is the maximum likelihood estimator and a is the maximal ancillary the representation makes use of two results the conditional distribution of the maximum likelihood estimator p 00 a barndorff nielsen 1983 and a result of masreliez 1975 it is shown that under a normal prior e ojy can be represented as a linear transformation of the score function of p 01 a where p 0la p 10 a p 0 do the representation can be viewed as a generalization of masreliez s result that deals with the model x 0 0 n m r2 and represents the posterior mean in terms of the derivative of the logarithm of the marginal density for x the appendix gives a sufficient regularity condition and a rigorous proof of an interchange of derivative and integral sign required for masreliez s theorem a representation is also developed in the case where the prior density is a normal scale mixture the results have a quantitative robustness appeal for example under any model and a normal prior the sensitivity of e oy with respect to aberrant observations is

exact scheffe type confidence intervals for output from groundwater flow models 2 combined use of hydrogeologic information and calibration data calibration data observed values corresponding to model computed values of dependent variables are incorporated into a general method of computing exact scheffe type confidence intervals analogous to the confidence intervals developed in part 1 cooley this issue for a function of parameters derived from a groundwater flow model parameter uncertainty is specified by a distribution of parameters conditioned on the calibration data this distribution was obtained as a posterior distribution by applying bayes theorem to the hydrogeologically derived prior distribution of parameters from part 1 and a distribution of differences between the calibration data and corresponding model computed dependent variables tests show that the new confidence intervals can be much smaller than the intervals of part 1 because the prior parameter variance covariance structure is altered so that combinations of parameters that give poor model fit to the data are unlikely the confidence intervals of part 1 and the new confidence intervals can be effectively employed in a sequential method of model construction whereby new information is used to reduce confidence interval widths at each stage

geometry of exponential family with competing risks and censored data abstract employing the differential geometrical methods in statistics suggested by amari 1985 and amari et al 1987 considering the exponential family with censored data and competing risks as a manifold of a statistical model the geometry of the manifold is investigated based on two information sources as an application of the geometry the asymptotic expansions of the bootstrap prediction bayesian prediction and their risk evaluations are investigated the results show that these expansions are related to the coefficients of connections and metric tensors and the predictive density function is the estimative density function in the asymptotic sense finally taking rayleigh distribution and prostatic cancer data as examples some computation and simulation results are presented to illustrate our main results

on efficient probability forecasting systems we study the asymptotic behaviour of probability forecasting systems and discuss their usefulness as inferential tools for statistical problems such as model verification and selection our theoretical setting is the prequential or predictive sequential framework proposed by dawid 1984 we study especially the notion of prequential efficiency of a forecasting system and present some new results we focus on plug in or estimative forecasting systems where the forecast distribution is generated by replacing the unknown parameter with an estimate

frequentist prediction intervals and predictive distributions summary we consider parametric frameworks for the prediction of future values of a random variable y based on previously observed data x simple pivotal methods for obtaining calibrated prediction intervals are presented and illustrated frequentist predictive distri butions are defined as confidence distributions and their utility is demonstrated a simple pivotal based approach that produces prediction intervals and predictive distributions with well calibrated frequentist probability interpretations is introduced and efficient simulation methods for producing predictive distributions are considered properties related to an average kullback leibler measure of goodness for predictive or estimated distributions are given the predictive distributions here are shown to be optimal in certain settings with invariance structure and to dominate plug in distributions under certain conditions

bayes prediction in regressions with elliptical errors abstract in this paper the prediction problem is considered for linear regression models with elliptical errors when the bayes prior is non informative we show that the bayes prediction density under the elliptical errors assumption is exactly the same as that obtained with normally distributed errors thus assuming that the errors have a normal distribution when the true distribution is elliptical will not lead to incorrect predictive inferences if the error variance structure is correctly specified this extends the results of zellner 1976 finally based on monte carlo numerical integration procedures computations are provided in a model with multiplicative heteroscedasticity

predictive likelihood inference with applications summary conditional and marginal predictive likelihood are defined and used in predicting un observed or missing data a distinction is made between predictive and estimative goals of a data analysis such methodology and distinctions are illustrated in the context of outlier theory and mixed model ano va the prediction of unobserved or missing data is considered from a likelihood perspective a general predictive likelihood is proposed for such purposes and contrasted with parametric likelihood the resulting maximum likelihood predictor is also compared with the limiting imputation from the em algorithm the new methodology is illustrated and contrasted with estimative likelihood analysis in the context of outlier theory and mixed model ano va predictions are proposed that are based on conditional and marginal predictive likelihoods for the unobserved data which are predictive adaptations of the methods developed by kalbfleisch and sprott 1970 1973 these authors were concerned with the elimination of nuisance parameters in order to make inference about another set of structural parameters prediction in the frame work of parametric distributions involves the elimination of all parameters in such a way that inference about the unobserved data is possible therefore in the predictive setting all para meters are nuisance parameters and the unobserved data now play a role similar to that of the structural parameters various predictive approaches to statistical inference have been advocated principally by geisser 1971 1975 stone 1974 aitchison and dunsmore 1975 and guttman 1970 a survey is provided by geisser 1980 and a brief overview in a parametric setting is presented in the next section following this a conditional likelihood approach is developed and shown to be closely related to the lauritzen 1974 hinkley 1979 likelihood method but specified in a manner allowing for a wider range of applicability a marginal predictive likelihood procedure is also specified which when applicable essentially agrees with the conditional likelihood procedure for the examples considered for a general account of likelihood based inference see fraser

robust predictive distributions for exponential families summary the bootstrap predictive distribution considered by harris 1989 is modified by replacing the maximum likelihood estimator of the unknown parameter by its minimum hellinger distance estimator the predictive distribution thus obtained is asymptotically superior to the corresponding estimative distribution in terms of the average kullback leibler divergence when the true distribution is in the natural exponential family simulation results are provided for the binomial and normal distributions which suggest that the proposed predictive distributions are robust and can perform better than the likelihood based methods under data contamination

approximate predictive likelihood summary a predictive likelihood is given which approximates both bayes and maximum likeli hood predictive inference by expansion of a posterior likelihood this synthesizes and extends previous results and is widely applicable the approximation usually differs from exact bayes posterior predictive density by op n 2 and from exact predictive likelihood by op n 1 but does not depend on the availability of prior information and is applicable when exact predictive likelihood cannot be found the results are applied to the prediction various treatments of the likelihood function form the basis of parametric inference but until recently few non bayesian attempts have been made to define versions of it suitable for prediction however lauritzen 1974 hinkley 1979 and more recently butler 1986 have proposed rather similar definitions of predictive likelihood based on the idea of conditioning on the value of a sufficient statistic unfortunately they apply only when sufficiency provides a genuine reduction of the data moreover even if for a given problem an exact predictive likelihood exists the calculations needed to derive it can be onerous this paper provides approximations to hinkley lauritzen predictive likelihood accur ate to op n 1 in many cases which are applicable in situations where maximum likelihood estimation is regular they also apply to problems which admit no exact predictive likelihood because sufficiency provides no real reduction of the data the approximations are derived from a posterior predictive density and may if prior informa tion is available be regarded as bayesian procedures their construction requires only the repeated maximization of a likelihood and evaluation of the observed information matrix at the maximum and is fast and accurate using usual numerical maximization methods there is a close connexion with the approximations to posterior moments and marginal densities given by tierney kadane 1986 who suggest the same approxima tion as used here but give no detailed results for prediction section 2 of this paper shows how laplace s method for integrals may be applied to a posterior predictive density to yield approximate predictive likelihood section 3 discusses a connection with conditional inference essentially the same approximation

nonlinear stability of a latitudinal ring of point vortices on a nonrotating sphere we study the nonlinear stability of relative equilibria of configurations of identical point vortices on the surface of a sphere in particular we study how the stability changes as a function of the colatitude theta and of the number of vortices n by using the integrals of motion we view the system in asuitable corotating frame where the polygonal vortex configuration is at rest then after a sufficient criterion due to dirichlet the stability ranges are the theta intervals for which the hessian of the hamiltonian evaluated at the equilibrium configuration is positive or negative definite we find that the stability intervals coincide with those for linear stability determined by polvani and dritschel j fluid mech 255 1993 pp 35 64 for n 3 we recover the result previously established by pekarsky and marsden j math phys 39 1998 pp 5894 5907

do inertial manifolds apply to turbulence abstract in this article we survey some recent results concerning attractors inertial manifolds and approximate inertial manifolds for dissipative evolution equations and in particular for the two dimensional navier stokes equation the role of inertial manifolds as a modelization of the interaction of small and large eddies in turbulent flows is described and discussed

standard errors of prediction in generalized linear mixed models abstract the unconditional mean squared error of prediction umsep is widely used as a measure of prediction variance for inferences concerning linear combinations of fixed and random effects in the classical normal theory mixed model but the umsep is inappropriate for generalized linear mixed models where the conditional variance of the random effects depends on the data when the random effects describe variation between independent small domains and domain specific prediction is of interest we propose a conditional mean squared error of prediction cmsep as a general measure of prediction variance the cmsep is shown to be the sum of the conditional variance and a positive correction that accounts for the sampling variability of parameter estimates we derive a second order correct estimate of the cmsep that consists of three components a a plug in estimate of the conditional variance b a plug in estimate of a taylor series approximation to the correction term and c a bootstrap estimate of

a framework for bayesian and likelihood approximations in statistics summary regular bayesian and frequentist approximations in statistics are studied within a unified framework in particular it is shown how some higher order likelihood based approximations arise from their bayesian counterparts via an unsmoothing argument this approach serves to shed new light on these formulae and to clarify relationships between bayesian and frequentist inferences for example bayesian interpretations of higher order approximations can give insights into the acceptability or otherwise of these approximations from the point of view of relevance to the actual data observed furthermore this approach is a very natural one for the study of more general nonregular problems models for dependent data and approximate conditional inference for ease of exposition the development is in terms of a single real parameter the main analytic development proceeds in terms of signed roots of log density ratios

three ways to solve the poisson equation on a sphere with gaussian forcing motivated by the needs of vortex methods we describe three different exact or approximate solutions to the poisson equation on the surface of a sphere when the forcing is a gaussian of the three dimensional distance 2 j exp 2 e 2 1 cos q c g a u s s e more precisely the forcing is a gaussian minus the gauss constraint constant c g a u s s this subtraction is necessary because j is bounded for any type of forcing only if the integral of the forcing over the sphere is zero y kimura h okamoto vortex on a sphere j phys soc jpn 56 1987 4203 4206 d g dritschel contour dynamics surgery on the sphere j comput phys 79 1988 477 483 the legendre polynomial series is simple and yields the exact value of the gauss constraint constant but converges slowly for large e the analytic solution involves nothing more exotic than the exponential integral but all four terms are singular at one or the other pole cancelling in pairs so that j is everywhere nice the method of matched asymptotic expansions yields simpler uniformly valid approximations as series of inverse even powers of e that converge very rapidly for the large values of e e 40 appropriate for geophysical vortex computations the series converges to a nonzero o exp 4 e 2 error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order

motion of three point vortices on a sphere abstract we solve the equations governing the relative motion of three point vortices of arbitrary strength moving on the surface of a sphere of radius r the system is more general than the corresponding one in the plane 1 3 5 19 28 33 34 37 38 and reduces to it in the limit r as long as the three vortices remain sufficiently close to each other during the course of their motion instead of using spherical coordinates we use cartesian coordinates in which the vector i points from the center of the sphere to the vortex with strength i an important conserved quantity is the center of vorticity vector c i i i which with no loss of generality we align with the z axis based on the size of this vector relative to the radius of the sphere we classify the motion into one of five types super radial radial sub radial degenerate or a limiting super radial case this categorization allows us to draw several conclusions about the qualitative motion of the vortices we then fully characterize all fixed and relative equilibria on the sphere for fixed equilibria the vortices must lie on great circles geodesics if the strengths are equal they form an equilateral triangle otherwise the triangle shape is specified once the strength of the vortices is given the relative equilibria are classified as either degenerate c 0 or non degenerate c 0 for each type the shape of the vortex triangle is described and the frequency of rotation is computed as in the planar problem it is possible to introduce trilinear coordinates and study the motion in a phase plane which allows us to locate all the equilibria as well as to characterize more complex relative dynamics we then describe self similar vortex collapse on the sphere stating necessary and sufficient conditions for collapse to occur computing the collapse times and vortex trajectories on the route towards collapse comparisons are made with the collapse formulas in the plane derived in 1 30

generalised information criteria in model selection summary the problem of evaluating the goodness of statistical models is investigated from an information theoretic point of view information criteria are proposed for evaluating models constructed by various estimation procedures when the specified family of probability distributions does not contain the distribution generating the data the proposed criteria are applied to the evaluation of models estimated by maximum likelihood robust penalised likelihood bayes procedures etc we also discuss the use of the bootstrap in model evaluation problems and present a variance reduction technique in the bootstrap simulation

approximate predictive pivots and densities summary this paper suggests two predictive likelihoods that can be applied in almost any parametric model setting the first can sometimes be interpreted as an approximate predictive pivot barnard 1986 while the second is often an approximation to a bayesian predictive density with a flat prior the issue of calibrating various predictive likelihoods in terms of long run predictive coverage is also discussed and a specific criterion by which these likelihoods can be compared is proposed

information geometry of generalized bayesian prediction using alpha divergences as loss functions in this paper the methods of information geometry are employed to investigate a generalized bayes rule for prediction taking alpha divergences as the loss functions optimality and asymptotic properties of the generalized bayesian predictive densities are considered we show that the bayesian predictive densities minimize a generalized bayes risk we also find that the asymptotic expansions of the densities are related to the coefficients of the alpha connections of a statistical manifold in addition we discuss the difference between two risk functions of the generalized bayesian predictions based on different priors finally using the non informative priors i e jeffreys and reference priors uniform prior and conjugate prior two examples are presented to illustrate the main results

predictive fit for natural exponential families summary the paper examines predictive distributions concentrating on measuring their fit to the true distribution by average kullback leibler divergence the notion of an averaged bootstrap predictive distribution is introduced this predictive distribution is shown to be asymptotically superior to the estimative distribution in terms of average kullbackleibler divergence when the true distribution is in a natural exponential family smallsample results are presented for the poisson and binomial distributions which suggest that the bootstrap distribution performs well in these cases

inertial manifolds for the 3d modified leray α model abstract in this paper we investigate the critical modified leray model in t 3 and prove the existence of an n dimensional inertial manifold for this problem

stability criterion for bright solitary waves of the perturbed cubic quintic schro dinger equation abstract the stability of the bright solitary wave solution to the perturbed cubic quintic schrodinger equation is considered it is shown that in a certain region of parameter space these solutions are unstable with the instability being manifested as a small positive eignevalue furthermore it is shown that in the complimentary region of parameter space there are no small unstable eigenvalues the proof involves a novel calculation of the evans function which is of interest in its own right as a consequence of the eigenvalue calculation it is additionally shown that n bump bright solitary waves bifurcate from the primary wave

singular manifolds and quasi periodic solutions of hamiltonians for vortex lattices abstract a general method for establishing the existence of quasi periodic solutions of hamiltonian systems for vortex lattices is illustrated in a simple example involving two degrees of freedom the geometry of intersecting singular manifolds of the hamiltonians introduces suitable canonical transformations which put the hamiltonian into the form of singular weakly coupled oscillators as by products of this procedure additional integrals of motion are found for the leading term in the transformed hamiltonian these extra integrals are approximate invariants for the full hamiltonians

on asymptotic properties of predictive distributions summary second order asymptotic theory of predictive distributions is investigated it is shown that estimative distributions with asymptotically efficient estimators can be improved to predictive distributions that do not belong to the model the model is assumed to be a multidimensional curved exponential family predictive distributions can be constructed by shifting estimative distributions in a direction orthogonal to the model the average kullback leibler divergence from the true distribution to a predictive distribution is represented as the sum of two components one of them depends on the choice of the estimative distribution and the other depends on the shift orthogonal to the model the optimal shift orthogonal to the model is obtained a quantity that can be regarded as the mixture mean curvature of the model in the space of all probability distributions is introduced the difference in average kullback leibler divergence between an estimative distribution and the optimal predictive distribution obtained by shifting it can be regarded as the mixture mean curvature of the model an asymptotic expression for bayesian predictive distributions is also obtained

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

improved minimax predictive densities under kullback leibler loss let x n p u x i and y n p v y i be independentp dimensional multivariate normal vectors with common unknown mean based on only observing x x we consider the problem of obtaining a predictive density p y x for y that is close to p y as measured by expected kullback leibler loss a natural procedure for this problem is the formal bayes predictive density p u y x under the uniform prior u 1 which is best invariant and minimax we show that any bayes predictive density will be minimax if it is obtained by a prior yielding a marginal that is superharmonic or whose square root is superharmonic this yields wide classes of minimax procedures that dominate p u y x including bayes predictive densities under superharmonic priors fundamental similarities and differences with the parallel theory of estimating a multivariate normal mean under quadratic loss are described

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

invariant dynamical systems embedded in the n vortex problem on a sphere with pole vortices abstract we are concerned with the system of n vortex points on a sphere with two fixed vortex points at the poles this article gives a reduction method of the system to invariant dynamical systems when all the vortex points have the same strength it is carried out by considering the invariant property of the system with respect to the shift and pole reversal transformations for which the polygonal ring configuration of the n vortex points at the line of latitude called the n ring remains unchanged we prove that there exists a 2 p dimensional invariant dynamical system reduced by the p shift transformation for an arbitrary factor p of n the p shift invariant system is equivalent to the p vortex points system generated by the averaged hamiltonian with the modified pole vortices it is also shown that the system can be reduced by the pole reversal transformation when the pole vortices are identical since the reduced dynamical systems are defined in the linear space spanned by the eigenvectors given in the linear stability analysis for the n ring we obtain the inclusion relation among the invariant reduced dynamical systems this allows us to decompose the system of a large number of vortex points into a collection of invariant reduced subsystems

a note on modified estimative prediction limits and distributions in this paper a modification of the estimative prediction limit is derived for which the coverage probability equals the target value to third order accuracy under ordinary repeated sampling it provides an explicit approximate solution to the equation given by barndorff nielsen cox 1996 which determines an upper prediction limit for an unobserved random variable the corresponding predictive density is specified in a simple closed form and its main properties are emphasised

likelihood and other approaches to prediction in dynamic models abstract in this paper we consider the problem of generating multi period predictions from two simple dynamic models an autoregressive model and a geometric random walk the autoregressive model constitutes a useful paradigm for many of the practical problems of prediction because it possesses a number of features that differentiate it sharply from the standard linear regression model the geometric random walk model is widely used in macroeconomics and finance and is fundamentally non normal the ideal situation for the prediction problem would be to know the true density of the future observations unfortunately that density depends on parameters that are unknown and must be estimated we analyze six prediction functions approximations of the true density that attempt to circumvent this problem we contrast the theoretical properties of the likelihood prediction function proposed by cooley and parke 1986 with certainty equivalence prediction functions and mean squared error prediction functions the results of a monte carlo study illustrate the relative performance of the alternative prediction functions for conditional predictions and for the analysis of policy interventions the results confirm the importance of accounting for parameter uncertainty and approximating the true shape of the future density

a simple predictive density based on the p formula the predictive density proposed by harris 1989 is based on integrating the density for a new observation with respect to the estimated sampling distribution of the maximum likelihood estimator of the unknown parameter this has good properties but is rather complicated to compute even for simple models an approximation to the harris proposal is considered which consists of approximating the sampling distribution of the maximum likelihood estimator of the unknown parameter by barndorff nielsen s 1983 p formula and then using a laplace approximation with o n 1 correction terms for integrating out the parameter the result can generally be expressed in terms of standard likelihood derivatives and takes a quite simple form for exponential families and for location models

relative equilibria of point vortices on the sphere we prove the existence of many different symmetry types of relative equilibria for systems of identical point vortices on a non rotating sphere the proofs use the rotational symmetry group so 3 and the resulting conservation laws the time reversing reflectional symmetries in o 3 and the finite symmetry group of permutations of identical vortices results include both global existence theorems and local results on bifurcations from equilibria a more detailed study is made of relative equilibria which consist of two parallel rings with n vortices in each rotating about a common axis the paper ends with discussions of the bifurcation diagrams for systems of 3 6 identical vortices

on singular hamiltonians the existence of quasi periodic solutions and nonlinear stability on considere l existence de solutions quasiperiodiques pour une classe de hamiltoniens singuliers et la stabilite non lineaire des singularites

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

estimability and likelihood inference for generalized linear mixed models using data cloning maximum likelihood estimation for generalized linear mixed models glmm an important class of statistical models with substantial applications in epidemiology medical statistics and many other fields poses significant computational difficulties in this article we use data cloning a simple computational method that exploits advances in bayesian computation in particular the markov chain monte carlo method to obtain maximum likelihood estimators of the parameters in these models this method also leads to a simple estimator of the asymptotic variance of the maximum likelihood estimators determining estimability of the parameters in a mixed model is in general a very difficult problem data cloning provides a simple graphical test to not only check if the full set of parameters is estimable but also and perhaps more importantly if a specified function of the parameters is estimable one of the goals of mixed models is to predict random effects we suggest a frequentist method to obtain predict

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system