the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

a paradox in the approximation of dirichlet control problems in curved domains in this paper we study the approximation of a dirichlet control problem governed by an elliptic equation defined on a curved domain omega to solve this problem numerically it is usually necessary to approximate omega by a typically polygonal new domain omega h the difference between the solutions of both infinite dimensional control problems one formulated in omega and the second in omega h was studied in e casas and j sokolowski siam j control optim 48 2010 pp 3746 3780 where an error of order o h was proved in k deckelnick a gunther and m hinze siam j control optim 48 2009 pp 2798 2819 the numerical approximation of the problem defined in omega was considered the authors used a finite element method such that omega h was the polygon formed by the union of all triangles of the mesh of parameter h they proved an error of order o h 3 2 for the difference between continuous and discrete optimal controls here we show that the estimate obtained in e casas and j sokolowski siam j control optim 48 2010 pp 3746 3780 cannot be improved which leads to the paradox that the numerical solution is a better approximation of the optimal control than the exact one obtained just by changing the domain from omega to omega h

a priori error estimates of the dtn fem for the transmission problem in acoustics abstract this paper is concerned with a variational approach solving the two dimensional acoustic transmission problems the original problem is reduced to an equivalent nonlocal boundary value problem by introducing an exact dirichlet to neumann dtn mapping in terms of fourier expansion series on an artificial boundary uniqueness and existence of solutions in appropriate sobolev spaces are established for the corresponding variational problem and its modification due to the truncation of dtn mapping a priori error estimates containing the effects of both element meshsize and truncation order of series for the finite element approximation are derived numerical experiments are also presented to illustrate the efficiency and accuracy of the numerical scheme

linking first occurrence polynomials over f2 by steenrod operations abstract it is proved that for every irreducible representation l of the full matrix semigroup mn f 2 the first occurrence of l as a composition factor in the polynomial algebra p f 2 x1 xn is linked by a steenrod operation to the first occurrence of l as a submodule in p this steenrod operation is given explicitly as the image of an admissible monomial in the steenrod squares sqr under the canonical anti automorphism of the mod 2 steenrod algebra a the first occurrences of both kinds are also linked to higher degree occurrences of l by elements of the milnor basis of a

an adaptive strategy for elliptic problems including a posteriori controlled boundary approximation we derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary the numerical solution is assumed to be defined on a finite element mesh whose boundary vertices are located on the boundary of the continuous problem no assumption is made on a geometrically fitting shape a posteriori error estimates are given in the energy norm and the l 2 norm and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation furthermore a strategy is presented to compute the effect of the non discretized part of the domain on the error starting from a coarse mesh this especially implies that parts of the domain where the measured error is small stay non discretized the presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary the reliable computation of the a posteriori estimates and a mesh adaptation based on delaunay techniques numerical examples illustrate that errors outside the initial discretization will be detected

criteria for the approximation property for multigrid methods in nonnested spaces we extend the abstract frameworks for the multigrid analysis for nonconforming finite elements to the case where the assumptions of the second strang lemma are violated the consistency error is studied in detail for finite element discretizations on domains with curved boundaries this is applied to prove the approximation property for conforming elements stabilized q1 p0elements and nonconforming elements for linear elasticity on nonpolygonal domains proving the approximation property for the multigrid analysis for nonconforming finite element discretizations is formalized in 7 4 15 for many cases it suffices to verify criteria on the approximation quality and the consistency error in these papers it is required that a continuous bilinear form can be extended to a nonconforming finite element space which is not valid for many interesting applications the purpose of this paper is to establish a full set of criteria which guarantees the approximation property for a wide range of nonnested discretizations where we do not assume that the discrete bilinear form coincides with the continuous bilinear form for all conforming functions in the notation we follow bramble 5 chap 4 and our results can be applied directly to the multigrid theory described there the results extend known results by brenner 7 and stevenson 15 and they provide a systematic and constructive way of studying nonnested multigrid algorithms for more general nonnested spaces and varying forms the paper is organized as follows first we introduce an abstract setting describing a multigrid hierarchy for nonconforming discretizations of an elliptic partial differential equation without full regularity as usual the multigrid approximation property is derived by comparison with the finite element approximation property which we formulate using an interpolation operator and its adjoint with respect to the energy scalar product in a second step section 1 9 we derive the approximation property from consistency assumptions on a conforming comparison space similar to the approach in 7 in section 2 we consider the case of conforming finite elements on a polygonal approximation of the computational domain here we choose a comparison space consisting of curved finite elements after introducing a suitable interpolation operator we use the equivalence of the operator norm scale used throughout section 1 received by the editor january 23 2001 and in revised form march 21 2003 2000 mathematics subject classification primary 65n55 65f10

convergence of the maximum a posteriori path estimator in hidden markov models in a hidden markov model hmm the underlying finite state markov chain cannot be observed directly but only by an additional process we are interested in estimating the unknown path of the markov chain the most widely used estimator is the maximum a posteriori path estimator map path estimator it can be calculated effectively by the viterbi 1967 algorithm as is e g frequently done in the field of coding theory correction of intersymbol interference and speech recognition we investigate component wise convergence of the map path estimator convergence is shown under the condition of unbounded likelihood ratios this condition is satisfied in the important case of hmms with additive white gaussian noise we also prove convergence if the markov chain has two states the so called viterbi paths are an important tool for obtaining these results

numerical analysis of quasilinear parabolic equations under low regularity assumptions in this paper we carry out the numerical analysis of a class of quasilinear parabolic equations where the diffusion coefficient depends on the solution of the partial differential equation the goal is to prove error estimates for the fully discrete equation using discontinuous galerkin discretization in time dg 0 combined with piecewise linear finite elements in space this analysis is performed under minimal regularity assumptions on the data in particular we omit any assumption regarding existence of a second derivative in time of the solution

testing for randomness in a random coefficient autoregression model abstract we propose a test to discern between an ordinary autoregressive model and a random coefficient one to this end we develop a full fledged estimation theory for the variances of the idiosyncratic innovation and of the random coefficient based on a two stage wls approach our results hold irrespective of whether the series is stationary or nonstationary and as an immediate result they afford the construction of a test for relevant randomness further building on these results we develop a randomised test statistic for the null that the coefficient is non random as opposed to the alternative of a standard r c a 1 model monte carlo evidence shows that the test has the correct size and very good power for all cases considered

the second generation shifted boundary method and its numerical analysis abstract recently the shifted boundary method sbm was proposed within the class of unfitted or immersed or embedded finite element methods by reformulating the original boundary value problem over a surrogate approximate computational domain the sbm avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning accuracy is maintained by modifying the original boundary conditions using taylor expansions hence the name of the method that shifts the location and values of the boundary conditions in this article we present enhanced variational sbm formulations for the poisson and stokes problems with improved flexibility and robustness these simplified variational forms allow to relax some of the assumptions required by the mathematical proofs of stability and convergence of earlier implementations first we show that these new sbm implementations can be proved asymptotically stable and convergent even without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive second we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at dirichlet boundaries therefore avoiding the calibration of an additional stabilization parameter finally we prove enhanced l 2 estimates without the cumbersome assumption of earlier proofs that the surrogate domain is convex instead we rely on a conventional assumption that the boundary of the true domain is smooth which can also be replaced by requiring convexity of the true domain the aforementioned improvements open the way to a more general and efficient implementation of the shifted boundary method particularly in complex three dimensional geometries we complement these theoretical developments with numerical experiments in two and three dimensions

approximations of very weak solutions to boundary value problems standard weak solutions to the poisson problem on a bounded domain have square integrable derivatives which limits the admissible regularity of inhomogeneous data the concept of solution may be further weakened in order to define solutions when data is rough such as for inhomogeneous dirichlet data that is only square integrable over the boundary such very weak solutions satisfy a nonstandard variational form u v g v a galerkin approximation combined with an approximation of the right hand side g defines a finite element approximation of the very weak solution applying conforming linear elements leads to a discrete solution equivalent to the text book finite element solutionto the poisson problem in which the boundary data is approximated by l2 projections the l2 convergence rate of the discrete solution is o hs for some s in 0 1 2 that depends on the shape of the domain assuming a polygonal two dimensional or polyhedral three dimensional domain without slits and only square integr

asymptotic normality of the maximum likelihood estimator for general hidden markov models hidden markov models hmms have during the last decade become a widespread tool for modeling sequences of dependent random variables inference for such models is usually based on the maximum likelihood estimator mle and consistency of the mle for general hmms was recently proved by leroux in this paper we show that under mild conditions the mle is also asymptotically normal and prove that the observed information matrix is a consistent estimator of the fisher information

the composition factors of fp x1 x2 x3 as a gl 3 p module let gl n p denote the general linear group over the prime field fp and let sd denote the right gl n p module consisting of all homogeneous polynomials of degree d in n variables x1 x2 x over f we wish to determine the simple composition factors of sd unfortunately satisfactory models for all the simple gl n p modules themselves are not known there are p p 1 of these simple modules up to isomorphism and all of them are known to occur in the symmetric algebra s cf sd the general problem is therefore a very difficult one in our present state of knowledge the purpose of this paper is to offer a solution to this problem in the case n 3 this extends the work of david carlisle who calculated in his manchester thesis 2 the occurrences of all the l dimensional modules and developed a method for dealing with the general case let us begin with some general remarks about the problem the case n 1 is trivial as s d detd the dth power of the determinant representation the case n 2 is also easily disposed of for instance by glover 9 the simple modules are the tensor products of the sd for d 1 and d p the module s is not simple for example the pth power sf of the natural module is a proper submodule of 411 0021 8693192 3 00

testing for serial correlation against an arma 1 1 process abstract this article is concerned with tests for serial correlation in time series and in the errors of regression models in particular the nonstandard problem of testing for white noise against autoregressive moving average model arma 1 1 alternatives is considered the likelihood ratio lr sup lagrange multiplier lm and exponential average lm and lr tests are shown to be asymptotically admissible for arma 1 1 alternatives in addition they are shown to be consistent against all weakly stationary strong mixing non white noise alternatives simulation results compare the tests to several tests in the literature these results show that the lr and exp lr tests have very good all around power properties for nonseasonal alternatives

a discontinuous galerkin based immersed boundary method with non homogeneous boundary conditions and its application to elasticity we propose a discontinuous galerkin based immersed boundary method for elasticity problems the resulting numerical scheme does not require boundary fitting meshes and avoids boundary locking by switching the elements intersected by the boundary to a discontinuous galerkin approximation special emphasis is placed on the construction of a method that retains an optimal convergence rate in the presence of non homogeneous essential and natural boundary conditions the role of each one of the approximations introduced is illustrated by analyzing an analog problem in one spatial dimension finally extensive two and three dimensional numerical experiments on linear and nonlinear elasticity problems verify that the proposed method leads to optimal convergence rates under combinations of essential and natural boundary conditions

asymptotic normality of the maximum likelihood estimator in state space models state space models is a very general class of time series models capable of modelling dependent observations in a natural and interpretable way inference in such models has been studied by bickel ritov and ryden who consider hidden markov models which are special kinds of state space models and prove that the maximum likelihood estimator is asymptotically normal under mild regularity conditions in this paper we generalize the results of bickel ritov and ryden to state space models where the latent process is a continuous state markov chain satisfying regularity conditions which are fulfilled if the latent process takes values in a compact space

properties of the maximum a posteriori path estimator in hidden markov models a hidden markov model hmm consists of a nonobservable markov chain x x0 x1 and a measured process y y 0 y1 whose distribution is determined by x to estimate the hidden path of x up to time n x0 x1 xn by the observations y 0 y1 yn usually the maximum a posteriori probability path estimator map path estimator is applied an effective means for calculating this estimator is the viterbi algorithm which is widely employed in the fields of coding theory correction of intersymbol interference and text recognition here properties of the map estimator are derived under a certain condition c it is shown that the limiting process u u0 u1 is a regenerative process particularly this means that u has an asymptotic distribution satisfies the central limit theorem and possesses a mean error furthermore condition c is satisfied for a broad class of hmms including the most important case for applications the hmm with additive white gaussian noise

subalgebras of the steenrod algebra and the action of matrices on truncated polynomial algebras let a t be the subalgebra of the steenrod algebra generated by sqi i 2t we determine which a module indecomposable summands of f2 x1 xn are free over a t by first showing this to be equivalent to the group theoretic question of determining which irreducible modular representations of mn f2 or gln f2 occur as composition factors in the truncated polynomial algebra f2 x1 xn x12 xn2 the second question is then answered

tests for parameter instability and structural change with unknown change point this paper considers tests for parameter instability and structural change with unknown change point the results apply to a wide class of parametric models that are suitable for estimation by generalized method of moments procedures the asymptotic distributions of the test statistics considered here are nonstandard because the change point parameter only appears under the alternative hypothesis and not under the null the tests considered here are shown to have nontrivial asymptotic local power against all alternatives for which the parameters are nonconstant the tests are found to perform quite well in a monte carlo experiment reported elsewhere copyright 1993 by the econometric society

gaussian mixture vector autoregression this paper proposes a new nonlinear vector autoregressive var model referred to as the gaussian mixture vector autoregressive gmvar model the gmvar model belongs to the family of mixture vector autoregressive models and is designed for analyzing time series that exhibit regime switching behavior the main difference between the gmvar model and previous mixture var models lies in the definition of the mixing weights that govern the regime probabilities in the gmvar model the mixing weights depend on past values of the series in a specific way that has very advantageous properties from both theoretical and practical point of view a practical advantage is that there is a wide diversity of ways in which a researcher can associate different regimes with specific economically meaningful characteristics of the phenomenon modeled a theoretical advantage is that stationarity and ergodicity of the underlying stochastic process are straightforward to establish and contrary to most other nonlinear autoregressive models explicit expressions of low order stationary marginal distributions are known these theoretical properties are used to develop an asymptotic theory of maximum likelihood estimation for the gmvar model whose practical usefulness is illustrated in a bivariate setting by examining the relationship between the eur usd exchange rate and a related interest rate data

finite element approximation of spectral problems with neumann boundary conditions on curved domains this paper deals with the finite element approximation of the spectral problem for the laplace equation with neumann boundary conditions on a curved nonconvex domain convergence and optimal order error estimates are proved for standard piecewise linear continuous elements on a discrete polygonal domain h in tile framework of the abstract spectral approximation theory

isogeometric analysis on v reps first results abstract inspired by the introduction of volumetric modeling via volumetric representations v reps by massarwi and elber in 2016 in this paper we present a novel approach for the construction of isogeometric numerical methods for elliptic pdes on trimmed geometries seen as a special class of more general v reps we develop tools for approximation and local re parameterization of trimmed elements for three dimensional problems and we provide a theoretical framework that fully justifies our algorithmic choices we validate our approach both on two and three dimensional problems for diffusion and linear elasticity

sharp estimates for finite element approximations to elliptic problems with neumann boundary data of low regularity consider a homogeneous parabolic problem on a smooth bounded domain in n but with initial data and neumann boundary data of low regularity sharp interior maximum norm error estimates are given for a semidiscrete c0 finite element approximation to this problem these estimates are obtained by first establishing a new localized l estimate for semidiscrete finite element approximations on interior subdomains numerical examples illustrate the findings

admissibility of the likelihood ratio test when a nuisance parameter is present only under the alternative this paper establishes the asymptotic admissibility of the likelihood ratio lr test for a general class of testing problems in which a nuisance parameter is present only under the alternative hypothesis the paper also establishes the finite sample admissibility of the lr test for testing problems of this sort that arise in gaussian linear regression models with known variance

on finite element approximations of problems having inhomogeneous essential boundary conditions abstract the analysis and implementation of finite element methods for problems with inhomogeneous essential boundary conditions are considered the results are given for linear second order elliptic partial differential equations and for the nonlinear stationary navier stokes equations for certain easily implemented boundary treatments optimal error estimates and numerical examples are provided for problems posed on polyhedral domains

minorization conditions and convergence rates for markov chain monte carlo abstract general methods are provided for analyzing the convergence of discrete time general state space markov chains such as those used in stochastic simulation algorithms including the gibbs sampler the methods provide rigorous a priori bounds on how long these simulations should be run to give satisfactory results results are applied to two models of the gibbs sampler a bivariate normal model and a hierarchical poisson model with gamma conditionals the methods use the notion of minorization conditions for markov chains

optimal tests when a nuisance parameter is present only under the alternative this paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null for example the results apply to tests of structural change with unknown changepoint the testing problem considered is nonstandard and the classical asymptotic optimality results for the lagrange multiplier wald and likelihood ratio do not apply a weighted average power criterion is used here to generate optimal tests this criterion is similar to that used by a wald 1943 to obtain the classical asymptotic optimality properties of wald tests in regular testing problems copyright 1994 by the econometric society

asymptotic properties of the maximum likelihood estimator in autoregressive models with markov regime an autoregressive process with markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable markov chain in this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden markov chain conditional on the observations

residual based rank specification tests for ar garch type models this paper derives the asymptotic distribution for a number of rank based and classical residual specification tests in ar garch type models we consider tests for the null hypotheses of no linear and quadratic serial residual autocorrelation residual symmetry and no structural breaks we also apply our method to backtesting value at risk for these tests we show that generally no size correction is needed in the asymptotic test distribution when applied to ar garch residuals obtained through gaussian quasi maximum likelihood estimation to be precise we give exact expressions for the limiting null distribution of the test statistics applied to standardized residuals and find that standard critical values often though not always lead to conservative tests for this result we give simple necessary and sufficient conditions simulations show that our asymptotic approximations work well for a large number of ar garch models and parameter values we also show that the rank based tests often though not always have superior power properties over the classical tests even if they are conservative an empirical application illustrates the relevance of these tests to the ar garch models for weekly stock market return indices of some major and emerging countries

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

optimal error exponents in hidden markov models order estimation we consider the estimation of the number of hidden states the order of a discrete time finite alphabet hidden markov model hmm the estimators we investigate are related to code based order estimators penalized maximum likelihood ml estimators and penalized versions of the mixture estimator introduced by liu and narayan 1994 we prove strong consistency of those estimators without assuming any a priori upper bound on the order and smaller penalties than previous works we prove a version of stein s lemma for hmm order estimation and derive an upper bound on underestimation exponents then we prove that this upper bound can be achieved by the penalized ml estimator and by the penalized mixture estimator the proof of the latter result gets around the elusive nature of the ml in hmm by resorting to large deviation techniques for empirical processes finally we prove that for any consistent hmm order estimator for most hmm the overestimation exponent is null

finite element approximations in a nonlipschitz domain in this paper we analyze the approximation by standard piecewise linear finite elements of a nonhomogeneous neumann problem in a cuspidal domain since the domain is not lipschitz many of the results on sobolev spaces which are fundamental in the usual error analysis do not apply therefore we need to work with weighted sobolev spaces and to develop some new theorems on traces and extensions we show that in the domain considered here suboptimal order can be obtained with quasi uniform meshes even when the exact solution is in h 2 and we prove that the optimal order with respect to the number of nodes can be recovered by using appropriate graded meshes

a priori error analysis for hdg methods using extensions from subdomains to achieve boundary conformity we present the first a priori error analysis of a technique that allows us to numerically solve steady state diffusion problems defined on curved domains by using finite element methods defined in polyhedral subdomains dh for a wide variety of hybridizable discontinuous galerkin and mixed methods we prove that the order of convergence in the l2 norm of the approximate flux and scalar unknowns is optimal as long as the distance between the boundary of the original domain and that of the computational domain h is of order h we also prove that the l 2 norm of a projection of the error of the scalar variable superconverges with a full additional order when the distance between and h is of order h 5 4 but with only half an additional order when such a distance is of order h finally we present numerical experiments confirming the theoretical results and showing that even when the distance between and h is of order h the above mentioned projection of the error of the scalar variable can still superconverge with a full additional order

galerkin approximations of abstract parabolic boundary value problems with rough boundary data l p theory etude des approximations de galerkin d un probleme aux valeurs limites parabolique abstrait avec des donnees aux limites grossieres determination des vitesses de convergence optimales dans les normes lp

bayesian predictive information criterion for the evaluation of hierarchical bayesian and empirical bayes models summary the problem of evaluating the goodness of the predictive distributions of hierarchical bayesian and empirical bayes models is investigated a bayesian predictive information criterion is proposed as an estimator of the posterior mean of the expected loglikelihood of the predictive distribution when the specified family of probability distributions does not contain the true distribution the proposed criterion is developed by correcting the asymptotic bias of the posterior mean of the loglikelihood as an estimator of its expected loglikelihood in the evaluation of hierarchical bayesian models with random effects regardless of our parametric focus the proposed criterion considers the bias correction of the posterior mean of the marginal loglikelihood because it requires a consistent parameter estimator the use of the bootstrap in model evaluation is also discussed

specification testing in markov switching time series models this paper develops a series of specification tests of markov switching time series models tests for omitted autocorrelation omitted arch misspecification of the markovian dynamics and omitted explanatory variables are proposed all of the tests can be constructed as a natural byproduct of the routine used to calculate the smoothed probability that a given observation came from a particular regime and do not require estimation of additional parameters the paper performs monte carlo analysis of the tests and briefly illustrates their use with an empirical application

weyl modules and the mod 2 steenrod algebra abstract this paper continues our study of the action of the mod 2 steenrod algebra a 2 on the polynomial algebra p n f 2 t 1 t n we obtain further partial results on the hit problem of f p peterson which asks for a minimal generating set for p n as an a 2 module we also study the structure of the quotient by the hit elements as a graded representation of the finite general linear group g n gl n f 2 i e as a module over the finite group algebra f 2 g n these results were obtained in previous work of the authors for the special case of the steinberg module for g n by extending the scalars to f 2 the algebraic closure of f 2 we obtain commuting actions of a 2 and g n on p n f 2 t 1 t n while this makes no essential difference to the representation theory of g n or to the hit problem it allows us to treat the action of g n on p n as the restriction of that of the algebraic group g n gl n f 2 in particular we make use of tilting modules for g n to show that for every irreducible representation l of g n a minimal set of a 2 generators of p n must contain a copy of the corresponding dual weyl module

calculating posterior distributions and modal estimates in markov mixture models abstract this paper is concerned with finite mixture models in which the populations from one observation to the next are selected according to an unobserved markov process a new full bayesian approach based on the method of gibbs sampling is developed calculations are simplified by data augmentation achieved by introducing a population index variable into the list of unknown parameters it is shown that the latent variables one for each observation can be simulated from their joint distribution given the data and the remaining parameters this result serves to accelerate the convergence of the gibbs sample modal estimates are also computed by stochastic versions of the em algorithm these provide an alternative to a full bayesian approach and to existing methods of locating the maximum likelihood estimate the ideas are applied in detail to poisson data mixtures of multivariate normal distributions and autoregressive time series

testing when a parameter is on the boundary of the maintained hypothesis this paper considers testing problems where several of the standard regularity conditions fail to hold we consider the case where i parameter vectors in the null hypothesis may lie on the boundary of the maintained hypothesis and ii there may be a nuisance parameter that appears under the alternative hypothesis but not under the null the paper establishes the asymptotic null and local alternative distributions of quasi likelihood ratio rescaled quasi likelihood ratio wald and score tests in this case the results apply to tests based on a wide variety of extremum estimators and apply to a wide variety of models examples treated in the paper are i tests of the null hypothesis of no conditional heteroskedasticity in a garch 1 1 regression model and ii tests of the null hypothesis that some random coefficients have variances equal to zero in a random coefficients regression model with possibly correlated random coefficients

a robust finite element method for nonhomogeneous dirichlet problems in domains with curved boundaries in this paper the authors consider a simple finite element method on an approximately polygonal domain using linear elements the dirichlet data are transferred in a natural way and the resulting linear system can be solved using multigrid techniques their analysis takes into account the change in domain and data transfer and optimal error estimates are obtained that are robust in the regularity of the boundary data provided they are at least square integrable it is proved that the natural extension of this finite element approximation to the original domain is optimal order accurate

testing for observation dependent regime switching in mixture autoregressive models abstract testing for regime switching when the regime switching probabilities are specified either as constants mixture models or are governed by a finite state markov chain markov switching models are long standing problems that have also attracted recent interest this paper considers testing for regime switching when the regime switching probabilities are time varying and depend on observed data observation dependent regime switching specifically we consider the likelihood ratio test for observation dependent regime switching in mixture autoregressive models the testing problem is highly nonstandard involving unidentified nuisance parameters under the null parameters on the boundary singular information matrices and higher order approximations of the log likelihood we derive the asymptotic null distribution of the likelihood ratio test statistic in a general mixture autoregressive setting using high level conditions that allow for various forms of dependence of the regime switching probabilities on past observations and we illustrate the theory using two particular mixture autoregressive models the likelihood ratio test has a nonstandard asymptotic distribution that can easily be simulated and monte carlo studies show the test to have good finite sample size and power properties

some estimates for a weighted l 2 projection this paper is devoted to the error estimates for some weighted l2 projections nearly optimal estimates are obtained these estimates can be applied to the analysis of the usual multigrid method multilevel preconditioner and domain decomposition method for solving elliptic boundary problems whose coefficients have large jump discontinuities

markov chain monte carlo estimation of classical and dynamic switching and mixture models bayesian estimation of a very general model class where the distribution of the observations depends on a latent process taking values in a discrete state space is discussed in this article this model class covers finite mixture modeling markov switching autoregressive modeling and dynamic linear models with switching the consequences the unidentifiability of this type of model has on markov chain monte carlo mcmc estimation are explicitly dealt with joint bayesian estimation of all latent variables model parameters and parameters that determine the probability law of the latent process is carried out by a new mcmc method called permutation sampling the permutation sampler first samples from the unconstrained posterior which often can be done in a convenient multimove manner and then applies a permutation of the current labeling of the states of the latent process in a first run the random permutation sampler used selected the permutation randomly the mcmc output of the random permutation s

testing for regime switching we analyze use of a quasi likelihood ratio statistic for a mixture model to test the null hypothesis of one regime versus the alternative of two regimes in a markov regime switching context this test exploits mixture properties implied by the regime switching process but ignores certain implied serial correlation properties when formulated in the natural way the setting is nonstandard involving nuisance parameters on the boundary of the parameter space nuisance parameters identified only under the alternative or approximations using derivatives higher than second order we exploit recent advances by andrews 2001 and contribute to the literature by extending the scope of mixture models obtaining asymptotic null distributions different from those in the literature we further provide critical values for popular models or bounds for tail probabilities that are useful in constructing conservative critical values for regime switching tests we compare the size and power of our statistics to other useful tests for regime switching via monte carlo methods and find relatively good performance we apply our methods to reexamine the classic cartel study of porter 1983 and reaffirm porter s findings

powell sabin splines with boundary conditions for polygonal and non polygonal domains powell sabin splines are piecewise quadratic polynomials with a global c 1 continuity defined on conforming triangulations imposing boundary conditions on such a spline leads to a set of constraints on the spline coefficients first we discuss boundary conditions defined on a polygonal domain before we treat boundary conditions on a general curved domain boundary we consider dirichlet and neumann conditions and we show that a particular choice of the ps triangles at the boundary can greatly simplify the corresponding constraints finally we consider an application where the techniques developed in this paper are used the numerical solution of a partial differential equation by the galerkin and collocation method

immersed finite element method for eigenvalue problem abstract we consider the approximation of elliptic eigenvalue problem with an interface the main aim of this paper is to prove the stability and convergence of an immersed finite element method ifem for eigenvalues using crouzeix raviart p 1 nonconforming approximation we show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem we analyze the ifem for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues numerical experiments demonstrate our theoretical results

optimal test for markov switching parameters this paper proposes a class of optimal tests for the constancy of parameters in random coefficients models our testing procedure covers the class of hamilton s models where the parameters vary according to an unobservable markov chain but also applies to nonlinear models where the random coefficients need not be markov we show that the contiguous alternatives converge to the null hypothesis at a rate that is slower than the standard rate therefore standard approaches do not apply we use bartlett type identities for the construction of the test statistics this has several desirable properties first it only requires estimating the model under the null hypothesis where the parameters are constant second the proposed test is asymptotically optimal in the sense that it maximizes a weighted power function we derive the asymptotic distribution of our test under the null and local alternatives asymptotically valid bootstrap critical values are also proposed

a locally modified parametric finite element method for interface problems we present a modified finite element method that is able to approximate interface problems with high accuracy we consider interface problems where the solution is continuous its derivatives however may be discontinuous across interface curves within the domain the proposed discretization is based on a local modification of the finite element basis functions using a fixed quadrilateral mesh instead of moving mesh nodes we resolve the interface locally by an adapted parametric approach all modifications are applied locally and in an implicit fashion the scheme is easy to implement and is well suited for time dependent moving interface problems we show optimal order of convergence for elliptic problems and further we give a bound on the condition number of the system matrix both estimates do not depend on the interface location relative to the mesh

laplace eigenvalues on regular polygons a series in 1 n abstract for regular polygons p n inscribed in a circle the eigenvalues of the laplacian converge as n to the known eigenvalues on a circle we compute the leading terms of n in a series in powers of 1 n by applying the calculus of moving surfaces to a piecewise smooth evolution from the circle to the polygon the o 1 n 2 term comes from hadamard s formula and reflects the change in area this term disappears if we transcribe the polygon scaling it to have the same area as the circle

new estimates for multilevel algorithms including the v cycle the purpose of this paper is to provide new estimates for certain multilevel algorithms in particular we are concerned with the simple additive multilevel algorithm discussed recently together with j xu and the standard v cycle algorithm with one smoothing step per grid we shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity for example the theory applies to the standard multigrid vcycle on the l shaped domain or a domain with a crack and yields a uniform convergence rate we also prove uniform convergence rates for the multigrid v cycle for problems with nonuniformly refined meshes finally we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid v cycle algorithms

asymptotic properties of the maximum likelihood estimator in regime switching econometric models abstract markov regime switching models have been widely used in numerous empirical applications in economics and finance however the asymptotic distribution of the maximum likelihood estimator mle has not been proven for some empirically popular markov regime switching models in particular the asymptotic distribution of the mle has been unknown for models in which some elements of the transition probability matrix have the value of zero as is commonly assumed in empirical applications with models with more than two regimes this also includes models in which the regime specific density depends on both the current and the lagged regimes such as the seminal model of hamilton 1989 and switching arch model of hamilton and susmel 1994 this paper shows the asymptotic normality of the mle and consistency of the asymptotic covariance matrix estimate of these models

analysis of a robust finite element approximation for a parabolic equation with rough boundary data the approximation of parabolic equations with nonhomogeneous dirichlet boundary data by a numerical method that consists of finite elements for the space discretization and the backward euler time discretization is studied the boundary values are assumed in a least squares sense it is shown that this method achieves an optimal rate of convergence for rough only l2 boundary data and for smooth data as well the results of numerical computations which confirm the robust theoretical error estimates are also presented

adaptive finite elements for exterior domain problems we present an adaptive finite element method for solving elliptic problems in exterior domains that is for problems in the exterior of a bounded closed domain in bbb r d d in 2 3 we describe a procedure to generate a sequence of bounded computational domains omega h k k 1 2 more precisely a sequence of successively finer and larger grids until the desired accuracy of the solution u h is reached to this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm by means of a residual based error estimator furthermore we prove convergence of the adaptive algorithm numerical examples show the optimal order of convergence

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

efficient likelihood estimation in state space models motivated by studying asymptotic properties of the maximum likelihood estimator mle in stochastic volatility sv models in this paper we investigate likelihood estimation in state space models we first prove under some regularity conditions there is a consistent sequence of roots of the likelihood equation that is asymptotically normal with the inverse of the fisher information as its variance with an extra assumption that the likelihood equation has a unique root for each n then there is a consistent sequence of estimators of the unknown parameters if in addition the supremum of the log likelihood function is integrable the mle exists and is strongly consistent edge worth expansion of the approximate solution of likelihood equation is also established several examples including markov switching models arma models g arch models and stochastic volatility sv models are given for illustration

finite element approximation of dirichlet boundary control for elliptic pdes on two and three dimensional curved domains we consider the variational discretization of elliptic dirichlet optimal control problems with constraints on the control the underlying state equation which is considered on smooth two and three dimensional domains is discretized by linear finite elements taking into account domain approximation the control variable is not discretized we obtain optimal error bounds for the optimal control in two and three space dimensions and prove a superconvergence result in two dimensions provided that the underlying mesh satisfies some additional condition we confirm our analytical findings by numerical experiments

specification testing when score test statistics are identically zero we investigate the problem of previous specification testing when the score vector evaluated at the restricted mle is identically zero several econometric examples are provided a general test procedure which generalizes the geometric principle of the score test is proposed the wald and the likelihood ratio tests are also analyzed even under such irregularities the usual asymptotic distribution of the likelihood ratio statistics is still valid however the wald type statistics need to be modified the generalized score the likelihood ratio and the modified wald tests are shown to be asymptotically equivalent the asymptotic efficiency of the tests is derived

approximations of elliptic optimal control problems with controls acting on a lower dimensional manifold in this paper we study finite element approximations to some elliptic optimal control problems with controls acting on a lower dimensional manifold which can be a point a curve or a surface we use piecewise linear finite elements to approximate state variables while utilizing the variational discretization to approximate control variables we derive several a priori error estimates for optimal controls from different cases depending on the dimensions of the computational domain and the manifold where controls act we ends up with extensive numerical experiments which confirm our theoretical findings