a posteriori error estimates for mixed finite element approximations of elliptic problems we derive residual based a posteriori error estimates of the flux in l 2 norm for a general class of mixed methods for elliptic problems the estimate is applicable to standard mixed methods such as the raviart thomas nedelec and brezzi douglas marini elements as well as stabilized methods such as the galerkin least squares method the element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order

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

discontinuous galerkin methods for incompressible and nearly incompressible elasticity by nitsche s method we propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes we show optimal error estimates that are uniform with respect to poisson s ratio the method is thus locking free we also introduce an equivalent mixed formulation allowing for completely incompressible elasticity problems numerical results are presented

an hp adaptive mixed discontinuous galerkin fem for nearly incompressible linear elasticity we develop the a posteriori error estimation of mixed hp version discontinuous galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions computable upper and lower bounds on the error measured in terms of a natural mesh dependent energy norm are derived the bounds are explicit in the local mesh sizes and approximation orders and are independent of the locking parameter a series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure

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

laplace transform analysis of a multiplicative asset transfer model abstract we analyze a simple asset transfer model in which the transfer amount is a fixed fraction f of the giver s wealth the model is analyzed in a new way by laplace transforming the master equation solving it analytically and numerically for the steady state distribution and exploring the solutions for various values of f 0 1 the laplace transform analysis is superior to agent based simulations as it does not depend on the number of agents enabling us to study entropy and inequality in regimes that are costly to address with simulations we demonstrate that boltzmann entropy is not a suitable e g non monotonic measure of disorder in a multiplicative asset transfer system and suggest an asymmetric stochastic process that is equivalent to the asset transfer model

on expected volumes of multidimensional confidence sets associated with the usual and adjusted likelihoods we consider a general multiparameter set up where both the interest and the nuisance parameters are possibly vector valued we derive an explicit higher order asymptotic formula to compare the expected volumes of confidence sets given by likelihood ratio statistics arising from the usual profile likelihood and various adjustments thereof our general framework also allows us to include highest posterior density regions with approximate frequentist validity in the study the fact that our interest parameter is possibly vector valued complicates the derivation and warrants the development of special tools and techniques

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

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

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 unifying theory of a posteriori finite element error control residual based a posteriori error estimates are derived within a unified setting for lowest order conforming nonconforming and mixed finite element schemes the various residuals are identified for all techniques and problems as the operator norm l of a linear functional of the form in the variable of a sobolev space v the main assumption is that the first order finite element space is included in the kernel ker l of l as a consequence any residual estimator that is a computable bound of l can be used within the proposed frame without further analysis for nonconforming or mixed fe schemes applications are given for the laplace stokes and navier lame equations

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

a unifying theory of a posteriori error control for nonconforming finite element methods residual based a posteriori error estimates were derived within one unifying framework for lowest order conforming nonconforming and mixed finite element schemes in carstensen numer math 100 617 637 2005 therein the key assumption is that the conforming first order finite element space v h c annulates the linear and bounded residual l written v h c subseteq rm ker ell that excludes particular nonconforming finite element methods ncfems on parallelograms in that v h c not subset rm ker ell the present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates also for mortar and discontinuous galerkin methods the key assumption is the existence of some bounded linear operator pi v h c rightarrow v h nc with some elementary properties it is conjectured that the more general hypothesis h1 h3 can be established for all known ncfems applications on various nonstandard finite element schemes for the laplace stokes and navier lame equations illustrate the presented unifying theory of a posteriori error control for ncfem

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

energy norm a posteriori error estimates for discontinuous galerkin approximations of the linear elasticity problem we present a residual based a posteriori error estimate in an energy norm of the error in a family of discontinuous galerkin approximations of linear elasticity problems the theory is developed in two and three spatial dimensions and general nonconvex polygonal domains are allowed we also present some illustrating numerical examples

a simple computational homogenization method for structures made of linear heterogeneous viscoelastic materials in this paper a numerical multiscale method is proposed for computing the response of structures made of linearly non aging viscoelastic and highly heterogeneous materials in contrast with most of the approaches reported in the literature the present one operates directly in the time domain and avoids both defining macroscopic internal variables and concurrent computations at micro and macro scales the macroscopic constitutive law takes the form of a convolution integral containing an effective relaxation tensor to numerically identify this tensor a representative volume element rve for the microstructure is first chosen relaxation tests are then numerically performed on the rve correspondingly the components of the effective relaxation tensor are determined and stored for different snapshots in time at the macroscopic scale a continuous representation of the effective relaxation tensor is obtained in the time domain by interpolating the data with the help of spline functions the convolution integral characterizing the time dependent macroscopic stress strain relation is evaluated numerically arbitrary local linear viscoelastic laws and microstructure morphologies can be dealt with implicit algorithms are provided to compute the time dependent response of a structure at the macroscopic scale by the finite element method accuracy and efficiency of the proposed approach are demonstrated through 2d and 3d numerical examples and applied to estimate the creep of structures made of concrete

likelihood ratio tests for fixed model terms using residual maximum likelihood likelihood ratio tests for fixed model terms are proposed for the analysis of linear mixed models when using residual maximum likelihood estimation bartlett type adjustments using an approximate decomposition of the data are developed for the test statistics a simulation study is used to compare properties of the test statistics proposed with or without adjustment with a ward test a proposed test statistic constructed by dropping fixed terms from the full fixed model is shown to give a better approximation to the asymptotic chi 2 distribution than the wald test for small data sets bartlett adjustment is shown to improve the chi 2 approximation for the proposed tests substantially

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

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

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

comparison of different numerical laplace inversion methods for engineering applications abstract laplace transform is a powerful method for enabling solving differential equation in engineering and science using the laplace transform for solving differential equations however sometimes leads to solutions in the laplace domain that are not readily invertible to the real domain by analytical means numerical inversion methods are then used to convert the obtained solution from the laplace domain into the real domain four inversion methods are evaluated in this paper several test functions which arise in engineering applications are used to evaluate the inversion methods we also show that each of the inversion methods is accurate for a particular case this study shows that among all these methods the fourier transform inversion technique is the most powerful but also the most computationally expensive stehfest s method which is used in many engineering applications is easy to implement and leads to accurate results for many problems including diffusion dominated ones and solutions that behave like e t type functions however this method fails to predict e t type functions or those with an oscillatory response such as sine and wave functions

locking free adaptive mixed finite element methods in linear elasticity mixed finite element methods such as peers or the bdms methods are designed to avoid locking for nearly incompressible materials in plane elasticity in this paper we establish a robust adaptive mesh refining algorithm that is rigorously based on a reliable and efficient a posteriori error estimate numerical evidence is provided for the independence of the constants in the a posteriori error bounds and for the efficiency of the adaptive mesh refining algorithm proposed

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

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 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

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

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

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

energy norm a posteriori error estimation for discontinuous galerkin methods in this paper we present a residual based a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of discontinuous galerkin approximations of elliptic problems the theory is developed for an elliptic model problem in two and three spatial dimensions and general nonconvex polygonal domains are allowed we also present some illustrating numerical examples

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

numerical inversion of a laplace transform solution of a diffusion equation with a mixed derivative term we consider a diffusion equation with a mixed derivative term modelling the transient flow of a second grade fluid the same diffusion equation with a mixed derivative term also comes up in the modelling of heat conduction in which the history of the temperature gradient is included the equations differ in the sign of the coefficient of the mixed derivative term we show that the method of lines solution is stable for the transient flow of a second grade fluid and unstable for heat conduction with the temperature gradient history included a laplace transform solution for the heat conduction case is then investigated the laplace transform solution is inverted numerically using a fourier based method a stable solution is obtained the solutions are compared and discussed

locking free adaptive discontinuous galerkin fem for linear elasticity problems an adaptive discontinuous galerkin finite element method for linear elasticity problems is presented we develop an a posteriori error estimate and prove its robustness with respect to nearly incompressible materials absence of volume locking furthermore we present some numerical experiments which illustrate the performance of the scheme on adaptively refined meshes

a posteriori error estimates for the morley plate bending element a local a posteriori error indicator for the well known morley element for the kirchhoff plate bending problem is presented the error indicator is proven to be both reliable and efficient the technique applied is general and it is shown to have also other applications