stability for quasi periodically perturbed hill s equations we consider a perturbed hill s equation of the form open image in new window p0 t p1 t 0 where p0 is real analytic and periodic p1 is real analytic and quasi periodic and is small assuming diophantine conditions on the frequencies of the decoupled system i e the frequencies of the external potentials p0 and p1 and the proper frequency of the unperturbed 0 hill s equation but without making any assumptions on the perturbing potential p1 other than analyticity we prove that quasi periodic solutions of the unperturbed equation can be continued into quasi periodic solutions if lies in a cantor set of relatively large measure in open image in new window where 0 is small enough our method is based on a resummation procedure of a formal lindstedt series obtained as a solution of a generalized riccati equation associated to hill s problem

higher order properties of gmm and generalized empirical likelihood estimators in an effort to improve the small sample properties of generalized method of moments gmm estimators a number of alternative estimators have been suggested these include empirical likelihood el continuous updating and exponential tilting estimators we show that these estimators share a common structure being members of a class of generalized empirical likelihood gel estimators we use this structure to compare their higher order asymptotic properties we find that gel has no asymptotic bias due to correlation of the moment functions with their jacobian eliminating an important source of bias for gmm in models with endogeneity we also find that el has no asymptotic bias from estimating the optimal weight matrix eliminating a further important source of bias for gmm in panel data models we give bias corrected gmm and gel estimators we also show that bias corrected el inherits the higher order property of maximum likelihood that it is higher order asymptotically efficient relative to the other bias corrected estimators

bayesian and frequentist bartlett corrections for likelihood ratio and conditional likelihood ratio tests summary this paper characterizes priors under which the bayesian and frequentist bartlett corrections for the likelihood ratio and the conditional likelihood ratio clr statistics differ by o 1 it is seen that except for sample points with negligible probability the clr statistic has a posterior distribution for which a posterior bartlett correction exists this observation leads to an alternative proof of the existence of a frequentist bartlett correction for the clr statistic

second order power comparisons for a class of nonparametric likelihood based tests summary this paper compares the second order power properties of a broad class of nonparametric likelihood tests recently introduced by baggerly 1998 as a generalisation of owen s 1988 empirical likelihood it is shown that in a multi parameter setting identity of power up to first order does not imply identity up to second order unless one considers the average power criterion it is also shown that the empirical likelihood ratio enjoys an optimality property in terms of local maximinity

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

approximate methods for assessing influence and sensitivity in bayesian analysis we use asymptotic approximations to posterior expectations in deriving diagnostics for case influence and prior sensitivity we obtain an easily interpreted expression for the change in posterior expectation following perturbation of the problem which also produces a useful interpretation of the likelihood displacement discussed by cook weisberg in addition we propose two simple diagnostics of posterior nonnormality the leukaemia data analysed by feigl zelen are used to illustrate the diagnostic measures we discuss

quasiperiodic motions in dynamical systems review of a renormalization group approach power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory in the case of quasiperiodic solutions the issue of convergence of the series is bedeviled by the so called small divisor problem in this paper we review a method recently introduced to deal with such a problem based on renormalization group ideas and multiscale techniques applications to both quasi integrable hamiltonian systems kolmogorov arnold moser kam theory and non hamiltonian dissipative systems are discussed the method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged leading to a solution which is not analytic in the perturbation parameter we consider explicitly examples of solutions which are only c in the perturbation parameter or even defined on a cantor set

lower dimensional invariant tori for quasiperiodically forced circle diffeomorphisms abstract for any analytic quasiperiodically forced circle diffeomorphisms p q e f where f is fixed and e is small we show that if is diophantine and the fibred rotation number of the diffeomorphism remains constant in a unilateral neighborhood of e 0 i e there is a unilateral phase locking at e 0 then the diffeomorphism has at least one analytic q invariant torus provided e is small enough

hamiltonian monte carlo sampling in bayesian empirical likelihood computation summary we consider bayesian empirical likelihood estimation and develop an efficient hamiltonian monte carlo method for sampling from the posterior distribution of the parameters of interest the method proposed uses hitherto unknown properties of the gradient of the underlying log empirical likelihood function we use results from convex analysis to show that these properties hold under minimal assumptions on the parameter space prior density and the functions used in the estimating equations determining the empirical likelihood our method employs a finite number of estimating equations and observations but produces valid semiparametric inference for a large class of statistical models including mixed effects models generalized linear models and hierarchical bayes models we overcome major challenges posed by complex non convex boundaries of the support routinely observed for empirical likelihood which prevent efficient implementation of traditional markov chain monte carlo methods like random walk metropolis hastings sampling etc with or without parallel tempering a simulation study confirms that our method converges quickly and draws samples from the posterior support efficiently we further illustrate its utility through an analysis of a discrete data set in small area estimation

bayesian estimation and comparison of moment condition models in this article we develop a bayesian semiparametric analysis of moment condition models by casting the problem within the exponentially tilted empirical likelihood etel framework we use this f

lower dimensional invariant tori in the regions of instability for nearly integrable hamiltonian systems consider a hamiltonian system of kam type h p q n p p p q with n degrees of freedom n 2 where the hessian of n is nondegenerate for one resonance condition 0 i n there is an immersed n 1 dimensional submanifold in action variable space where almost every point corresponds to a resonant torus for the unperturbed system which is foliated by n 1 dimensional ergodic components it is shown in this paper that there is a subset of with positive n 1 dim lebesgue measure such that for each resonant torus corresponding to a point in this set at least two n 1 dimensional tori can survive perturbations generically one is hyperbolic and the other one is elliptic

bayesian prediction with approximate frequentist validity we characterize priors which asymptotically match the posterior coverage probability of a bayesian prediction region with the corresponding frequentist coverage probability this is done considering both posterior quantiles and highest predictive density regions with reference to a future observation the resulting priors are shown to be invariant under reparameterization the role of jeffreys prior in this regard is also investigated it is further shown that for any given prior it may be possible to choose an interval whose bayesian predictive and frequentist coverage probabilities are asymptotically matched

empirical type likelihoods allowing posterior credible sets with frequentist validity higher order asymptotics summary with reference to a general class of empirical type likelihoods we develop higher order asymptotics for the frequentist coverage of bayesian credible sets based on posterior quantiles and highest posterior density these asymptotics in turn characterise members of the class that allow approximate frequentist validity of such sets it is seen that the usual empirical likelihood does not enjoy this property up to the order of approximation considered here

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

objective bayes conditional inference and the signed root likelihood ratio statistic barndorffnielsen oe 1986 biometrika v73 p307 barndorff nielsen o e 1994 inference asymptotic datta g s 2004 probability matching diciccio tj 1993 j roy stat soc b met v55 p305 diciccio tj 1993 biometrika v80 p731 doi 10 1093 biomet 80 4 731 diciccio tj 1994 j roy stat soc b met v56 p397 diciccio tj 1994 scand j stat v21 p447 diciccio tj 2010 biometrika v97 p497 doi 10 1093 biomet asq002 hall p 1992 bootstrap edgeworth lawless jf 1982 stat models methods mukerjee r 1993 biometrika v80 p499 doi 10 1093 biomet 80 3 499 mukerjee r 1997 biometrika v84 p970 doi 10 1093 biomet 84 4 970 nicolaou a 1993 j roy stat soc b met v55 p377 peers hw 1965 j roy stat soc b v27 p9 tibshirani r 1989 biometrika v76 p604 doi 10 1093 biomet 76 3 604 tierney l 1986 j am stat assoc v81 p82 doi 10 2307 2287970 welch b l 1963 j r stat soc b v62 p159

properties of sufficiency and statistical tests 1 in a previous paper dealing with the importance of properties of sufficiency in the statistical theory of small samples attention was mainly confined to the theory of estimation in the present paper the structure of small sample tests whether these are related to problems of estimation and fiducial distributions or are of the nature of tests of goodness of fit is considered further

bartlett adjustment of empirical discrepancy statistics summary we introduce two classes of empirical discrepancy statistics that extend empirical likelihood and establish simple conditions under which they admit bartlett adjustment

adjusted versions of profile likelihood and directed likelihood and extended likelihood several adjusted and invariant versions of profile likelihood and of directed likelihood short for signed log likelihood ratio are discussed these adjusted versions which fall into two groups are closely related to modified profile likelihood and modified directed likelihood the first group has the advantage of not requiring an explicit specification of an ancillary but partly at a cost of a loss of asymptotic accuracy in comparison with the modified versions the second group consists of some asymptotically sharpened versions of modified profile likelihood furthermore a concept of extended likelihood is proposed which associates a likelihood like function to certain combinants

a simple method for the adjustment of profile likelihoods summary we propose a simple adjustment for profile likelihoods the aim of the adjustment is to alleviate some of the problems inherent in the use of profile likelihoods such as bias inconsistency and overoptimistic variance estimates the adjustment is applied to the profile loglikelihood score function at each parameter value so that its mean is zero and its variance is the negative expected derivative matrix of the adjusted score function for cases in which explicit calculation of the adjustments is difficult we give two methods to simplify their computation an automatic simulation method that requires as input only the profile loglikelihood and its first few derivatives first order asymptotic expressions some examples are provided and a comparison is made with the conditional profile log likelihood of cox and reid

bartlett type adjustment for the conditional likelihood ratio statistic of cox and reid summary this paper explicitly derives a bartlett type adjustment for the conditional likelihood ratio statistic of cox reid via that for the usual likelihood ratio statistic

on the influence of the variable ordering for algorithmic learning using obdds obdds with a fixed variable ordering are used successfully as data structure in experiments with learning heuristics based on examples in this paper it is shown that for some functions it is necessary to develop an algorithm to learn also a good obdd variable ordering there are functions with the following properties they have obdds of linear size for optimal variable orderings but for all but a small fraction of all variable orderings one needs large size to represent a list of randomly chosen examples these properties are shown for simple functions like the multiplexer and the inner product

a quasi periodic poincare s theorem we study the persistence of invariant tori on resonant surfaces of a nearly integrable hamiltonian system under the usual kolmogorov non degenerate condition by introducing a quasi linear iterative scheme to deal with small divisors we generalize the poincare theorem on the maximal resonance case i e the periodic case to the general resonance case i e the quasi periodic case by showing the persistence of majority of invariant tori associated to non degenerate relative equilibria on any resonant surface

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

on the relationship between bayesian and non bayesian interval estimates summary let y1 y denote independent observations each distributed according to a density depending on a scalar parameter 0 suppose that we are interested in constructing an interval estimate for 0 one approach is to construct a confidence region with a specified coverage probability based on the likelihood ratio test statistic another approach is to construct a highest posterior density region with a specified posterior probability the goal of this paper is to study the relationship between these two approaches in particular we derive asymptotic expansions for the posterior probability of confidence regions based on the likelihood ratio test statistic and for the coverage probability of highest posterior density regions conditions under which the two methods lead to identical regions at least approximately are also given

degenerate elliptic resonances quasi periodic motions on invariant tori of an integrable system of dimension smaller than half the phase space dimension may continue to exist after small perturbations the parametric equations of the invariant tori can often be computed as a formal power series in the perturbation parameter and can be given a meaning via resummations here we prove that for a class of elliptic tori a resummation algorithm can be devised and proved to be convergent thus extending to such lower dimensional invariant tori the methods employed to prove convergence of the lindstedt series either for the maximal i e kam tori or for the hyperbolic lower dimensional invariant tori

fractional lindstedt series the parametric equations of the surfaces on which highly resonant quasiperiodic motions develop lower dimensional tori cannot be analytically continued in general in the perturbation parameter e i e they are not analytic functions of e however rather generally quasiperiodic motions whose frequencies satisfy only one rational relation resonances of order 1 admit formal perturbation expansions in terms of a fractional power of e depending on the degeneration of the resonance we find conditions for this to happen and in such a case we prove that the formal expansion is convergent after suitable resummation

approximate marginal densities of nonlinear functions this paper presents an asymptotic approximation for the marginal density of a nonlinear function g 0 that is applicable when the joint density of 0 is dominated by a single mode and the jacobian of g is of full rank near that mode the approximation is based on laplace s method and its asymptotic properties are similar to those of the saddlepoint approximation the approximation is applied to the computation of a marginal posterior density a marginal sampling density and a marginal density based on a multivariate saddlepoint approximation to a joint density

gevrey smoothness of invariant tori for analytic nearly integrable hamiltonian systems under russmann s non degeneracy condition abstract in this paper we prove gevrey smoothness of the persisting invariant tori for small perturbations of an analytic integrable hamiltonian system with russmann s non degeneracy condition by an improved kam iteration method with parameters

bayesian exponentially tilted empirical likelihood summary while empirical likelihood has been shown to exhibit many of the properties of conven tional parametric likelihoods a formal probabilistic interpretation has so far been lacking we show that a likelihood function very closely related to empirical likelihood naturally arises from a nonparametric bayesian procedure which places a type of noninformative prior on the space of distributions this prior gives preference to distributions having a small support and among those sharing the same support it favours entropy maximising distributions the resulting nonparametric bayesian procedure admits a computationally convenient representation as an empirical likelihood type likelihood where the probability weights are obtained via exponential tilting the proposed methodology provides an attractive alternative to the bayesian bootstrap as a nonparametric limit of a bayesian procedure for moment condition models

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

bayesian and frequentist confidence intervals arising from empirical type likelihoods summary for a general class of empirical type likelihoods for the population mean higher order asymp totics are developed with a view to characterizing its members which allow for any given prior the existence of a confidence interval that has approximately correct posterior as well as fre quentist coverage in particular it is seen that the usual empirical likelihood always allows such a confidence interval while many of its variants proposed in the literature do not enjoy this property an explicit form of the confidence interval is also given

improving the variable ordering of obdds is np complete ordered binary decision diagrams are a useful representation of boolean functions if a good variable ordering is known variable orderings are computed by heuristic algorithms and then improved with local search and simulated annealing algorithms this approach is based on the conjecture that the following problem is np complete given an obdd g representing f and a size bound s does there exist an obdd g respecting an arbitrary variable ordering representing f with at most s nodes this conjecture is proved

bayesian empirical likelihood summary research has shown that empirical likelihood tests have many of the same asymptotic properties as those derived from parametric likelihoods this leads naturally to the possibility of using empirical likelihood as the basis for bayesian inference different ways in which this goal might be accomplished are considered the validity of the resultant posterior inferences is examined as are frequentist properties of the bayesian empirical likelihood intervals

on the level error after bartlett adjustment of the likelihood ratio statistic summary it is well known that bartlett adjustment reduces level error of the likelihood ratio statistic from order n l to order n3 2 in the present note we show that level error of the adjusted statistic is actually order n2

bayesian pseudo empirical likelihood intervals for complex surveys summary bayesian methods for inference on finite population means and other parameters by using sample survey data face hurdles in all three phases of the inferential procedure the formulation of a likelihood function the choice of a prior distribution and the validity of posterior inferences under the design based frequentist framework in the case of independent and identically distributed observations the profile empirical likelihood function of the mean and a non informative prior on the mean can be used as the basis for inference on the mean and the resulting bayesian empirical likelihood intervals are also asymptotically valid under the frequentist set up for complex survey data we show that a pseudo empirical likelihood approach can be used to construct bayesian pseudo empirical likelihood intervals that are asymptotically valid under the design based set up the approach proposed compares favourably with a full bayesian analysis under simple random sampling without replacement it is also valid under general single stage unequal probability sampling designs unlike a full bayesian analysis moreover the approach is very flexible in using auxiliary population information and can accommodate two scenarios which are practically important incorporation of known auxiliary population information for the construction of intervals by using the basic design weights calculation of intervals by using calibration weights based on known auxiliary population means or totals

expected lengths of confidence intervals based on empirical discrepancy statistics we consider a very general class of empirical discrepancy statistics that includes the cressie read discrepancy statistics and in particular the empirical likelihood ratio statistic higher order asymptotics for expected lengths of associated confidence intervals are investigated an explicit formula is worked out and its use for comparative purposes is discussed it is seen that the empirical likelihood ratio statistic which enjoys interesting second order power properties loses much of its edge under the present criterion

on a property of probability matching priors matching the alternative coverage probabilities summary in frequentist inference based on confidence sets both the true coverage and the probability for a confidence set to include an alternative value of the parameter of interest are important thus if probability matching priors also match such alternative coverage probabilites there is perhaps a stronger justification for calling them noninformative considering contiguous alternatives we obtain the relevant necessary and sufficient conditions in particular with n denoting the sample size it is seen that probability matching priors up to o n 2 also match the alternative coverage probabilities up to that order while this is not necessarily the case with probability matching priors up to o n

empirical likelihood for small area estimation abstract small areaestimation has become a topic of growingimportance inrecent years due to increasing demand for small area statistics fromboth the public and private sectors the current methodologies inthis area with possibly a handful of exceptions are all parametric be it bayesian or frequentist even the so called nonparamet ric procedures based on empirical best linear unbiased predictors eblup assume linearity of the estimators of small area means resampling methods like jacknife and bootstrap are used mostlyfor mean squared error estimation in this paper we propose analternative bayesian approach based on empirical likelihood whichdoes not require the assumption of either a parametric likelihoodor linearity of estimators the proposed method can also handleboth discrete and continuous data in a uni ed manner empiricallikelihood for both area and unit level models are introduced theimproved performance of our estimator is illustrated through twoexamples involving both discrete and continuous data key words and phrases area level exponentially tilted t prior unit level

a kam theorem for hyperbolic type degenerate lower dimensional tori in hamiltonian systems a kam theorem for degenerate lower dimensional tori in nearly integrable hamiltonian systems is given in this paper for the non degenerate cases both hyperbolic and elliptic the kam theorem has been well established by many authors 8 9 11 13 14 17

empirical likelihood as a goodness of fit measure summary the method of empirical likelihood can be viewed as one of allocating probabilities to an n cell contingency table so as to minimise a goodness of fit criterion it is shown that when the cressie read power divergence statistic is used as the criterion confidence regions enjoying the same convergence rates as those found for empirical likelihood can be obtained for the entire range of values of the cressie read parameter including 1 maximum entropy 0 empirical likelihood and 1 pearson s z2 it is noted that in the power divergence family empirical likelihood is the only member which is bartlettcorrectable however simulation results suggest that for the mean using a scaled f distribution yields more accurate coverage levels for moderate sample sizes

on bartlett adjustments for approximate bayesian inference summary in wide generality the posterior distributions of the likelihood ratio statistic and the posterior ratio statistic are chi squared to error of order o n where n is sample size the error in the chi squared approximation can be reduced to order o n 2 by bartlett correction in this paper explicit formulae are derived for the bartlett adjustment factors of both statistics and the derivations are based on the tierney kass kadane 1989 asymptotic approximation for marginal posterior probability density functions the use of numerical differentiation to facilitate calculation of the bartlett adjustments is also described some applications are considered that concern inference about regression models from both complete and right censored data

resummation of perturbation series and reducibility for bryuno skew product flows we consider skew product systems on d l sl 2 for bryuno base flows close to constant coefficients depending on a parameter in any dimension d and we prove reducibility for a large measure set of values of the parameter the proof is based on a resummation procedure of the formal power series for the conjugation and uses techniques of renormalisation group in quantum field theory

on confidence intervals associated with the usual and adjusted likelihoods in the presence of nuisance parameters we derive an explicit higher order asymptotic formula to compare the expected lengths of confidence intervals given by likelihood ratio statistics arising from the usual profile likelihood and various adjustments thereof highest posterior density regions with approximate frequentist validity are also included in the study

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

on the implementation of local probability matching priors for interest parameters probability matching priors are priors for which the posterior probabilities of certain specified sets are exactly or approximately equal to their coverage probabilities these priors arise as solutions of partial differential equations that may be difficult to solve either analytically or numerically recently levine casella 2003 presented an algorithm for the implementation of probability matching priors for an interest parameter in the presence of a single nuisance parameter in this paper we develop a local implementation that is very much more easily computed a local probability matching prior is a data dependent approximation to a probability matching prior and is such that the asymptotic order of approximation of the frequentist coverage probability is not degraded we illustrate the theory with a number of examples including three discussed in levine casella 2003

on an exact probability matching property of right invariant priors summary the paper considers priors which are right invariant with respect to the haar measure it is shown that the posterior coverage probabilities of certain invariant bayesian predictive regions exactly match the corresponding frequentist probabilities several examples are given to illustrate the main result

periodic solutions for a class of nonlinear partial differential equations in higher dimension we prove the existence of periodic solutions in a class of nonlinear partial differential equations including the nonlinear schrodinger equation the nonlinear wave equation and the nonlinear beam equation in higher dimension our result covers cases of completely resonant equations where the bifurcation equation is infinite dimensional such as the nonlinear schrodinger equation with zero mass for which solutions which at leading order are wave packets are shown to exist

a decomposition for the likelihood ratio statistic and the bartlett correction a bayesian argument let l n 1 log p x be the log likelihood of an n dimensional x under a p dimensional let j be the mle under h j 1 1 0 j j 0 and 0 be the unrestricted mle define t j as 2n l j 1 1 2 sgn j j 1 j 0 let t t 1 t p then under regularity conditions the following theorem is proved under 0 t is asymptotically n n 1 2 a 0 n 1 a j n 1 o n 3 2 where j is the identity matrix the result is proved by first establishing an analogous result when is random and then making the prior converge to a degenerate distribution the existence of the bartlett correction to order n 3 2 follows from the theorem we show that an edgeworth expansion with error o n 2 for t involves only polynomials of degree less than or equal to 3 and hence verify rigorously lawley s 1956 result giving the order of the error in the bartlett correction as o n 2

oscillator synchronisation under arbitrary quasi periodic forcing we study the problem of existence of response solutions for a real analytic one dimensional system consisting of a rotator subject to a small quasi periodic forcing with bryuno frequency vector we prove that at least one response solution always exists without any assumption on the forcing besides smallness and analyticity this strengthens the results available in the literature where generic non degeneracy conditions are assumed the proof is based on a diagrammatic formalism and relies on renormalisation group techniques which exploit the formal analogy with problems of quantum field theory a crucial role is played by remarkable identities between classes of diagrams

on lower dimensional invariant tori in reversible systems abstract in this paper a result on the persistence of lower dimensional invariant tori in reversible systems is obtained under some weak non degenerate conditions such a result can be applied to the case where normal frequencies are not simple and some of them may be zero we also give an answer of a conjecture of m b sevryuk 1986 reversible systems lecture notes in math vol 1211 springer verlag new york berlin about the existence of lower dimensional invariant tori in resonant zones

bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator for rather general parametric models a simple connection is established between the bartlett adjustment factor of the log likelihood ratio statistic and the normalizing constant c of the formula c i i 1 2l for the conditional distribution of a maximum likelihood estimator as applied to the full model and the model of the hypothesis tested this leads to a relatively simple demonstration that division of the likelihood ratio statistic by a suitable constant or estimated factor improves the chi squared approximation to its distribution various expressions for these quantities are discussed in particular for the case of a one dimensional parameter an approximation to the constants involved is derived which does not require integration over the sample space

a symbolic algorithm for studying adjustments to the profile likelihood summary a symbolic algorithm for the derivation of the asymptotic expansion of the maximum likelihood estimate was first introduced by andrews stafford 1993 this algorithm is quite general and can be modified to give asymptotic expansions for the maximum likelihood estimate based on either the conditional modified or adjusted profile likelihoods expansions are obtained quickly with no chance for human clerical error as a result these three adjustments can be easily studied and some new statistical results are obtained bartlett corrections of each adjusted likelihood and asymptotic expansions for the first four moments of the maximum likelihood estimate based on the conditional and adjusted profile likelihoods are derived

frequentist and bayesian bartlett correction of test statistics based on adjusted profile likelihoods summary for inference about a multidimensional parameter b in the presence of nuisance param

minimal invariant tori in the resonant regions for nearly integrable hamiltonian systems consider a real analytical hamiltonian system of kam type h p q n p p p q that has n degrees of freedom n 2 and is positive definite in p let r n 0 k z n in this paper we show that for most rotation vectors in in the sense of n 1 dimensional lebesgue measure there is at least one n 1 dimensional invariant torus these tori are the support of corresponding minimal measures the lebesgue measure estimate on this set is uniformly valid for any perturbation

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

information bias and adjusted profile likelihoods summary the bias and information bias of the ordinary profile score statistic are both typically of order 0 1 several additive adjustments to the profile score statistic that reduce its bias to order o n have been proposed in certain situations the information bias of these adjusted profile score statistics is also reduced to order o n q specifically the modified profile likelihood of barndorff nielsen yields an adjusted profile score statistic having both reduced bias and reduced information bias in general however a bias reducing adjustment to the profile score statistic will not automatically reduce the order of the information bias as well in this paper an analytical formula is obtained for the information bias of a general bias adjusted profile score statistic this formula is used to compare bias reducing adjustments to the profile score statistic as well as to construct further adjustments that reduce the information bias to o n 1 several examples are presented to illustrate use of the formula for information bias in particular the information bias formula may be utilized in a criterion for choosing between orthogonal parameterizations for the conditional profile likelihood of cox and reid

on the equivalence of prospective and retrospective likelihood methods in case control studies summary we present new approaches to analyzing case control studies using prospective likelihood methods in the classical framework we extend the equality of the profile likelihoods to the barndorff nielsen modified profile likelihoods for prospective and retrospective models this enables simple and accurate approximate conditional inference for stratified case control studies of moderate stratum size in the bayesian framework we provide sufficient conditions on priors for the prospective model parameters to yield a prospective marginal posterior density equal to its retrospective counterpart our results extend the prospective retrospective equivalence in the bayesian paradigm with a more general class of priors than has previously been investigated

improved small sample inference in the mixed linear model bartlett correction and adjusted likelihood summary the mixed linear model is a popular method for analysing unbalanced repeated measurement data the classical statistical tests for parameters in this model are based on asymptotic theory that is unreliable in the small samples that are often encountered in practice for testing a given fixed effect parameter with a small sample we develop and investigate refined likelihood ratio lr tests the refinements considered are the bartlett correction and use of the cox reid adjusted likelihood these are examined separately and in combination we illustrate the various lr tests on an actual data set and compare them in two simulation studies the conventional lr test yields type i error rates that are higher than nominal the adjusted lr test yields rates that are lower than nominal with absolute accuracy similar to that of the conventional lr test in the first simulation study and better in the second the bartlett correction substantially improves the accuracy of the type i error rates with either the conventional or the adjusted lr test in many cases error rates that are very close to nominal are achieved with the refined methods

birkhoff periodic orbits for small perturbations of completely integrable hamiltonian systems with convex hamiltonians pour des systemes a n degres de liberte on montre l existence d au moins n orbites periodiques distinctes avec un vecteur frequence rationnel admissible proche du tore correspondant du systeme non perturbe sous l hypothese que le hamiltonien du systeme non perturbe a des surfaces d energie convexes pour les variables action angle

birkhoff kolmogorov arnold moser tori in convex hamiltonian systems for the hamiltonian systems of kam type it is proved that some lower dimensional invariant tori always exist in the resonance gaps although those maximum tori can not survive small perturbations in the generic case

proper likelihoods for bayesian analysis summary the validity of posterior probability statements follows from probability calculus when the likelihood is the density of the observations to investigate other cases a second more intuitive definition of validity is introduced based on coverage of posterior sets this notion of validity suggests that the likelihood must be the density of a statistic not necessarily sufficient for posterior probability statements to be valid a convenient numerical method is proposed to invalidate the use of certain likelihoods for bayesian analysis integrated marginal and conditional likelihoods derived to avoid nuisance parameters are also discussed

perturbations of lower dimensional tori in the resonant zone for reversible systems consider a nearly integrable system which is reversible with respect to an involution of n n type for a given l resonant surface 1 l n we show that there is a subset of it with positive measure such that in each resonant torus corresponding to a point of this set at least one dimensional invariant submanifold can survive any small perturbations

on the nonapproximability of boolean function by obdds and read k times branching programs branching programs are considered as a nonuniform model of computation in complexity theory as well as a data structure for boolean functions in several applications in many applications e g verification exact representations are required for learning boolean functions f on the basis of classified examples it is sufficient to produce the representation of a function g approximating f this motivates the investigation of the size of the smallest branching program approximating f although several nonapproximability results are contained in the papers on randomized branching programs these results often do not work for the uniform distribution which is the most important one in applications here the following nonapproximability results are presented 1 it is proven that two simple and well known functions from the branching program literature require exponential size to be approximated with respect to the uniform distribution by obdds which are the most important type of branching programs in applications 2 the first truly exponential lower bound on the size of approximating syntactic read k times branching programs with respect to the uniform distribution and error probability 1 2 2 n n the input size is shown in order to improve upon the best previous results for error probabilities smaller than 1 3 a strong combinatorial lemma from a paper of ajtai on linear length branching programs is exploited