on the approximate solutions of constrained inverse lipschitz function problems and inverse function theorems abstract we present a principle of approximate solutions of constrained inverse lipschitz function problems as corollaries and applications of the principle we obtain a result of convergence of an approximate solutions sequence for the constrained problems a conclusion relating direct and inverse images of upper and lower limits of a sequence of subsets and several versions of inverse lipschitz function theorems finally we give local uniqueness criteria for solutions to constrained nonlinear problems in finite dimension spaces

reference bayesian methods for generalized linear mixed models abstract bayesian methods furnish an attractive approach to inference in generalized linear mixed models in the absence of subjective prior information for the random effect variance components these analyses are typically conducted using either the standard invariant prior for normal responses or diffuse conjugate priors previous work has pointed out serious difficulties with both strategies and we show here that as in normal mixed models the standard invariant prior leads to an improper posterior distribution for generalized linear mixed models this article proposes and investigates two alternative reference i e objective or noninformative priors an approximate uniform shrinkage prior and an approximate jeffreys s prior we give conditions for the existence of the posterior distribution under any prior for the variance components in conjunction with a uniform prior for the fixed effects the approximate uniform shrinkage prior is shown to satisfy these conditions for several families of d

a note on the existence of the posterior distribution for a class of mixed models for binomial responses summary necessary and sufficient conditions are given for the existence of the posterior distribution of the variance components in a class of mixed models for binomial responses the implications of our results are illustrated through an example

bayes factor covariance testing in item response models two marginal one parameter item response theory models are introduced by integrating out the latent variable or random item parameter it is shown that both marginal response models are multivariate probit models with a compound symmetry covariance structure several common hypotheses concerning the underlying covariance structure are evaluated using fractional bayes factor tests the support for a unidimensional factor i e assumption of local independence and differential item functioning are evaluated by testing the covariance components the posterior distribution of common covariance components is obtained in closed form by transforming latent responses with an orthogonal helmert matrix this posterior distribution is defined as a shifted inverse gamma thereby introducing a default prior and a balanced prior distribution based on that an mcmc algorithm is described to estimate all model parameters and to compute fractional bayes factor tests simulation studies are used to show that the fractional bayes factor tests have good properties for testing the underlying covariance structure of binary response data the method is illustrated with two real data studies

variance component testing in generalised linear models with random effects summary there is considerable interest in testing for overdispersion correlation and heterogeneity across groups in biomedical studies in this paper we cast the problem in the framework of generalised linear models with random effects we propose a global score test for the null hypothesis that all the variance components are zero this test is a locally asymptotically most stringent test and is robust in the special sense that the test does not require specifying the joint distribution of the random effects we also propose individual score tests and their approximations for testing the variance components separately both tests can be easily implemented using existing statistical software we illustrate these tests with an application to the study of heterogeneity of mating success across males and females in an experiment on salamander matings and evaluate their performance through simulation

misspecifying the covariance structure in a linear mixed model under mar drop out misspecification of the covariance structure in a linear mixed model lmm can lead to biased population parameters estimates under mar drop out in our motivating example of modeling cd4 cell counts during untreated hiv infection random intercept and slope lmms are frequently used in this article we evaluate the performance of lmms with specific covariance structures in terms of bias in the fixed effects estimates under specific mar drop out mechanisms and adopt a bayesian model comparison criterion to discriminate between the examined approaches in real data applications we analytically show that using a random intercept and slope structure when the true one is more complex can lead to seriously biased estimates with the degree of bias depending on the magnitude of the mar drop out under misspecified covariance structure we compare in terms of induced bias the approach of adding a fractional brownian motion bm process on top of random intercepts and slopes with the approach of using splines for the random effects in general the performance of both approaches was satisfactory with the bm model leading to smaller bias in most cases a simulation study is carried out to evaluate the performance of the proposed bayesian criterion in identifying the model with the correct covariance structure overall the proposed method performs better than the aic and bic criteria under our specific simulation setting the models under consideration are applied to real data from the cascade study the most plausible model is identified by all examined criteria

convex composite non lipschitz programming in this paper necessary and sufficient optimality conditions are established without lipschitz continuity for convex composite continuous optimization model problems subject to inequality constraints necessary conditions for the special case of the optimization model involving max min constraints which frequently arise in many engineering applications are also given optimality conditions in the presence of lipschitz continuity are routinely obtained using chain rule formulas of the clarke generalized jacobian which is a bounded set of matrices however the lack of derivative of a continuous map in the absence of lipschitz continuity is often replaced by a locally unbounded generalized jacobian map for which the standard form of the chain rule formulas fails to hold in this paper we overcome this situation by constructing approximate jacobians for the convex composite function involved in the model problem using e perturbations of the subdifferential of the convex function and the flexible generalized calculus of unbounded approximate jacobians examples are discussed to illustrate the nature of the optimality conditions

the fermat rule for multifunctions on banach spaces using variational analysis we study vector optimization problems with objectives being closed multifunctions on banach spaces or in asplund spaces in particular in terms of the coderivatives we present fermat s rules as necessary conditions for an optimal solution of the above problems as applications we also provide some necessary conditions in terms of clarke s normal cones or the limiting normal cones for pareto efficient points

robust wald type tests of one sided hypotheses in the linear model abstract let us consider the linear model y x e in the usual matrix notation where the errors are iid our main objective is to develop robust wald type tests for a large class of hypotheses on by robust we mean robustness in terms of size and power against long tailed error distributions first assume that the error distribution is symmetric about the origin let l and be the least squares and a robust estimator of assume that they are asymptotically normal about with covariance matrices 2 xtx 1 and 2 xtx 1 respectively so could be an m estimator or a high breakdown point estimator robust wald type tests based on denoted by rw are studied here for testing a large class of one sided hypotheses on it is shown that the asymptotic null distribution of rw and that of the usual wald type statistic based on l denoted by w are the same this is a useful result since the critical values and procedures for computing the p values for w are directly applicable to rw as well a more impo

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

sharp variational conditions for convex composite nonsmooth functions in this paper we present first and second order variational conditions for a convex composite function g circ f where g is a nonsmooth convex function and f is a vector valued map the first order results which apply to not necessarily locally lipschitz continuous maps f not only recapture the results of the special cases where f is locally lipschitz or g teaux differentiable but also yield sharp necessary variational conditions in these cases the results are achieved by applying a new strengthened notion of approximate jacobian called a g teaux g approximate jacobian without the use of the upper semicontinuity of the approximate jacobian these variational results are generally derived by using a chain rule formula or by constructing upper convex approximations to the composite function these approaches often need the upper semicontinuity requirement of a generalized jacobian map such a requirement not only limits the derivation of sharp optimality conditions as the small approximate jacobians or generalized subdifferentials lack an upper semi continuity property but also restricts the treatment of g teaux differentiable maps f this situation is overcome by the use of g approximate jacobians the second order variational conditions are shown to hold in particular in the case where f is continuously g teaux differentiable

bayesian covariance selection in generalized linear mixed models repeated measures and longitudinal data are commonly collected for analysis in epidemiology clinical trials biology sociology and economic sciences in such studies a response is measured repeatedly over time for each subject under study and the number and timing of the observations often varies among subjects in contrast to cross sectional studies that collect a single measurement per subject longitudinal studies have the extra complication of within subject dependence in the repeated measures such dependence can be thought to arise due to the impact of unmeasured predictors main effects of unmeasured predictors lead to variation in the average level of response among subjects while interactions with measured predictors lead to heterogeneity in the regression coefficients this justification has motivated random effects models which allow the intercept and slopes in a regression model to be subject specific random effects models are broadly useful for modeling of dependence not only for longitudinal data but also in multicenter studies meta analysis and functional data analysis

laplace approximations for posterior expectations when the mode occurs at the boundary of the parameter space abstract this article gives asymptotic expansions for posterior expectations when the mode is on the boundary of the parameter space the idea based on the divergence theorem is to reduce the high dimensional integrals over the parameters space to surface integrals over the boundary of the parameter space and then apply the usual interior mode laplace method to the latter integrals it is shown that these approximations have second order accuracy the method is illustrated with applications to a two sample binomial problem and a random eflects model

on the solution of two coupled wiener hopf equations vector wiener hopi equations previously solved in closed form by using different techniques involve matrices having a particular structure a general procedure is given to factorize these matrices it leads to an approach which even if sometimes cumbersome is uniform and allows the solution of new problems in addition a new method is proposed which transforms simultaneous wiener hopf equations in the form of fredholm integral equation of the second kind

on the commutative factorization of n n matrix wiener hopf kernels with distinct eigenvalues in this article we present a method for factorizing n x n matrix wiener hopf kernels where n 2 and the factors commute we are motivated by a method posed by jones jones 1984a proc r soc a 393 185 192 to tackle a narrower class of matrix kernels however no matrix of jones form has yet been found to arise in physical wiener hopf models in contrast the technique proposed herein should find broad application to illustrate the approach we consider a 3 x 3 matrix kernel arising in a problem from elastostatics while this kernel is not of jones form we shall show how it can be factorized commutatively we discuss the essential difference between our method and that of jones and explain why our method is a generalization the majority of wiener hopf kernels that occur in canonical diffraction problems are however strictly non commutative for 2x2 matrices abrahams has shown that one can overcome this difficulty using pade approximants to rearrange a non commutative kernel into a partial commutative form an approximate factorization can then be derived by considering the dynamic analogue of antipov s model we show for the first time that abrahams pade approximant method can also be employed within a 3x3 commutative matrix form

the wiener hopf technique for impenetrable wedges having arbitrary aperture angle diffraction by impenetrable wedges having arbitrary aperture angle is studied by means of the wiener hopf w h technique a system of functional equations called generalized wiener hopf equations gwhe is obtained only for certain values of the aperture angle are these equations recognizable as standard or classical wiener hopf equations cwhe however in all cases a mapping is found that reduces the gwhe to cwhe it means that the diffraction by an impenetrable wedge always reduces to a standard w h factorization the solution for the diffraction by a wedge with given face impedances the malyuzhinets problem is obtained in closed form by an explicit factorization of the kernel

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

fixed and random effects selection in linear and logistic models we address the problem of selecting which variables should be included in the fixed and random components of logistic mixed effects models for correlated data a fully bayesian variable selection is implemented using a stochastic search gibbs sampler to estimate the exact model averaged posterior distribution this approach automatically identifies subsets of predictors having nonzero fixed effect coefficients or nonzero random effects variance while allowing uncertainty in the model selection process default priors are proposed for the variance components and an efficient parameter expansion gibbs sampler is developed for posterior computation the approach is illustrated using simulated data and an epidemiologic example

a nondegenerate fuzzy optimality condition for constrained optimization problems without qualification conditions abstract fuzzy optimization conditions in terms of the frechet subdifferential for reflexive spaces were investigated by borwein treiman and zhu 1998 in 1 to achieve the nondegenerate form it is well known that some qualification conditions should be assumed in this paper we are going to prove that the nondegenerate fuzzy optimality condition even holds with no qualification conditions in asplund spaces in particular reflexive spaces for optimization problems with semi continuous and continuous data the results are even new in finite dimensional frameworks

wiener hopf solution for impenetrable wedges at skew incidence a new wiener hopf approach for the solution of impenetrable wedges at skew incidence is presented mathematical aspects are described in a unified and consistent theory for angular region problems solutions are obtained using analytical and numerical analytical approaches several numerical tests from the scientific literature validate the new technique and new solutions for anisotropic surface impedance wedges are solved at skew incidence the solutions are presented considering the geometrical and uniform theory of diffraction coefficients total fields and possible surface wave contributions

robust stochastic dominance and its application to risk averse optimization we introduce a new preference relation in the space of random variables which we call robust stochastic dominance we consider stochastic optimization problems where risk aversion is expressed by a robust stochastic dominance constraint these are composite semi infinite optimization problems with constraints on compositions of measures of risk and utility functions we develop necessary and sufficient conditions of optimality for such optimization problems in the convex case in the nonconvex case we derive necessary conditions of optimality under additional smoothness assumptions of some mappings involved in the problem

global invertibility of nonsmooth mappings on etend le theoreme de hadamard d inversibilite globale a la classe des applications localement continues de lipschitz

convexity and generalized second order derivatives for locally lipschitz functions abstract we define the generalized second order directional derivatives by means of the clarke generalized gradient for locally lipschitz functions then we give characterization of convexity and state a new sufficient optimality condition

on the non commutative factorization of wiener hopf kernels of khrapkov type there are a great many problems in mathematical physics and engineering which when modelled mathematically are reduced to wiener hopf functional equations defined in some region of a complex plane in simple models this equation is scalar but in more complicated situations inherent coupling etc can often give rise to coupled systems of equations and so the resulting wiener hopf kernels are of matrix form the crucial step in the solution of any wiener hopf equation is to decompose the kernel which can be a quite general function of the complex transform variable into a product of two factors which are regular in overlapping half planes this can be accomplished explicitly for scalar kernels and may be generalized to a particular class of matrix kernels namely those which permit a commutative decomposition although this class is wide and therefore contains some valuable physical problems it is found that many such kernels yield factorization elements with exponentially large behaviour at infinity in their half planes of analyticity this prevents a later step in the wiener hopf procedure being carried through and therefore introduces a serious limitation to the usefulness of khrapkov s approach previous researchers have offered schemes to eliminate the exponential behaviour of the khrapkov factors but for reasons discussed herein these have problems in their successful implementation this article offers a new procedure for quenching the exponential growth which is based on a method proposed recently by the author for fully coupled kernels which do not have commutative factors the approach is two step first a commutative factorization is obtained which is algebraic at infinity as required but which has spurious branch cuts in the regions of intended analyticity second an approximate procedure is used to eliminate the offending branch points by introduction of pade approximants for a certain scalar function and by inclusion of a regularizing meromorphic matrix for one particular example kernel in which an exact factorization happens to be known the explicit approximate kernel factors thus derived are shown to yield very close agreement to the exact factors the application of this work to problems in the area of fluid structural interaction is discussed

bayes factors and approximations for variance component models abstract in this article we consider tests of variance components using bayes factors such tests arise in many fields of application including medicine agriculture and engineering when using bayes factors the choice of prior distribution on the parameter of interest is of great importance we propose a unit information reference method for variance component models the calculation of bayes factors in this context is not straightforward there are well documented difficulties with markov chain monte carlo approaches such as gibbs sampling and the usual laplace approximation is not appropriate due to the boundary null hypothesis we describe both an importance sampling approach and an analytical approximation for calculating the numerator and denominator of the bayes factor the importance sampling approach is straightforward to implement and also forms the basis for a rejection algorithm that allows generation of samples from the posterior distributions under the null and alternative hypotheses

approximate mean value theorem for upper subderivatives the purpose of this paper is to give a mean value theorem for a lower semicontinuous function f on a banach space using the upper subderivatives defined by rockafellar l as will be shown by example section 4 it may happen that the subdifferential af x is empty for all x in the closed line segment a 61 where a and b belong to x therefore we cannot formulate the mean value theorem in such a form as that obtained in 2 and 3 for other generalized derivatives we prove among other factors that there exists a sequence xk converging to some point x e a 61 x b such that the sets df xj are nonempty and

on the solution of wiener hopf problems involving noncommutative matrix kernel decompositions many problems in physics and engineering with semi infinite boundaries or interfaces are exactly solvable by the wiener hopf technique it has been used successfully in a multitude of different disciplines when the wiener hopf functional equation contains a single scalar kernel for complex boundary value problems however the procedure often leads to coupled equations which therefore have a kernel of matrix form the key step in the technique is to decompose the kernel into a product of two functions one analytic in an upper region of a complex transform plane and the other analytic in an overlapping lower half plane this is straightforward for scalar kernels but no method has yet been proposed for general matrices in this article a new procedure is introduced whereby pade approximants are employed to obtain an approximate but explicit noncommutative factorization of a matrix kernel as well as being simple to apply the use of approximants allows the accuracy of the factorization to be increased

differentiation of sets in measure abstract suppose f e for each e 0 1 is a bounded borel subset of r d and f e f 0 as e 0 let a e f e f 0 be symmetric difference and p be an absolutely continuous measure on r d we introduce the notion of derivative of f e with respect to e d f e d e d a e d e such that d d e p a e e 0 q d d e a e e 0 where q is another explicitly described measure although not in r d we discuss why this sort of derivative is needed to study local point processes in neighbourhood of a set in short if sequence of point processes n n n 1 2 is given on the class of set valued mappings f f such that all f e converge to the same f f 0 then the weak limit of the local processes n n a e f e f lives on the class of derivative sets d f e d e e 0 f f we compare this notion of the derivative set valued mapping with other existing notions

the differentiability of real functions on normed linear space using generalized subgradients abstract the modification of the clarke generalized subdifferential due to michel and penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space the g teaux differentiability of any real function can be deduced from the g teaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative for any distance function on a space with uniformly g teaux differentiable norm the clarke and michel penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for g teaux differentiability however on a banach space with rotund dual the frechet differentiability of a distance function implies that it is a convex function a mean value theorem for the modified generalized subdifferential has implications for g teaux differentiability

composite nonsmooth programming with gâteaux differentiability this paper examines constrained nonsmooth optimization problems where the objective function and the constraints are compositions of locally lipschitz functions and g teaux differentiable functions but are not necessarily frechet differentiable or strict differentiable lagrangian necessary and sufficient optimality conditions are presented for various classes of composite programs these are obtained by constructing appropriate convex approximations for composite functions the lagrange multipliers are also characterized in terms of subgradients of the value function under appropriate conditions

homeomorphisms and local c 1 approximations soient 7b h et 7b y des espaces de banach v 7b h ouvert et f v 7b y continue on demontre que les extensions de john du theoreme de hadamard sont applicables a des fonctions entre des espaces de banach avec des conteneurs de derivees locaux inversibles

scaling regression inputs by dividing by two standard deviations interpretation of regression coefficients is sensitive to the scale of the inputs one method often used to place input variables on a common scale is to divide each numeric variable by its standard deviation here we propose dividing each numeric variable by two times its standard deviation so that the generic comparison is with inputs equal to the mean 1 standard deviation the resulting coefficients are then directly comparable for untransformed binary predictors we have implemented the procedure as a function in r we illustrate the method with two simple analyses that are typical of applied modeling a linear regression of data from the national election study and a multilevel logistic regression of data on the prevalence of rodents in new york city apartments we recommend our rescaling as a default option an improvement upon the usual approach of including variables in whatever way they are coded in the data file so that the magnitudes of coefficients can be directly compared as a matter of routine statistical practice copyright 2007 john wiley sons ltd

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

a sharp lagrange multiplier rule for nonsmooth mathematical programming problems involving equality constraints it is shown that a lagrange multiplier rule that uses approximate jacobians holds for mathematical programming problems involving lipschitzian functions finitely many equality constraints and convex set constraints it is sharper than the corresponding lagrange multiplier rules for the convex valued subdifferentials such as those of clarke optimization and nonsmooth analysis 2nd ed siam 1990 and michel and penot differential integral equations 5 1992 pp 433 454 the lagrange multiplier result is obtained by means of a controllability criterion and the theory of fans developed by a d ioffe math oper res 9 1984 pp 159 189 math programming 58 1993 pp 137 145 as an application necessary optimality conditions are derived for a class of constrained minimax problems an example is discussed to illustrate the nature of the multiplier rule

scattering by a semi infinite sandwich panel perforated on one side the scattering of sound waves by the edge of a sandwich panel which consists of two thin elastic plates containing a light interior cellular structure is analysed the wiener hopf method is used to examine the interaction effects of a semi infinite panel clamped to a semi infinite rigid screen one major difficulty is that the presence of two different velocity potentials on either side of the plane y 0 results in a matrix wiener hopf equation an approximate solution is given in the asymptotic limit of small values of a parameter tau which accounts for the perforations

on the multiplicity of dini subgradients in separable spaces abstract for an extended valued function on a separable banach space the dini subdifferential is either single valued or empty on a large subset of the domain because the set of points at which it contains two or more elements can be covered by a countable family of lipschitz hypersurfaces this generalizes a finite dimensional result of benoist and has applications to the hadamard differentiability of regularly locally convex functions

set separation approximating multicones and the lipschitz maximum principle abstract we present a general necessary condition for separation of the reachable set of a lipschitz control system from another given set of states expressed in terms of an approximating multicone to the set in a sense that contains as special cases the clarke and mordukhovich cones we then show how this separation result implies a strengthened form of the usual sufficient condition for local controllability along the reference curve and the necessary condition for optimality

convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints sample average approximation saa method has recently been applied to solve stochastic programs with second order stochastic dominance ssd constraints in particular hu et al math program 133 171 201 2012 presented a detailed convergence analysis of epsilon optimal values and epsilon optimal solutions of sample average approximated stochastic programs with polyhedral ssd constraints in this paper we complement the existing research by presenting convergence analysis of stationary points when saa is applied to a class of stochastic minimization problems with ssd constraints specifically under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts moreover by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations we derive exponential rate of convergence of stationary points

algebraic univalence theorems for nonsmooth functions abstract a well known univalence result due to d gale and h nikaido 1965 math ann 159 81 93 asserts that if the jacobian matrix of a differentiable function from a closed rectangle k in r n into r n is a p matrix at each point of k then f is one to one on k in this paper by introducing the concepts of h differentiability and h differential of a function as a set of matrices we generalize the gale nikaido result to nonsmooth functions our results further extend those of other authors valid for compact rectangles we show that our results are applicable when the h differential is any one of the following the jacobian matrix of a differentiable function the generalized jacobian of a locally lipschitzian function the bouligand subdifferential of a semismooth function and the c differential of l qi 1993 math oper res 18 227 244

infinite dimensional generalized jacobian properties and calculus rules the extension to infinite dimensional domains of clarke s generalized jacobian is the focus of this paper first a generalization of a fabian preiss theorem to the infinite dimensional setting is obtained as a consequence a new formula relating the clarke s generalized jacobians corresponding to finite dimensional spaces k l with k l is established furthermore in the infinite dimensional case basic properties pertaining the generalized jacobian are developed and then an identification of this set valued map is produced applications of these results in the form of chain rules including sum and product rules and a computational formula for continuous selections are derived

generalized score test of homogeneity based on correlated random effects models summary we present a family of tests based on correlated random effects models which provides a synthesis and a generalization of recent work on homogeneity testing in these models each subject has a particular random effect but the random effects between subjects are correlated we derive the general form of the score statistic for testing that the random effects have a variance equal to 0 we apply this result to both parametric and semiparametric models in both cases we show that under certain conditions the score statistic has an asymptotic normal distribution we consider several applications of this theory including overdispersion heterogeneity between groups spatial correlations and genetic linkage

weak directional closedness and generalized subdifferentials abstract spingarn introduced the notion of a submonotone operator and showed that the clarke subdifferential is submonotone if and only if it is semismooth in the sense of mifflin and regular in the sense of clarke in this article a property of operators referred to as weak directional closedness wdc is introduced the wdc property is used to extend spingarn s result to a broad class of generalized subdifferentials for locally lipschitz functions two members of this class of subdifferentials are the clarke subdifferential which is always wdc and the michel penot subdifferential which may or may not be wdc a subdifferential that is wdc and is contained in the clarke subdifferential is constructed it is shown that this subdifferential coincides with the michel penot subdifferential whenever the michel penot subdifferential is wdc and submonotone

permutation tests for random effects in linear mixed models summary inference regarding the inclusion or exclusion of random effects in linear mixed models is challenging because the variance components are located on the boundary of their parameter space under the usual null hypothesis as a result the asymptotic null distribution of the wald score and likelihood ratio tests will not have the typical 2 distribution although it has been proved that the correct asymptotic distribution is a mixture of 2 distributions the appropriate mixture distribution is rather cumbersome and nonintuitive when the null and alternative hypotheses differ by more than one random effect as alternatives we present two permutation tests one that is based on the best linear unbiased predictors and one that is based on the restricted likelihood ratio test statistic both methods involve weighted residuals with the weights determined by the among and within subject variance components the null permutation distributions of our statistics are computed by permuting the residuals both within and among subjects and are valid both asymptotically and in small samples we examine the size and power of our tests via simulation under a variety of settings and apply our test to a published data set of chronic myelogenous leukemia patients

asymmetric channel divider in stokes flow this article examines the classic problem of stokes flow into a divided channel with in contrast to previous literature the divider barrier asymmetrically placed with respect to the moving parallel channel walls the boundary value problem is reduced to a wiener hopf equation that is of matrix form and of a class for which no exact solution is known an explicit approximate solution in general accurate to any specified degree is obtained by a recent method which employs pade approximants numerical results exhibit the flows due to moving walls or various combinations of downstream pressure gradients

the effective boundary conditions for a perforated elastic sandwich panel in a compressible fluid the transmission of sound through a sandwich panel consisting of a honeycomb cellular structure bounded by two thin elastic plates is considerably reduced at a certain frequency if one or both of the plates is perforated so as to link a significant number of the cells to the exterior fluid this effect occurs at or near the helmholtz resonance frequency for the cells with the acoustic wavelength large compared with the cell dimensions an analysis is given for the problem of plane waves incident upon a plane sandwich plate of infinite extent using matched expansions for the cases of acoustically hard or acoustically transparent cell walls the compound panel is shown to be acoustically equivalent to that of a hypothetical surface with different normal velocities on either side and effective boundary conditions are derived with generalizations to deal with more complicated structures finite plates and more general incident fields

mode conversion corner singularities and matrix wiener hopf factorization in diffraction theory the diffraction of acoustic electromagnetic and elastic waves by an edge is a subject of continuing mathematical and physical importance the physical motivation for research is the vast number of technological and geophysical applications of the fundamental models of linear wave propagation in continuous media while the mathematical drive is to understand and devise new analytical and numerical techniques for their quantification this paper is concerned with certain physical aspects of diffraction which have proven to give the most troublesome of mathematical difficulties these are mode conversion that is when the incident wave is scattered among different modes supported by the propagating medium and edge or corner singularities when the diffracting obstacle creates a stress singularity it is shown by way of a particular example that the exact solution of diffraction problems involving these phenomena will in general give rise to a vector wiener hopf equation with a matrix kernel whose factorization is non commutative the main result of this paper is a new procedure motivated by a precise treatment of both the corner singularity and mode conversion for determining the factors in this case

mean value theorems and sufficient optimality conditions for nonsmooth functions in the last few years the mean value theorem for various classes of nondifferentiable functions has been the subject of many investigations in particular hiriart urruty 4 sect ii proved some variants of this theorem for extended real valued functions defined on locally convex spaces in this paper we present two other theorems which are to some extent generalizations of the above mentioned results more precisely we introduce the notion of an upper convex approximation for a function being a modification of the first order convex approximation of ioffe 51 and formulate the mean value theorems in terms of subdifferentials of these approximations we first consider the class of functions f x t r where x is a locally convex space such that the restriction offto the closed line segment a 61 is finite and lower semicontinuous our result for this case theorem 4 2 has a weaker form than the classical mean value theorem the difference f h f a is expressed by means of the subdifferential of an upper convex approximation for f at a point ce a b if f is discontinuous c may be equal to a or h however if the restriction fl a h is continuous we obtain a stronger result theorem 4 3 stating that c belongs to the open line segment a h this theorem is a generalization of theorem 1 of 9 and in part of theorem 2 of 4 finally we obtain as a corollary from theorem 4 3 a sufficient condition for a strict local minimum of a continuous function on a convex subset of x theorem 5 1 this theorem extends the result of lo to the case of functions which may not satisfy the lipschitz condition in the paper we make use of some notions and theorems of convex analysis which can be found in chapter 6 of 6

sequential normal compactness versus topological normal compactness in variational analysis abstract we study relationships between two normal compactness properties of sets in banach spaces that play an essential role in many aspects of variational analysis and its applications particularly in calculus rules for generalized differentiation necessary optimality and suboptimality conditions for optimization problems etc both properties automatically hold in finite dimensional spaces and reveal principal features of the infinite dimensional variational theory similar formulations of these properties involve the weak convergence of sequences and nets respectively containing generalized normal cones in duals to banach spaces we prove that these properties agree for a large class of banach spaces that include weakly compactly generated spaces we also show that they are always different in banach spaces whose unit dual ball is not weak sequentially compact moreover the sequential and topological normal compactness properties may not coincide even in non separable asplund spaces that admit an equivalent c smooth norm

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

differentiation of sets the general case abstract in recent work by khmaladze and weil 2008 and by einmahl and khmaladze 2011 limit theorems were established for local empirical processes near the boundary of compact convex sets k in r d the limit processes were shown to live on the normal cylinder of k respectively on a class of set valued derivatives in the latter result was based on the concept of differentiation of sets at the boundary k of k which was developed in khmaladze 2007 here we extend the theory of set valued derivatives to boundaries f of rather general closed sets f r d making use of a local steiner formula for closed sets established in hug last and weil 2004

approximate bayes factors when a mode occurs on the boundary abstract bayes factors measuring the strength of evidence in favor of the null are often used in the bayesian approach to testing hypotheses laplace approximations to bayes factors are convenient and quite accurate in many contexts however one usual assumption the existence of an interior mode does not always hold the posterior mode can occur at the boundary of the parameter space this article discusses the boundary mode laplace s method for boundary modes uses it to approximate the bayes factor and presents a modification to the schwarz criterion the sensitivity of bayes factor to the choice of prior on the nuisance parameter in testing nested models is also investigated results are illustrated for the case of testing extrabinomial variability

approximate jacobian matrices for nonsmooth continuous maps and c 1 optimization the notion of approximate jacobian matrices is introduced for a continuous vector valued map it is shown for instance that the clarke generalized jacobian is an approximate jacobian for a locally lipschitz map the approach is based on the idea of convexificators of real valued functions mean value conditions for continuous vector valued maps and taylor s expansions for continuously g teaux differentiable functions i e c1 functions are presented in terms of approximate jacobians and approximate hessians respectively second order necessary and sufficient conditions for optimality and convexity of c1 functions are also given

testing random effects in the linear mixed model using approximate bayes factors deciding which predictor effects may vary across subjects is a difficult issue standard model selection criteria and test procedures are often inappropriate for comparing models with different numbers of random effects due to constraints on the parameter space of the variance components testing on the boundary of the parameter space changes the asymptotic distribution of some classical test statistics and causes problems in approximating bayes factors we propose a simple approach for testing random effects in the linear mixed model using bayes factors we scale each random effect to the residual variance and introduce a parameter that controls the relative contribution of each random effect free of the scale of the data we integrate out the random effects and the variance components using closed form solutions the resulting integrals needed to calculate the bayes factor are low dimensional integrals lacking variance components and can be efficiently approximated with laplace s method we propose a default prior distribution on the parameter controlling the contribution of each random effect and conduct simulations to show that our method has good properties for model selection problems finally we illustrate our methods on data from a clinical trial of patients with bipolar disorder and on data from an environmental study of water disinfection by products and male reproductive outcomes

convex composite multi objective nonsmooth programming this paper examines nonsmooth constrained multi objective optimization problems where the objective function and the constraints are compositions of convex functions and locally lipschitz and g teaux differentiable functions lagrangian necessary conditions and new sufficient optimality conditions for efficient and properly efficient solutions are presented multi objective duality results are given for convex composite problems which are not necessarily convex programming problems applications of the results to new and some special classes of nonlinear programming problems are discussed a scalarization result and a characterization of the set of all properly efficient solutions for convex composite problems are also discussed under appropriate conditions

variance components testing in the longitudinal mixed effects model this article discusses the asymptotic behavior of likelihood ratio tests for nonzero variance components in the longitudinal mixed effects linear model described by laird and ware 1982 biometrics 38 963 974 our discussion of the large sample behavior of likelihood ratio tests for nonzero variance components is based on the results for nonstandard testing situations by self and liang 1987 journal of the american statistical association 82 605 610

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

the lagrange multiplier rule for super efficiency in vector optimization in this paper we study constrained multiobjective optimization problems with objectives being closed graph multifunctions in banach spaces in terms of the coderivatives and clarke s normal cones we establish lagrange multiplier rules for super efficiency as necessary or sufficient optimality conditions of the above problems

expressions for subdifferential and optimization problems defined by nonsmooth homogeneous functions abstract in this paper we prove a theoretical expression for subdifferentials of lower semicontinuous and homogeneous functions the theoretical expression is a generalization of the euler formula for differentiable homogeneous functions as applications of the generalized euler formula we consider constrained optimization problems defined by nonsmooth positively homogeneous functions in smooth banach spaces some results concerning karush kuhn tucker points and necessary optimality conditions for the optimization problems are obtained

random effects selection in linear mixed models nous presentons le probleme pratique important de la selection des effets aleatoires dans un modele lineaire mixte un modele bayesien hierarchique est utilise pour identifier tout effet aleatoire devariance nulle l approche proposee reparametrise le modele mixte de telle sorte que les fonctions des parametres de covariance de la distribution des effets aleatoires soient incorporees comme des coefficients de regression pour des variables latentes normales nous permettons aux effets aleatoires de sortir du modele en choisissant des melanges d a priori avec des masses ponctuelles en zero pour les variances des effets aleatoires du fait de la reparametrisation le modele prend une structure lineaire conditionnelle qui facilite l utilisation de lois a priori conjuguees et normales nous demontrons que les calculs a posteriori peuvent etre effectues via un algorithme simple et efficace de monte carlo par chaine de markov les methodes sont illustrees par des donnees simulees et par des donnees reelles d exposition prenatale au polychlorobiphenile et des donnees sur le developpement psychomoteur des enfants

nonsmooth analysis on smooth manifolds we study infinitesimal properties of nonsmooth nondifferentiable functions on smooth manifolds the eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function a subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems nonclassical problems of calculus of variations and generalized solutions of first order partial differential equations on manifolds we also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions