analysis of multilevel methods for eddy current problems in papers by arnold falk and winther and by hiptmair novel multigrid methods for discrete h curl elliptic boundary value problems have been proposed such problems frequently occur in computational electromagnetism particularly in the context of eddy current simulation this paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods it provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains in particular asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems the principal idea is to use approximate helmholtz decompositions of the function space h curl into an h1 regular subspace and gradients the main results of standard multilevel theory for h1 elliptic problems can then be applied to both subspaces this yields preliminary decompositions still outside the edge element spaces judicious alterations can cure this

reducing spurious dispersion anisotropy and reflection in finite element analysis of time harmonic acoustics abstract galerkin and stabilized galerkin least squares and galerkin gradient least squares finite element methods for solving problems of time harmonic acoustics are presented and analyzed the error of discretization is related to spurious representation of physical phenomena namely dispersion anisotropy and reflection all of which decrease with mesh refinement the performance of the stabilized methods in terms of dispersion and anisotropy is identical both are superior to the galerkin method relaxing wave resolution requirements further insight into the design of these methods is provided by the analyses the stabilized methods differ in the degree of spurious reflection that is engendered by transitions in mesh size the performance of galerkin gradient least squares deteriorates on non uniform meshes however galerkin least squares maintains its enhanced performance exhibiting low sensitivity to transitions in mesh size thereby reducing spurious dispersion anisotropy and reflection in finite element analysis of time harmonic acoustics guidelines for discretization pertaining to mesh orientation and wave resolution are presented

extremes of moving averages of random variables with finite endpoint where zj are iid and nonnegative random variables and c 0 are constants satisfying summability conditions at least sufficient to make the random series above converge we suppose that the distribution of zj is regularly varying near 0 and discuss lower tail behavior of finite and infinite linear combinations the behavior is quite different in the two cases for finite linear combinations the lower tail is again regularly varying but for infinite moving averages the lower tail is f varying i e it is in the domain of attraction of a type i extreme value distribution in the sense of minima convergence of point processes based on the moving averages is shown to hold in both the finite and infinite order cases and suitable conclusions are drawn from such convergences a useful analytic tool is asymptotic normality of the esscher transform of the common distribution of the z s the extreme value results of this paper are in terms of minima of the moving average processes but results can be adapted to study maxima of moving averages of random variables in the domain of attraction of the type iii extreme value distribution for maxima

asymptotics for in sample density forecasting this paper generalizes recent proposals of density forecasting models and it develops theory for this class of models in density forecasting the density of observations is estimated in regions where the density is not observed identification of the density in such regions is guaranteed by structural assumptions on the density that allows exact extrapolation in this paper the structural assumption is made that the density is a product of one dimensional functions the theory is quite general in assuming the shape of the region where the density is observed such models naturally arise when the time point of an observation can be written as the sum of two terms e g onset and incubation period of a disease the developed theory also allows for a multiplicative factor of seasonal effects seasonal effects are present in many actuarial biostatistical econometric and statistical studies smoothing estimators are proposed that are based on backfitting full asymptotic theory is derived for them a practical example from the insurance business is given producing a within year budget of reported insurance claims a small sample study supports the theoretical results

a high precision integration scheme for the spectral element time domain method in electromagnetic simulation a high precision integration hpi scheme combined with the spectral element time domain setd method is presented to solve the time dependent maxwell s equations spatial discretization by spectral elements will lead to block diagonal mass matrices thus greatly alleviating the computational burden of inverting the mass matrices a high precision time integration method based on the matrix exponential is then employed to solve the discretized setd system numerical examples demonstrate that this algorithm is unconditionally stable and very accurate

high spatial order finite element method to solve maxwell s equations in time domain this paper presents a finite element method with high spatial order for solving the maxwell equations in the time domain in the first part we provide the mathematical background of the method then we discuss the advantages of the new scheme compared to a classical finite difference time domain fdtd method several examples show the advantages of using the new method for different kinds of problems comparisons in terms of accuracy and cpu time between this method the fdtd and the finite volume time domain methods are given as well

the gaussian measure of shifted balls let be a centered gaussian measure on a hilbert spaceh and let b r subseteq h be the centered ball of radiusr 0 fora h and mathop lim limits t mathbf to mathbf infty mathbf r t t mathbf a we give the exact asymptotics of b r t t a ast also upper and lower bounds are given when is defined on an arbitrary separable banach space our results range from small deviation estimates to large deviation estimates

regularity of laws and ergodicity of hypoelliptic sdes driven by rough paths we consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior in particular we focus on the case when the driving noise is a rough path valued fractional brownian motion with hurst parameter h 2 1 1 our contribution in this work is twofold first when the driving vector fields satisfy hormander s c elebrated lie bracket condition we derive explicit quantitative bounds on the i nverse of the malliavin ma trix en route to this we provide a novel deterministic version of norris s lemma for differential equations driven by rough paths this result with the added assumption that the linearized equation has moments will then yield that the transition laws have a smooth density with respect to lebesgue measure our second main result states that under hormander s condi tion the solutions to rough differential equations driven by fractional brownian motion with h 2 1 1 enjoy a suitable version of the strong feller property under a standard controllability condition this implies that they admit a unique stationary solution that is physical in the sense that it does not look into the future

on the gaussian approximation of convolutions under multidimensional analogues of s n bernstein s inequality conditions on etudie le taux d approximation de distribution de sommes de vecteurs aleatoires independants par des distributions gaussiennes correspondantes

time domain finite difference and finite element methods for maxwell equations in complex media extensions of finite difference time domain fdtd and finite element time domain fetd algorithms are reviewed for solving transient maxwell equations in complex media also provided are a few representative examples to illustrate the modeling capabilities of fdtd and fetd for complex media the term complex media refers here to media with dispersive bi anisotropic inhomogeneous and or nonlinear properties present in the constitutive tensors

rates of clustering in strassen s lil for partial sum processes we provide a detailed description of the rate of clustering in strassen s functional law of the iterated logarithm for partial sum processes necessary and sufficient conditions are given for certain clustering rates in terms of moment type conditions in the process we also derive a new strong approximation of sums of iid random variables by a wiener process

rates of convergence for bayes and maximum likelihood estimation for mixture of normal densities we study the rates of convergence of the maximum likelihood estimator mle and posterior distribution in density estimation problems where the densities are location or location scale mixtures of normal distributions with the scale parameter lying between two positive numbers the true density is also assumed to lie in this class with the true mixing distribution either compactly supported or having sub gaussian tails we obtain bounds for hellinger bracketing entropies for this class and from these bounds we deduce the convergence rates of sieve mles in hellinger distance the rate turns out to be log n n where k 1 is a constant that depends on the type of mixtures and the choice of the sieve next we consider a dirichlet mixture of normals as a prior on the unknown density we estimate the prior probability of a certain kullback leibler type neighborhood and then invoke a general theorem that computes the posterior convergence rate in terms the growth rate of the hellinger entropy and the concentration rate of the prior the posterior distribution is also seen to converge at the rate log n n in where k now depends on the tail behavior of the base measure of the dirichlet process

hp a priori error estimates for a non dissipative spectral discontinuous galerkin method to solve the maxwell equations in the time domain in this paper we present the hp convergence analysis of a non dissipative high order discontinuous galerkin method on unstructured hexahedral meshes using a mass lumping technique to solve the time dependent maxwell equations in particular we underline the spectral convergence of the method in the sense that when the solutions and the data are very smooth the discretization is of unlimited order moreover we see that the choice of a non standard approximate space for a discontinuous formulation with the absence of dissipation can imply a loss of spatial convergence finally we present a numerical result which seems to confirm this property

the existence and asymptotic properties of a backfitting projection algorithm under weak conditions we derive the asymptotic distribution of a new backfitting procedure for estimating the closest additive approximation to a nonparametric regression function the procedure employs a recent projection interpretation of popular kernel estimators provided by mammen marron turlach and wand 1997 and the asymptotic theory of our estimators is derived using the theory of additive projections reviewed in bickel klaassen ritov and wellner 1993 our procedure achieves the same bias and variance as the oracle estimator based on knowing the other components and in this sense improves on the method analysed in opsomer and ruppert 1997 we provide high level conditions independent of the sampling scheme we then verify that these conditions are satisfied in a regression and a time series autoregression under weak conditions

clustering behavior of finite variance partial sum processes clustering rates in strassen s functional law of the iterated logarithm are determined for finite variance partial sum processes in one dimension a general characterization of these rates similar to one recently obtained for onedimensional brownian motion shows that relatively mild moment conditions on a partial sum process lead to high order clustering rates at certain points of the strassen set

the logistic normal distribution for bayesian nonparametric predictive densities abstract this article models the common density of an exchangeable sequence of observations by a generalization of the process derived from a logistic transform of a gaussian process the support of the logistic normal includes all distributions that are absolutely continuous with respect to the dominating measure of the observations the logistic normal family is closed in the prior to posterior bayes analysis with the observations entering the posterior distribution through the covariance function of the gaussian process the covariance of the gaussian process plays the role of a smoothing kernel three features of the model provide a flexible structure for computing the predictive density a the mean of the gaussian process corresponds to the prior mean of the random density b the prior variance of the gaussian process controls the influence of the data in the posterior process as the variance increases the predictive density has greater fidelity to the data c the prior covariance of the gau

efficient estimation of the partly linear additive cox model the partly linear additive cox model is an extension of the linear cox model and allows flexible modeling of covariate effects semiparametrically we study asymptotic properties of the maximum partial likelihood estimator of this model with right censored data using polynomial splines we show that with a range of choices of the smoothing parameter the number of spline basis functions required for estimation of the nonparametric components the estimator of the finite dimensional regression parameter is root n consistent asymptotically normal and achieves the semiparametric information bound rates of convergence for the estimators of the nonparametric components are obtained they are comparable to the rates in nonparametric regression implementation of the estimation approach can be done easily and is illustrated by using a simulated example

rates of clustering for some gaussian self similar processes the analogue of strassen s functional law of the iterated logarithm in known for many gaussian processes which have suitable scaling properties and here we establish rates at which this convergence takes place we provide a new proof of the best upper bound for the convergence tok by suitably normalized brownian motion and then continue with this method to get similar bounds for the brownian sheet and other self similar gaussian processes the previous method which produced these results for brownian motion in 1 was highly dependent on many special properties unavailable when dealing with other gaussian processes

lim inf results for gaussian samples and chung s functional lil lim inf results for gaussian samples are obtained in particular there are applications to rates of convergence for the functional form of chung s law of the iterated logarithm for brownian motion

bayesian inference for semiparametric regression using a fourier representation summary this paper presents the bayesian analysis of a semiparametric regression model that consists of parametric and nonparametric components the nonparametric component is represented with a fourier series where the fourier coefficients are assumed a priori to have zero means and to decay to 0 in probability at either algebraic or geometric rates the rate of decay controls the smoothness of the response function the posterior analysis automatically selects the amount of smoothing that is coherent with the model and data posterior probabilities of the parametric and semiparametric models provide a method for testing the parametric model against a non specific alternative the bayes estimator s mean integrated squared error compares favourably with the theoretically optimal estimator for kernel regression

estimating multiplicative and additive hazard functions by kernel methods we propose new procedures for estimating the component functions in both additive and multiplicativen onparametricm arker dependenht azard models we work with a full counting process framework that allows for left truncation and right censoring and time varying covariates our procedures are based on kernel hazard estimation as developed by nielsen and linton and on the idea of marginal integration we provide a central limit theorem for the marginal integration estimator we then define estimators based on finite step backfitting in both additive and multiplicative cases and prove that these estimators are asymptotically normal and have smaller variance than the marginal integration method

on the lower tail probabilities of some random series the behavior of tail probabilities p s r r 0 is investigated where s is a series s j z j generated by some sequence of positive numbers j and by a sequence z j of independent copies of a positive random variable z we present the exact asymptotic expression for p s r by means of laplace transform eexp s under weak assumptions on the behavior of the tail probabilities of z in the vicinity of zero the bounds of accuracy are also given and under weak supplementary smoothness conditions the asymptotic properties of the density of s are investigated

kernel estimation in a nonparametric marker dependent hazard model we introduce a new kernel hazard estimator in a nonparametric model where the stochastic hazard depends on the current value of time and on the current value of a time dependent covariate or marker we establish the pointwise and global convergence of our estimator

a lim inf result in strassen s law of the iterated logarithm for a functionf from strassen s class we investigate the lim inf behaviour of its distance from the normalized trajectories of a wiener process the lim inf rate is expressed in terms of a certain functional off in addition we give a result on the lim inf behaviour of the distance of the normed trajectories from strassen s class as a whole

posterior consistency of gaussian process prior for nonparametric binary regression consider binary observations whose response probability is an unknown smooth function of a set of covariates suppose that a prior on the response probability function is induced by a gaussian process mapped to the unit interval through a link function in this paper we study consistency of the resulting posterior distribution if the covariance kernel has derivatives up to a desired order and the bandwidth parameter of the kernel is allowed to take arbitrarily small values we show that the posterior distribution is consistent in the l 1 distance as an auxiliary result to our proofs we show that under certain conditions a gaussian process assigns positive probabilities to the uniform neighborhoods of a continuous function this result may be of independent interest in the literature for small ball probabilities of gaussian processes

adaptive bayesian estimation using a gaussian random field with inverse gamma bandwidth we consider nonparametric bayesian estimation inference using a rescaled smooth gaussian field as a prior for a multidimensional function the rescaling is achieved using a gamma variable and the procedure can be viewed as choosing an inverse gamma bandwidth the procedure is studied from a frequentist perspective in three statistical settings involving replicated observations density estimation regression and classification we prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax optimal up to a logarithmic factor whatever the regularity level of the data generating distribution thus the hierachical bayesian procedure with a fixed prior is shown to be fully adaptive

exploring the nature of covariate effects in the proportional hazards model we discuss an exploratory technique for investigating the nature of covariate effects in cox s proportional hazards model this technique features an additive term sigma p1 fj chi ij in place of the usual linear term sigma p1 chi ij beta j where chi i1 chi i2 chi ip are covariate values for the ith individual the fj are unspecified smooth functions that are estimated using scatterplot smoothers these functions can be used for descriptive purposes or to suggest transformations of the covariates the estimation technique is a variation of the local scoring algorithm for generalized additive models hastie and tibshirani 1986 statistical science 1 297 318

rates of contraction of posterior distributions based on gaussian process priors we derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on gaussian processes the rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel hilbert space of the gaussian process and the small ball probabilities of the gaussian process we determine these quantities for a range of examples of gaussian priors and in several statistical settings for instance we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a fractionally integrated brownian motion we also consider regression with gaussian errors and smooth classification under a logistic or probit link function combined with various priors

diagnostic plots to reveal functional form for covariates in multiplicative intensity models abstract we show how plots based on the residuals from a proportional hazards model may be used to reveal the correct functional form for covariates in the model a smoothed plot of the martingale residues was suggested for this purpose by therneau grambsch and fleming 1990 biometrika 77 147 160 however its consistency required that the covariates be independent they also noted that the plot could be biased for large covariate effects we introduce two refinements which overcome these difficulties the first is based on a ratio of scatter plot smooths where the numerator is the smooth of the observed count plotted against the covariate and the denominator is a smooth of the expected count this is related to the arjas goodness of fit plot 1988 journal of the american statistical association 83 204 212 the second technique smooths the martingale residuals divided by the expected count using expected count as a weight this latter approach is related to a glm partial residual plot as well as to the iterative methods of hastie and tibshirani 1990 biometrics 46 1005 1016 and gentleman and crowley 1991 biometrics 47 1283 1296 applications to survival data sets are given

a series expansion of fractional brownian motion let b be a fractional brownian motion with hurst index h 0 1 denote by the positive real zeros of the bessel function j h of the first kind of order h and let be the positive zeros of j 1 h in this paper we prove the series representation where x 1 x 2 and y 1 y 2 are independent gaussian random variables with mean zero and and the constant c h 2 is defined by c h 2 1 1 2h sin h we show that with probability 1 both random series converge absolutely and uniformly in t 0 1 and we investigate the rate of convergence

analysis and application of an equivalent berenger s pml model abstract the perfectly matched layer pml is a technique initially proposed by berenger for solving unbounded electromagnetic problems with the finite difference time domain method in this work we first formulate an equivalent pml model from the original berenger pml model in the corner region and then establish its stability we further develop a discontinuous galerkin method to solve this pml model and discrete stability similar to the continuous case is proved to demonstrate the absorbing property of this pml model we apply it to simulate wave propagation in metamaterials

misspecification in infinite dimensional bayesian statistics we consider the asymptotic behavior of posterior distributions if the model is misspecified given a prior distribution and a random sample from a distribution p 0 which may not be in the support of the prior we show that the posterior concentrates its mass near the points in the support of the prior that minimize the kullback leibler divergence with respect to p 0 an entropy condition and a prior mass condition determine the rate of convergence the method is applied to several examples with special interest for infinite dimensional models these include gaussian mixtures nonparametric regression and parametric models

smooth backfitting in practice summary compared with the classical backfitting of buja hastie and tibshirani the smooth backfitting estimator sbe of mammen linton and nielsen not only provides complete asymptotic theory under weaker conditions but is also more efficient robust and easier to calculate however the original paper describing the sbe method is complex and the practical as well as the theoretical advantages of the method have still neither been recognized nor accepted by the statistical community we focus on a clear presentation of the idea the main theoretical results and practical aspects like implementation and simplification of the algorithm we introduce a feasible cross validation procedure and apply it to the problem of data driven bandwidth choice for the sbe by simulations it is shown that the sbe and our cross validation work very well indeed in particular the sbe is less affected by sparseness of data in high dimensional regression problems or strongly correlated designs the sbe has reasonable performance even in 100 dimensional additive regression problems

regression splines in the cox model with application to covariate effects in liver disease abstract the cox proportional hazards model restricts the log hazard ratio to be linear in the covariates a smooth nonlinear covariate effect may go undetected in this model but can be well approximated by a spline function a survival model based on data from a clinical trial of primary biliary cirrhosis is developed using regression splines and the resulting log hazard ratio estimates are compared with those from nonparametric methods we remove the linear restriction on the log hazard ratio by transforming a continuous covariate into a vector of fixed knot basis splines b splines b splines are known to produce better conditioned systems of equations than the truncated power basis when used as interpolants and show similar behavior when fitting proportional hazards models we describe the procedures for and the issues arising in the estimation and the testing of the b spline coefficients although inference is not well developed for some nonparametric methods that estimate covariate effects the

limit theorems for moving averages of independent random vectors let xj be i i d r v s taking values in a separable banach space and let s n sum limits j 1 n x j the almost sure limiting behavior of the normalized random set of increments bn 1 sk h sk 1 k n 1 h an for a broad class of sequences an with corresponding normalizations bnis determined when an log n the results generalize and improve some recent work of csorg and revesz and some previous work of lai when an clog n the results generalize and refine in a certain sense the erdos renyi law of large numbers the case an logn 0 is also studied

a note on multivariate gaussian estimates an upper bound of multivariate gaussian probability for a general convex domain d is given based on a geometric observation the bound is sharper than known ones on multivariate mills ratio in many cases

smooth backfitting in generalized additive models generalized additive models have been popular among statisticians and data analysts in multivariate nonparametric regression with non gaussian responses including binary and count data in this paper a new likelihood approach for fitting generalized additive models is proposed it aims to maximize a smoothed likelihood the additive functions are estimated by solving a system of nonlinear integral equations an iterative algorithm based on smooth backfitting is developed from the newton kantorovich theorem asymptotic properties of the estimator and convergence of the algorithm are discussed it is shown that our proposal based on local linear fit achieves the same bias and variance as the oracle estimator that uses knowledge of the other components numerical comparison with the recently proposed two stage estimator ann statist 32 2004 2412 2443 is also made

functional limit theorems for the increments of d dimensional gaussian processes in a holder type norm in this paper estimating upper bounds of large deviation probabilities for the increments of d dimensional gaussian processes in holder type norm we obtain functional limit results for the increments of d dimensional gaussian processes in the holder type norm

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

dispersive and dissipative behavior of the spectral element method if the nodes for the spectral element method are chosen to be the gauss legendre lobatto points and a lagrange basis is used then the resulting mass matrix is diagonal and the method is sometimes then described as the gauss point mass lumped finite element scheme we study the dispersive behavior of the scheme in detail and provide both a qualitative description of the nature of the dispersive and dissipative behavior of the scheme along with precise quantitative statements of the accuracy in terms of the mesh size and the order of the scheme we prove that a the gauss point mass lumped scheme i e spectral element method tends to exhibit phase lag whereas the consistent finite element scheme tends to exhibit phase lead b the absolute accuracy of the spectral element scheme is 1 p times better than that of the finite element scheme despite the use of numerical integration c when the order p the mesh size h and the frequency of the wave omega satisfy 2p 1 approx omega h the true wave is essentially fully resolved as a consequence one obtains a proof of the general rule of thumb sometimes quoted in the context of spectral element methods pi modes per wavelength are needed to resolve a wave

mur nedelec finite element schemes for maxwell s equations we study mapped mass lumped edge elements for approximating a three dimensional scattering problem the elements are mapped mur nedelec second kind elements and the scheme is designed to handle anisotropic inhomogeneous scatterers with moderately complicated geometry a novel aspect of the scheme is the use of an unusual choice of the magnetic field finite element that results in a very efficient discrete curl operator using a non standard dispersion analysis we show that the phase accuracy of the cubic edge element does not deteriorate disastrously when the mesh is deformed our numerical experiments in two dimensions confirm that the scheme performs well on a distorted mesh and that it is efficient compared to the standard yee scheme

an almost sure functional limit theorem at zero for a class of levy processes normed by the square root function and applications a recent result of bertoin doney and maller ann prob 2007 gives an integral condition to characterize the class of levy processes x t for which lim sup t downarrow 0 x t sqrt t in 0 infty occurs almost surely a s for such processes we have a kind of almost sure iterated logarithm result but without the logs in the present paper we prove a functional version of this result which then opens the way to various interesting applications obtained via a continuous mapping theorem we set these out in a rigorous framework including a characterisation of the existence of an a s cluster set for the interpolated process appropriate to the continuous time situation the applications relate to functional laws for the supremum reflected and a variety of other processes including a class of stochastic differential equations where we aim to give as informative a description as we can of the functional limit sets

some remarks on a question of strassen strassen s original functional law of the iterated logarithm for partial sums and brownian motion examined convergence and clustering in the sup norm here we address what happens if we use the much larger h norm we provide the answer to a query which appeared at the end of strassen s original paper and also present several contrasting results which are shown to be essentially best possible

metric entropy and the small ball problem for gaussian measures abstract we establish a precise link between the small ball problem for a gaussian measure on a separable banach space and the metric entropy of the unit ball of the hubert space h generating this link allows us to compute small ball probabilities from metric entropy results and vice versa

in sample forecasting with local linear survival densities in this paper in sample forecasting is defined as forecasting a structured density to sets where it is unobserved the structured density consists of one dimensional in sample components that identify the density on such sets we focus on the multiplicative density structure which has recently been seen as the underlying structure of non life insurance forecasts in non life insurance the in sample area is defined as one triangle and the forecasting area as the triangle that 20 added to the first triangle produces a square recent approaches estimate two one dimensional components by projecting an unstructured two dimensional density estimator onto the space of multiplicatively separable functions we show that time reversal reduces the problem to two one dimensional problems where the one dimensional data are left truncated and a one dimensional survival density estimator is needed this paper then uses the local linear density smoother with 25 weighted cross validated and do validated bandwidth selectors full asymptotic theory is provided with and without time reversal finite sample studies and an application to non life insurance are included

cluster sets for partial sums and partial sum processes let x x1 x2 be i i d mean zero random vectors with values in a separable banach space b sn x1 xn for n 1 and assume cn n 1 is a suitably regular sequence of constants furthermore let s n t 0 t 1 be the corresponding linearly interpolated partial sum processes we study the cluster sets a c sn cn and a c s n cn in particular a and a are shown to be non random and we derive criteria when elements in b and continuous functions f 0 1 b belong to a and a respectively when b rd we refine our clustering criteria to show both a and a are compact symmetric and star like and also obtain both upper and lower bound sets for a when the coordinates of x in rd are independent random variables we are able to represent a in terms of a and the classical strassen set k and except for degenerate cases show a is strictly larger than the lower bound set whenever d 2 in addition we show that for any compact symmetric star like subset a of rd there exists an x such that the corresponding functional cluster set a is always the lower bound subset if d 2 then additional refinements identify a as a subset of x1g1 x2g2 x1 x2 2 a g1 g2 2 k which is the functional cluster set obtained when the coordinates are assumed to be independent short title cluster sets ams 2000 subject classifications 60f15 60f17

optimizing the perfectly matched layer the berenger perfectly matched layer or pml 4 3 is used in computational electromagnetism as a sponge layer to terminate finite element approximations of scattering problems an infinite perfectly matched layer creates no reflection for incident waves of any frequency or any incident direction and the waves decay exponentially in magnitude into the layer this property is lost when the layer is truncated since the truncation boundary generates a reflected wave discretization of the differential equations further changes the reflection coefficient of the layer in this talk we shall describe the pml in the simple case of a two dimensional electromagnetic problem the first part of this abstract is expository we shall show how to derive the layer using a change of variable approach 5 9 6 after this derivation we describe some new analysis to show the effects of truncating and discretizing the pml 9 we end by showing how to compute an optimal layer and make some remarks about pml for the full maxwell system

towards a practicable bayesian nonparametric density estimator summary nonparametric density estimators smooth the empirical distribution function and are sensitive to the choice of smoothing parameters this paper develops an hierarchical bayes formulation for the smoothing problem the prior distribution for the density function is the logistic normal process which is a logistic transform of a gaussian process the covariance of the gaussian process is a smoothing kernel and has parameters that control the degree of smoothness the likelihood function for the smoothing parameters and their posterior density are computed from an approximation of the joint moments of the logistic normal process the marginal predictive density mixes the conditional predictive density given the smoothing parameters with their posterior distribution this hierarchical bayes analysis provides a fully automated data dependent method for smoothing and selects the amount of smoothing that is coherent with its prior specification a bayesian approach to nonparametric problems is to construct a probability distribution on the relevant space of distribution functions and to update this distribution as observations become available families of prior distributions must not only have stochastic properties that can adequately model prior information but also must provide tractable models the dirichlet process ferguson 1973 has been quite successful on both counts however it is not directly applicable to absolutely continuous distributions because the posterior probability of the space of absolutely continuous distributions is zero lo 1984 convolves the dirichlet process with a smoothing kernel to model a density function the computational complexity of the posterior mean increases exponentially with the sample size an alternative to the dirichlet model is provided by the logistic normal process

information rates of nonparametric gaussian process methods we consider the quality of learning a response function by a nonparametric bayesian approach using a gaussian process gp prior on the response function we upper bound the quadratic risk of the learning procedure which in turn is an upper bound on the kullback leibler information between the predictive and true data distribution the upper bound is expressed in small ball probabilities and concentration measures of the gp prior we illustrate the computation of the upper bound for the matern and squared exponential kernels for these priors the risk and hence the information criterion tends to zero for all continuous response functions however the rate at which this happens depends on the combination of true response function and gaussian prior and is expressible in a certain concentration function in particular the results show that for good performance the regularity of the gp prior should match the regularity of the unknown response function

functional limit theorems for the infinite series of ou processes in holder norm in this paper estimating large deviation probabilities for the infinite series of independent ou processes in holder norm we obtain functional limit theorems for this process in the holder norm which include functional modulus of continuity and functional lil for the infinite series of independent ou processes in the holder norm

minima of h valued gaussian processes we study low local extremes of gaussian random fields with values in a separable hilbert space and constant variance our results are sharp for certain stationary processes on the line and for these processes we also prove global limits

a galerkin least squares method for time harmonic maxwell equations using nedelec elements a galerkin least squares finite element method for the solution of the time harmonic maxwell s equations using nedelec elements is proposed this method appends a least squares term evaluated within element interiors to the standard galerkin method for the case of lowest order hexahedral element the numerical parameter multiplying this term is determined so as to optimize the dispersion properties of the resulting formulation in particular explicit expressions for this parameter are derived that lead to methods with no dispersion error for propagation along a specified direction and reduced dispersion error over all directions it is noted that this method is easy to implement and does not add to the computational costs of the standard galerkin method the performance of this method is tested on problems of practical interest

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

quadratic functionals and small ball probabilities for the m fold integrated brownian motion let the gaussian process xm t be the m fold integrated brownian motion for positive integer m the laplace transform of the quadratic functional of xm t is found by using an appropriate self adjoint integral operator the result is then used to show the power of a general connection between small ball probabilities for the gaussian process the connection is discovered by introducing an independent random shift the interplay between our results and the principal eigenvalues for nonuniform elliptic generators on an unbounded domain is also discussed

martingale based residuals for survival models summary graphical methods based on the analysis of residuals are considered for the setting of the highly used cox 1972 regression model and for the andersen gill 1982 generalization of that model we start with a class of martingale based residuals as proposed by barlow prentice 1988 these residuals and or their transforms are useful for investigating the functional form of a covariate the proportional hazards assumption the leverage of each subject upon the estimates of 13 and the lack of model fit to a given subject 1 1 model consider a set of n independent subjects such that the counting process ni ni t t o for the ith subject in the set indicates the number of observed events experienced over time t the sample paths of the ni are step functions with jumps of size 1 and with ni 0 0 we assume that the intensity function for ni t is given by yi t da t zi t yi t ep z t dao t 1 where yi t is a 0 1 process indicating whether the ith subject is a risk at time t 13 is a vector of regression coefficients zi t is a p dimensional vector of covariate processes and ao is the baseline cumulative hazard function several familar survival models fit into this framework the andersen gill 1982 generalization of the cox 1972 model arises when ao t is completely unspecified the further restriction that yi t 1 until the first event or censoring and is 0 thereafter yields the cox model the parametric form ao t t yields a poisson model or an exponential if restricted to a single event per subject and ao t tp a weibull model our attention will focus primarily on the andersen gill and cox models however the

posterior convergence rates of dirichlet mixtures at smooth densities we study the rates of convergence of the posterior distribution for bayesian density estimation with dirichlet mixtures of normal distributions as the prior the true density is assumed to be twice continuously differentiable the bandwidth is given a sequence of priors which is obtained by scaling a single prior by an appropriate order in order to handle this problem we derive a new general rate theorem by considering a countable covering of the parameter space whose prior probabilities satisfy a summability condition together with certain individual bounds on the hellinger metric entropy we apply this new general theorem on posterior convergence rates by computing bounds for hellinger bracketing entropy numbers for the involved class of densities the error in the approximation of a smooth density by normal mixtures and the concentration rate of the prior the best obtainable rate of convergence of the posterior turns out to be equivalent to the well known frequentist rate for integrated mean squared error n 2 5 up to a logarithmic factor

on the rate of convergence in strassen s law of the iterated logarithm let w t be a standard wiener process and let l be the compact class figuring in strassen s law of the iterated logarithm we investigate the rate of convergence to zero of the variable mathop inf limits f in mathfrak l text mathop text sup limits text 0 leqq x leqq 1 text w xt 2t log log t frac 1 2 f x it is shown that as t log log t belongs to the upper class of this variable if 2 3

krein s spectral theory and the paley wiener expansion for fractional brownian motion in this paper we develop the spectral theory of the fractional brownian motion fbm using the ideas of krein s work on continuous analogous of orthogonal polynomials on the unit circle we exhibit the functions which are orthogonal with respect to the spectral measure of the fbm and obtain an explicit reproducing kernel in the frequency domain we use these results to derive an extension of the classical paley wiener expansion of the ordinary brownian motion to the fractional case

some exact equivalents for the brownian motion in holder norm some exact equivalents of small probabilities are given for the wiener measure on spaces of holder paths it turns out that most of them are easier to derive than their counterparts in the uniform norm because of a classical result of z ciesielski which makes the brownian motion on these spaces easy to handle in particular we study the equivalents of the probability of eb in a fixed ball ofb in a small ball and we give applications to the speed of clustering in strassen law

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

a spatial high order hexahedral discontinuous galerkin method to solve maxwell s equations in time domain in this paper we present a non dissipative spatial high order discontinuous galerkin method to solve the maxwell equations in the time domain the non intuitive choice of the space of approximation and the basis functions induce an important gain for mass stiffness and jump matrices in terms of memory this spatial approximation combined with a leapfrog scheme in time leads also to a fast explicit and accurate method a study of the dispersive error is carried out and a stability condition for the proposed scheme is established some comparisons with other schemes are presented to validate the new scheme and to point out its advantages finally in order to improve the efficiency of the method in terms of cpu time on general unstructured meshes a strategy of local time stepping is proposed

small values of gaussian processes and functional laws of the iterated logarithm we estimate small ball probabilities for locally nondeterministic gaussian processes with stationary increments a class of processes that includes the fractional brownian motions these estimates are used to prove chung type laws of the iterated logarithm

smooth backfitting of proportional hazards with multiplicative components smooth backfitting has proven to have a number of theoretical and practical advantages in structured regression by projecting the data down onto the structured space of interest smooth backfitting

operational time and in sample density forecasting in this paper we consider a new structural model for in sample density forecasting in sample density forecasting is to estimate a structured density on a region where data are observed and then re use the estimated structured density on some region where data are not observed our structural assumption is that the density is a product of one dimensional functions with one function sitting on the scale of a transformed space of observations the transformation involves another unknown one dimensional function so that our model is formulated via a known smooth function of three underlying unknown one dimensional functions we present an innovative way of estimating the one dimensional functions and show that all the estimators of the three components achieve the optimal one dimensional rate of convergence we illustrate how one can use our approach by analyzing a real dataset and also verify the tractable finite sample performance of the method via a simulation study

a connection between strassen s and donsker varadhan s laws of the iterated logarithm a connection is given between various functional laws of the iterated logarithm for brownian motion due to donsker and varadhan csaki chung strassen and the author