On the Local Time of the Weighted Bootstrap and Compound Empirical Processes

This article is mainly concerned with the local times of the weighted bootstrap process. We prove a strong approximation theorem for the local time of the weighted bootstrap process by the local time of a Brownian bridge. We consider also the local time of the compound empirical processes that can be seen, asymptotically, as the local time of the convolution of two independent Gaussian processes.     

Multivariate orthogonal polynomials, homogeneous Padé approximants and Gaussian cubature

The connection between orthogonal polynomials, Padé approximants and Gaussian quadrature is well known and will be repeated in section 1. In the past, several generalizations to the multivariate case have been suggested for all three concepts [4,6,9,...], however without reestablishing a fundamental and clear link. In sections 2 and 3 we will elaborate definitions for multivariate Padé and Padé-type approximation, multivariate polynomial orthogonality and multivariate Gaussian integration in order to bridge the gap between these concepts. We will show that the new m-point Gaussian cubature rules allow the exact integration of homogeneous polynomials of degree 2m−1, in any number of variables. A numerical application of the new integration rules can be found in sections 4 and 5. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exact convergence rate of the Wong–Zakai approximation to RDEs driven by Gaussian rough paths

We consider a solution to a stochastic differential equation driven by a Gaussian process in the rough differential equation sense and the Wong–Zakai approximation to the solution. We give an upper bound of the error of the Wong–Zakai approximation. We also show that the upper bound is optimal in a particular case.     

带有误差变量混合分布模型的估计

在许多实际同题中，存在一些不可直接观测的变量，对此统计学家们提出了反卷积和混合分布模型来解决这一变量的分布的估计问题。本文对这一问题采用bootstrap模拟方法得出分布函数的估计，并进一步建立该分布函数的非参bootstrap百分位区间，在数值试验中将我们的处理方式与传统的EM算法得到的分布估计和正态逼近区间作比较，数值结果表明用bootstrap模拟方法得到的准确度更好，数值效果更理想。     

Modeling and analysis of power-tail distributions via classical teletraffic methods

We propose a new methodology for modeling and analyzing power-tail distributions, such as the Pareto distribution, in communication networks. The basis of our approach is a fitting algorithm which approximates a power-tail distribution by a hyperexponential distribution. This algorithm possesses several key properties. First, the approximation can be achieved within any desired degree of accuracy. Second, the fitted hyperexponential distribution depends only on a few parameters. Third, only a small number of exponentials are required in order to obtain an accurate approximation over many time scales. Once equipped with a fitted hyperexponential distribution, we have an integrated framework for analyzing queueing systems with power-tail distributions. We consider the GI/G/1 queue with Pareto distributed service time and show how our approach allows to derive both quantitative numerical results and asymptotic closed-form results. This derivation shows that classical teletraffic methods can be employed for the analysis of power-tail distributions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Adaptive radial basis function interpolation using an error indicator

In some approximation problems, sampling from the target function can be both expensive and time-consuming. It would be convenient to have a method for indicating where approximation quality is poor, so that generation of new data provides the user with greater accuracy where needed. In this paper, we propose a new adaptive algorithm for radial basis function (RBF) interpolation which aims to assess the local approximation quality, and add or remove points as required to improve the error in the specified region. For Gaussian and multiquadric approximation, we have the flexibility of a shape parameter which we can use to keep the condition number of interpolation matrix at a moderate size. Numerical results for test functions which appear in the literature are given for dimensions 1 and 2, to show that our method performs well. We also give a three-dimensional example from the finance world, since we would like to advertise RBF techniques as useful tools for approximation in the high-dimensional settings one often meets in finance.     

Bootstrap approximation of tail dependence function

For estimating a rare event via the multivariate extreme value theory, the so-called tail dependence function has to be investigated (see [L. de Haan, J. de Ronde, Sea and wind: Multivariate extremes at work, Extremes 1 (1998) 7-45]). A simple, but effective estimator for the tail dependence function is the tail empirical distribution function, see [X. Huang, Statistics of Bivariate Extreme Values, Ph.D. Thesis, Tinbergen Institute Research Series, 1992] or [R. Schmidt, U. Stadtmüller, Nonparametric estimation of tail dependence, Scand. J. Stat. 33 (2006) 307-335]. In this paper, we first derive a bootstrap approximation for a tail dependence function with an approximation rate via the construction approach developed by [K. Chen, S.H. Lo, On a mapping approach to investigating the bootstrap accuracy, Probab. Theory Relat. Fields 107 (1997) 197-217], and then apply it to construct a confidence band for the tail dependence function. A simulation study is conducted to assess the accuracy of the bootstrap approach.     

Bootstrapping continuous-time autoregressive processes

We develop a bootstrap procedure for Lévy-driven continuous-time autoregressive (CAR) processes observed at discrete regularly-spaced times. It is well known that a regularly sampled stationary Ornstein–Uhlenbeck process [i.e. a CAR(1) process] has a discrete-time autoregressive representation with i.i.d. noise. Based on this representation a simple bootstrap procedure can be found. Since regularly sampled CAR processes of higher order satisfy ARMA equations with uncorrelated (but in general dependent) noise, a more general bootstrap procedure is needed for such processes. We consider statistics depending on observations of the CAR process at the uniformly-spaced times, together with auxiliary observations on a finer grid, which give approximations to the derivatives of the continuous time process. This enables us to approximate the state-vector of the CAR process which is a vector-valued CAR(1) process, and whose sampled version, on the uniformly-spaced grid, is a multivariate AR(1) process with i.i.d. noise. This leads to a valid residual-based bootstrap which allows replication of CAR $(p)$ processes on the underlying discrete time grid. We show that this approach is consistent for empirical autocovariances and autocorrelations.     

Generalization bounds for function approximation from scattered noisy data

We consider the problem of approximating functions from scattered data using linear superpositions of non-linearly parameterized functions. We show how the total error (generalization error) can be decomposed into two parts: an approximation part that is due to the finite number of parameters of the approximation scheme used; and an estimation part that is due to the finite number of data available. We bound each of these two parts under certain assumptions and prove a general bound for a class of approximation schemes that include radial basis functions and multilayer perceptrons. This revised version was published online in June 2006 with corrections to the Cover Date.     

Adaptive computation of smallest eigenvalues of self‐adjoint elliptic partial differential equations

We consider a new adaptive finite element (AFEM) algorithm for self‐adjoint elliptic PDE eigenvalue problems. In contrast to other approaches we incorporate the inexact solutions of the resulting finite‐dimensional algebraic eigenvalue problems into the adaptation process. In this way we can balance the costs of the adaptive refinement of the mesh with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors, and use these for the adaptation process. We show that it is also possible to restrict to very few iterations of a Krylov subspace solver for the eigenvalue problem on coarse meshes. Several examples are presented to show that this new approach achieves much better complexity than the previous AFEM approaches which assume that the algebraic eigenvalue problem is solved to full accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimating the error distribution in nonparametric multiple regression with applications to model testing

In this paper we consider the estimation of the error distribution in a heteroscedastic nonparametric regression model with multivariate covariates. As estimator we consider the empirical distribution function of residuals, which are obtained from multivariate local polynomial fits of the regression and variance functions, respectively. Weak convergence of the empirical residual process to a Gaussian process is proved. We also consider various applications for testing model assumptions in nonparametric multiple regression. The model tests obtained are able to detect local alternatives that converge to zero at an n^−1/2-rate, independent of the covariate dimension. We consider in detail a test for additivity of the regression function.     

Error rates in classification of multivariate Gaussian random field observation

We consider the problem of supervised classifying the multivariate Gaussian random field (GRF) single observation into one of two populations in case of given training sample. The populations are specified by different regression mean models and by common factorized covariance function. For completely specified populations, we derive a formula for Bayes error rate. In the case of unknown regression parameters and feature covariance matrix, the plug-in Bayes discriminant function based on ML estimators of parameters is used for classification. We derive the actual error rate and the asymptotic expansion of the expected error rate associated with plug-in Bayes discriminant function. These results are multivariate generalizations of previous ones. Numerical analysis of the derived formulas is implemented for the bivariate GRF observations at locations belonging to the two-dimensional lattice with unit spacing.     

Sensitivities for topological graph changes in finite element meshes and application to adaptive refinement

We propose a novel approach to adaptivity in FEM based on local sensitivities for topological mesh changes. To this end, we consider refinement as a continuous operation on the edge graph of the finite element discretization, for instance by splitting nodes along edges and expanding edges to elements. Thereby, we introduce the concept of a topological mesh derivative for a given objective function that depends on the discrete solution of the underlying PDE. These sensitivities may in turn be used as refinement indicators within an adaptive algorithm. For their calculation, we rely on the first-order asymptotic expansion of the Galerkin solution with respect to the topological mesh change. As a proof of concept, we consider the total potential energy of a linear symmetric second-order elliptic PDE, minimization of which is known to decrease the approximation error in the energy norm. In this case, our approach yields local sensitivities that are closely related to the reduction of the energy error upon refinement and may therefore be used as refinement indicators in an adaptive algorithm. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on bootstrap approximations for the empirical copula process

It is well known that the empirical copula process converges weakly to a centered Gaussian field. Because the covariance structure of the limiting process depends on the partial derivatives of the unknown copula several bootstrap approximations for the empirical copula process have been proposed in the literature. We present a brief review of these procedures. Because some of these procedures also require the estimation of the derivatives of the unknown copula we propose an alternative approach which circumvents this problem. Finally a simulation study is presented in order to compare the different bootstrap approximations for the empirical copula process.     

Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

A univariate Hawkes process is a simple point process that is self-exciting and has a clustering effect. The intensity of this point process is given by the sum of a baseline intensity and another term that depends on the entire past history of the point process. Hawkes processes have wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we prove a functional central limit theorem for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large. The limit is a non-Markovian Gaussian process with dependent increments. We use the resulting approximation to study an infinite-server queue with high-volume Hawkes traffic. We show that the queue length process can be approximated by a Gaussian process, for which we compute explicitly the covariance function and the steady-state distribution. We also extend our results to multivariate stationary Hawkes processes and establish limit theorems for infinite-server queues with multivariate Hawkes traffic.     

Characterization of the Existence of an Enriched Linear Finite Element Approximation Using Biorthogonal Systems

The present paper is intended to give a characterization of the existence of an enriched linear finite element approximation based on biorthogonal systems. It is shown that the enriched element exists if and only if a certain multivariate generalized trapezoidal type cubature formula has a nonzero approximation error. Furthermore, for such an enriched element we derive simple explicit formulas for the basis functions, and show that the approximation error can be written as the sum of the error of the (non-enriched) element plus a perturbation that depends on the enrichment function. Finally, we estimate the approximation error in L² norm. We also give an alternative approach to estimate the approximation error, which relies on an appropriate use of the Poincaré inequality.     

Second-order accuracy of depth-based bootstrap confidence regions

We consider the problem of setting bootstrap confidence regions for multivariate parameters based on data depth functions. We prove, under mild regularity conditions, that depth-based bootstrap confidence regions are second-order accurate in the sense that their coverage error is of order n⁻¹, given a random sample of size n. The results hold in general for depth functions of types A and D, which cover as special cases the Tukey depth, the majority depth, and the simplicial depth. A simulation study is also provided to investigate empirically the bootstrap confidence regions constructed using these three depth functions.     

Inference for robust canonical variate analysis

We consider the problem of optimally separating two multivariate populations. Robust linear discriminant rules can be obtained by replacing the empirical means and covariance in the classical discriminant rules by S or MM-estimates of location and scatter. We propose to use a fast and robust bootstrap method to obtain inference for such a robust discriminant analysis. This is useful since classical bootstrap methods may be unstable as well as extremely time-consuming when robust estimates such as S or MM-estimates are involved. In particular, fast and robust bootstrap can be used to investigate which variables contribute significantly to the canonical variate, and thus the discrimination of the classes. Through bootstrap, we can also examine the stability of the canonical variate. We illustrate the method on some real data examples.     

Kernel-based goodness-of-fit tests for copulas with fixed smoothing parameters

We study a test statistic based on the integrated squared difference between a kernel estimator of the copula density and a kernel smoothed estimator of the parametric copula density. We show for fixed smoothing parameters that the test is consistent and that the asymptotic properties are driven by a U-statistic of order 4 with degeneracy of order 1. For practical implementation we suggest to compute the critical values through a semiparametric bootstrap. Monte Carlo results show that the bootstrap procedure performs well in small samples. In particular, size and power are less sensitive to smoothing parameter choice than they are under the asymptotic approximation obtained for a vanishing bandwidth.     

The Direct Method of Lines for Forward and Inverse Linear Elasticity Problems of Composite Materials in Star-Shaped Domains

In this paper, we generalize the direct method of lines for linear elasticity problems of composite materials in star-shaped domains and consider its application to inverse elasticity problems. We assume that the boundary of the star-shaped domain can be described by an explicit $C^1$ parametric curve in the polar coordinate. We introduce the curvilinear coordinate, in which the irregular star-shaped domain is converted to a regular semi-infinite strip. The equations of linear elasticity are discretized with respect to the angular variable and we solve the resulting semi-discrete approximation analytically using a direct method. The eigenvalues of the semi-discrete approximation converge quickly to the true eigenvalues of the elliptic operator, which helps capture the singularities naturally. Moreover, an optimal error estimate of our method is given. For the inverse elasticity problems, we determine the Lamé coefficients from measurement data by minimizing a regularized energy functional. We apply the direct method of lines as the forward solver in order to cope with the irregularity of the domain and possible singularities in the forward solutions. Several numerical examples are presented to show the effectiveness and accuracy of our method for both forward and inverse elasticity problems of composite materials.