projected score methods for approximating conditional scores summary this paper extends the projected score methods of small mcleish 1989 it is shown that the conditional score function may be approximated with arbitrarily small stochastic error in terms of a natural basis for the space of centred likelihood ratios the utility of using this basis is established by identifying a u statistic representation theorem and a class of expectation identities for the basis elements making higher order asymptotics more tractable the results are applied to a canonical exponential family model where it is shown that the projected scores with estimated nuisance parameters can provide an accurate approximation to the conditional score function

simple relaxed conditional likelihood when the data are sparse but not exceedingly so we face a trade off between bias and precision that makes the usual choice between conducting either a fully unconditional inference or a fully conditional inference unduly restrictive we propose a method to relax the conditional inference that relies upon commonly available computer outputs in the rectangular array asymptotic setting the relaxed conditional maximum likelihood estimator has smaller bias than the unconditional estimator and smaller mean square error than the conditional estimator

invariant cantor manifolds of quasi periodic oscillations for a nonlinear schrodinger equation this paper was written while both authors were guests of the forschungsinstitut fur mathematik at the eth zurich and we thank the institute for its hospitality pleasant working atmosphere and helpful staff in particular we thank jiirgen moser for numerous stimulating discussions on the subject we also benefitted from a remark by sigurd angenent concerning the analyticity of the solutions the second author also thanks the deutsche forschungsgemeinschaft for their financial support through a heisenberg grant

newton s method and periodic solutions of nonlinear wave equations we prove the existence of periodic solutions of the nonlinear wave equation satisfying either dirichlet or periodic boundary conditions on the interval o the coefficients of the eigenfunction expansion of this equation satisfy a nonlinear functional equation using a version of newton s method we show that this equation has solutions provided the nonlinearity g x u satisfies certain generic conditions of nonresonance and genuine nonlinearity 1993 john wiley sons inc

theoretical and computational issues in modelling material failure in strong discontinuity scenarios the paper deals with several aspects related to numerical modelling of material failure in strong discontinuity settings a the onset and development of local material failure in terms of continuum constitutive models equipped with strain softening closed forms formulas for the solutions of the discontinuous material bifurcation problem are given for a class of those models b finite elements with embedded discontinuities nodal and elemental enrichments families are formulated in the continuum strong discontinuity approach csda c instability treatment a discrete viscous perturbation method at the failure surfaces is presented as a way to substantially improve the robustness of the numerical simulations

modified profile likelihoods in models with stratum nuisance parameters it is well known at least through many examples that when there are many nuisance parameters modified profile likelihoods often perform much better than the profile likelihood ordinary asymptotics almost totally fail to deal with this issue for this reason we study asymptotic properties of the profile and modified profile likelihoods in models for stratified data in a two index asymptotics setting this means that both the sample size of the strata m and the dimension of the nuisance parameter q may increase to infinity it is shown that in this asymptotic setting modified profile likelihoods give improvements with respect to the profile likelihood in terms of consistency of estimators and of asymptotic distributional properties in particular the modified profile likelihood based statistics have the usual asymptotic distribution provided that l m o q 1 3 while the analogous condition for the profile likelihood is l m o q 1

adjusted profile estimating function in settings where the full probability model is not specified consider a general estimating function g 0 y that involves not only the parameters of interest 0 but also some nuisance parameters we consider methods for reducing the effects on g of fitting nuisance parameters we propose cox reid type adjustment to the profile estimating function g y that reduces its bias by two orders typically only the first two moments of the response variable are needed to form the adjustment important applications of this method include the estimation of the pairwise association and main effects in stratified clustered data and estimation of the main effects in a matched pair study a brief simulation study shows that the proposed method considerably reduces the impact of the nuisance parameters

the use of displacement discontinuities in a rate dependent medium abstract this paper illustrates the use of displacement discontinuities in a rate dependent elastoplastic damage model rate dependency is considered in the framework of perzyna viscoplasticity displacement discontinuities are incorporated using the partition of unity property of finite element shape functions a combined continuous discontinuous approach is proposed in which a discontinuity is only inserted after a certain amount of damage various examples illustrate the applicability of the combined approach to granular and quasi brittle materials with and without fibre reinforcement

modified estimating functions in a parametric model the maximum likelihood estimator of a parameter of interest may be viewed as the solution to the equation l p 0 where l p denotes the profile loglikelihood function it is well known that the estimating function l p is not unbiased and that this bias can in some cases lead to poor estimates of an alternative approach is to use the modified profile likelihood function or an approximation to the modified profile likelihood function which yields an estimating function that is approximately unbiased in many cases the maximum likelihood estimating functions are unbiased under more general assumptions than those used to construct the likelihood function for example under first or second moment conditions although the likelihood function itself may provide valid estimates under moment conditions alone the modified profile likelihood requires a full parametric model in this paper modifications to l p are presented that yield an approximately unbiased estimating function under more general conditions

the asymptotic efficiency of conditional likelihood methods summary this paper concerns the efficiency of the conditional likelihood method for inference in models which include nuisance parameters a new concept of ancillarity asymptotic weak ancillarity is introduced it is shown that the conditional maximum likelihood estimator and the conditional score test of 0 the parameter of interest are asymptotically equivalent to their unconditional counterparts and hence are asymptotically efficient provided that the conditioning statistic is asymptotically weakly ancillary the key assumption that the conditioning statistic is asymptotically weakly ancillary is verified when the underlying distribution is from exponential families some illustrative examples are given

conditioning to reduce the sensitivity of general estimating functions to nuisance parameters summary a conditional method is presented that renders an estimating function insensitive to nuisance parameters the approach is a generalisation of the conditional score method to a general estimating function context and does not require complete specification of the probability model we exploit the informal relationship between general estimating functions and score functions to derive simple generalisations of sufficient and partially ancillary statistics referred to as g sufficient and g ancillary statistics respectively these two types of statistic are defined in a manner that does not require complete knowledge of the probability model and thus are more suitable for use with estimating functions if we condition on a g sufficient statistic for the nuisance parameters the resulting conditional estimating function is insensitive to nuisance parameters and in particular achieves the plug in unbiasedness property furthermore if the conditioning argument is also g ancillary for the parameters of interest then the conditional estimating function possesses an attractive optimality property

projection as a method for increasing sensitivity and eliminating nuisance parameters l article analyse des methodes statistiques basees sur les fonctions d estimation qui sont des fonctions de parametres et de donnees a la fois destinees a permettre une inference sur un parametre inconnu une methode de projection peut etre utilisee pour la reduction de ces fonctions afin d accroitre la puissance d un test et l efficacite d un estimateur ponctuel ou bien de rendre une fonction d inference insensible a la valeur d un parametre de nuisance

jackknife and analytical bias reduction for nonlinear panel models fixed effects estimators of panel models can be severely biased because of the well known incidental parameters problem we show that this bias can be reduced by using a panel jackknife or an analytical bias correction motivated by large t we give bias corrections for averages over the fixed effects as well as model parameters we find large bias reductions from using these approaches in examples we consider asymptotics where t grows with n as an approximation to the properties of the estimators in econometric applications we show that if t grows at the same rate as n the fixed effects estimator is asymptotically biased so that asymptotic confidence intervals are incorrect but that they are correct for the panel jackknife we show t growing faster than n 1 3 suffices for correctness of the analytic correction a property we also conjecture for the jackknife

periodic solutions of nonlinear wave equations with general nonlinearities we present a variational principle for small amplitude periodic solutions with fixed frequency of a completely resonant nonlinear wave equation existence and multiplicity results follow by min max variational arguments

conditional score functions some optimality results summary the conditional score function has previously been shown to generate the optimal estimating equation for a parameter of interest when the conditioning statistic is complete and sufficient for the nuisance parameters godambe 1976 the present paper generalizes these results to partial likelihood factorizations and then examines the nature of the problem when the appropriate conditioning statistics depend on the parameter of interest in this case globally optimal estimating functions are impossible a weaker criterion of optimal weighting leads to a class of estimated conditional score functions which satisfy an information equality the purpose of this paper is to develop and justify an approach based on conditional likelihood to estimation in the presence of nuisance parameters the arguments focus on the case when there is a complete and sufficient statistic s 0 for the nuisance parameters when the parameter of interest 0 is fixed an extended notion of completeness will be presented in the context of partial likelihoods the problem takes on two separate natures depending on whether the statistic s 0 truly depends on 0 or not if s s 0 is free of 0 then the conditional likelihood of the data given s has been shown to generate optimal inferential procedures giving a best estimating equation godambe 1976 and a best asymptotically normal estimator in a mixture setting lindsay 1980 if s 0 depends on 0 then it will be shown that the best estimating equation also a conditional likelihood score will generally be only locally optimal leaving no single best strategy consideration of an optimal weighting strategy yields estimators which have second order properties similar to the conditional likelihoods of the first type

orthogonality of estimating functions and nuisance parameters summary cox reid 1987 proposed the technique of orthogonalizing parameters to deal with the general problem of nuisance parameters within fully parametric models they obtained a large sample approximation to the conditional likelihood along the same lines davison 1988 studied generalized linear models in the present paper we deal with the problem of nuisance parameters within a semiparametric setup which includes the class of distributions associated with generalized linear models the technique used is that of optimum orthogonal estimating functions godambe thompson 1989 the results are related to those of cox reid 1987

multiplicity of periodic solutions of nonlinear wave equations abstract we prove multiplicity of small amplitude periodic solutions with fixed frequency of completely resonant wave equations with general nonlinearities as 1 the number n of 2 periodic solutions u1 un un tends to the minimal period of the nth solution un is 2 n the proofs are based on the variational lyapunov schmidt reduction comm math phys to appear and minimax arguments 1

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

on ancillarityu and fisher information in the presence of a nuisance parameter summary godambe 1976 put forward two concepts of ancillarity in the presence of nuisance parameters in this paper they are unified and extended with an extended concept of fisher information

kam tori for 1d nonlinear wave equations with periodic boundary conditions in this paper one dimensional 1d nonlinear wave equations with periodic boundary conditions are considered v is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u 0 it is proved that for most potentials v x the above equation admits small amplitude periodic or quasi periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system the proof is based on an infinite dimensional kam theorem which allows for multiple normal frequencies

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

nekhoroshev stability of elliptic equilibria of hamiltonian systems we prove a conjecture by n n nekhoroshev about the long time stability of elliptic equilibria of hamiltonian systems without any diophantine condition on the frequencies higher order terms of the hamiltonian are used to provide convexity the singularity of the action angle coordinates at the origin is overcome by working in cartesian coordinates

efficiency of projected score methods in rectangular array asymptotics summary the paper considers a rectangular array asymptotic embedding for multistratum data sets in which both the number of strata and the number of within stratum replications increase and at the same rate it is shown that under this embedding the maximum likelihood estimator is consistent but not efficient owing to a non zero mean in its asymptotic normal distribution by using a projection operator on the score function an adjusted maximum likelihood estimator can be obtained that is asymptotically unbiased and has a variance that attains the cram6r rao lower bound the adjusted maximum likelihood estimator can be viewed as an approximation to the conditional maximum likelihood estimator

stability of some solutions for elliptic equations on a cylindrical domain we study analytically a class of solutions for the elliptic equation formula 1 where 0 and e is a small parameter this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every 0 it contains solutions that are defined for large values of time and they are very close of order o e to a linear torus for long times of order o e 1 the proof uses the fact that the equation leaves invariant a smooth centre manifold and for the restriction of the system to the centre manifold uses arguments of classical perturbation theory by considering the hamiltonian formulation of the problem the birkhoff normal form and neckhoroshev type estimates 1 embed graphic 1 gif

a regularized xfem framework for embedded cohesive interfaces this work focuses on xfem modeling of elastodamaging interfaces characterized by regularized kinematics after regularization of the displacement discontinuity a process zone of finite width with continuous strain fields emerges the stress fields are independently energy deduced modeled via specific constitutive laws and subsequently introduced within the virtual work principle the methodology can address cohesive interfaces with vanishing and finite thickness in a unified way two dimensional results for a strain localization problem and several delamination tests are discussed in detail and compared with previous numerical experimental results