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13
Nonparametric Methods for Inference in the Presence of Instrumental Variables
 Annals of Statistics
, 2005
"... We suggest two nonparametric approaches, based on kernel methods and orthogonal series to estimating regression functions in the presence of instrumental variables. For the first time in this class of problems, we derive optimal convergence rates, and show that they are attained by particular estima ..."
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Cited by 39 (10 self)
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We suggest two nonparametric approaches, based on kernel methods and orthogonal series to estimating regression functions in the presence of instrumental variables. For the first time in this class of problems, we derive optimal convergence rates, and show that they are attained by particular estimators. In the presence of instrumental variables the relation that identifies the regression function also defines an illposed inverse problem, the “difficulty ” of which depends on eigenvalues of a certain integral operator which is determined by the joint density of endogenous and instrumental variables. We delineate the role played by problem difficulty in determining both the optimal convergence rate and the appropriate choice of smoothing parameter. 1. Introduction. Data (Xi,Yi
Unification of the probe and singular sources methods for the inverse boundary value problem by the noresponse test
 Comm. Partial Differential Equations
"... In this article, we use the noresponse test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient � of the equation � · ��x� � + c�x� with piecewise regular � and bounded function c�x ..."
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Cited by 8 (5 self)
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In this article, we use the noresponse test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient � of the equation � · ��x� � + c�x� with piecewise regular � and bounded function c�x�. We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the noresponse test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the noresponse test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of ��x�, x ∈ �D at the boundary. A second consequence of this formula is that the blowup rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.
Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian
, 2007
"... We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz ..."
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Cited by 6 (0 self)
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We propose a new method to analyze and represent data recorded on a domain of general shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the KarhunenLoève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.
Computing acoustic waves in an inhomogeneous medium of the plane by a coupling of spectral and finite elements
 SIAM J. Numer. Anal
, 2003
"... Abstract. In this paper we analyze a Galerkin procedure, based on a combination of finite and spectral elements, for approximating a timeharmonic acoustic wave scattered by a bounded inhomogeneity. The finite element method used to approximate the near field in the region of inhomogeneity is couple ..."
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Cited by 4 (1 self)
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Abstract. In this paper we analyze a Galerkin procedure, based on a combination of finite and spectral elements, for approximating a timeharmonic acoustic wave scattered by a bounded inhomogeneity. The finite element method used to approximate the near field in the region of inhomogeneity is coupled with a nonlocal boundary condition, which consists in a linear integral equation. This integral equation is discretized by a spectral Galerkin approximation method. We provide error estimates for the Galerkin method, propose fully discrete schemes based on elementary quadrature formulas, and show that the perturbation due to this numerical integration gives rise to a quasioptimal rate of convergence. We also suggest a method for implementing the algorithm using the preconditioned GMRES method and provide some numerical results.
DISCRETE APPROXIMATION OF NONCOMPACT OPERATORS DESCRIBING CONTINUUMOFALLELES MODELS
, 2004
"... Abstract We consider the eigenvalue equation for the largest eigenvalue of certain kinds of noncompact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite ..."
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Cited by 3 (0 self)
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Abstract We consider the eigenvalue equation for the largest eigenvalue of certain kinds of noncompact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite rank, which constitutes a discretization procedure. For this purpose, two standard methods of approximation theory, the Nyström and the Galerkin method, are generalized. The operators considered describe models for mutation and selection of an infinitely large population of individuals that are labeled by real numbers, commonly called continuumofalleles (COA) models.
Efficient Estimation in Marginal Partially Linear Models for Longitudinal/Clustered Data Using Splines,” Scandinavian
 Journal of Statistics
, 2007
"... ABSTRACT. We consider marginal semiparametric partially linear models for longitudinal/clustered data and propose an estimation procedure based on a spline approximation of the nonparametric part of the model and an extension of the parametric marginal generalized estimating equations (GEE). Our e ..."
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Cited by 3 (2 self)
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ABSTRACT. We consider marginal semiparametric partially linear models for longitudinal/clustered data and propose an estimation procedure based on a spline approximation of the nonparametric part of the model and an extension of the parametric marginal generalized estimating equations (GEE). Our estimates of both parametric part and nonparametric part of the model have properties parallel to those of parametric GEE, that is, the estimates are efficient if the covariance structure is correctly specified and they are still consistent and asymptotically normal even if the covariance structure is misspecified. By showing that our estimate achieves the semiparametric information bound, we actually establish the efficiency of estimating the parametric part of the model in a stronger sense than what is typically considered for GEE. The semiparametric efficiency of our estimate is obtained by assuming only conditional moment restrictions instead of the strict multivariate Gaussian error assumption. Key words: clustered data, generalized estimating equations (GEE), longitudinal data, marginal model, nonparametric regression, partially linear models, polynomial splines, semiparametric efficiency.
A FreeSpace Adaptive FMMBased PDE Solver in Three Dimensions
, 2008
"... We present a kernelindependent, adaptive fast multipole method (FMM) of arbitrary order accuracy for solving elliptic PDEs in three dimensions with radiation boundary conditions. The algorithm requires only a Green’s function evaluation routine for the governing equation and a representation of the ..."
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Cited by 2 (1 self)
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We present a kernelindependent, adaptive fast multipole method (FMM) of arbitrary order accuracy for solving elliptic PDEs in three dimensions with radiation boundary conditions. The algorithm requires only a Green’s function evaluation routine for the governing equation and a representation of the source distribution (the righthand side) that can be evaluated at arbitrary points. The performance of the FMM is accelerated in two ways. First, we construct a piecewise polynomial approximation of the righthand side and compute farfield expansions in the FMM from the coefficients of this approximation. Second, we precompute tables of quadratures to handle the nearfield interactions on adaptive octree data structures, keeping the total storage requirements in check through the exploitation of symmetries. We present numerical examples for the Laplace, modified Helmholtz and Stokes equations. 1
The No Response Test for the Reconstruction of Polyhedral Objects in Electromagnetics
"... We develope a No Response Test for the reconstruction of some polyhedral obstacle from one or few timeharmonic electromagnetic incident waves in electromagnetics. The basic idea of the test is to probe some region in space with waves which are small on some test domain and, thus, do not generate a ..."
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We develope a No Response Test for the reconstruction of some polyhedral obstacle from one or few timeharmonic electromagnetic incident waves in electromagnetics. The basic idea of the test is to probe some region in space with waves which are small on some test domain and, thus, do not generate a response when the scatterer is inside of this test domain. This is the first formulation of the No Response Test for electromagnetics. We will prove convergence of the method for testing a nonvibrating domain B whether the far field pattern of some scattered timeharmonic field is analytically extendable into the interior of B. We will describe algorithmical realizations of the No Response Test. Finally, we will show the feasibility of the method by reconstruction of polygonal objects in three dimensions.
cemmap working paper CWP15/03 SemiNonparametric IV Estimation of ShapeInvariant
"... This paper concerns the identification and estimation of a shapeinvariant Engel curve system with endogenous total expenditure. The shapeinvariant specification involves a common shift parameter for each demographic group in a pooled system of Engel curves. Our focus is on the identification and ..."
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This paper concerns the identification and estimation of a shapeinvariant Engel curve system with endogenous total expenditure. The shapeinvariant specification involves a common shift parameter for each demographic group in a pooled system of Engel curves. Our focus is on the identification and estimation of both the nonparametric shape of the Engel curve and the parametric specification of the demographic scaling parameters. We present a new identification condition, closely related to the concept of bounded completeness in statistics. The estimation procedure applies the sieve minimum distance estimation of conditional moment restrictions allowing for endogeneity. We establish a new root mean squared convergence rate for the nonparametric IV regression when the endogenous regressor has unbounded support. Rootn asymptotic normality and semiparametric efficiency of the parametric components are also given under a set of ‘lowlevel ’ sufficient conditions. Monte Carlo simulations shed lights on the choice of smoothing parameters and demonstrate that the sieve IV estimator performs well. An application is made to the estimation of Engel curves using the UK Family Expenditure Survey and shows the importance of adjusting for endogeneity in terms of both the curvature and demographic parameters of systems of Engel curves.
Printed in Great Britain Temporal process regression
"... We consider regression for response and covariates which are temporal processes observed over intervals. A functional generalised linear model is proposed which includes extensions of standard models in multistate survival analysis. Simple nonparametric estimators of timeindexed parameters are dev ..."
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We consider regression for response and covariates which are temporal processes observed over intervals. A functional generalised linear model is proposed which includes extensions of standard models in multistate survival analysis. Simple nonparametric estimators of timeindexed parameters are developed using ‘working independence ’ estimating equations and are shown to be uniformly consistent and to converge weakly to Gaussian processes. The procedure does not require smoothing or a Markov assumption, unlike approaches based on transition intensities. The usual definition of optimal estimating equations for parametric models is then generalised to the functional model and the optimum is identified in a class of functional generalised estimating equations. Simulations demonstrate large efficiency gains relative to working independence at times where censoring is heavy. The estimators are the basis for new tests of the covariate effects and for the estimation of models in which greater structure is imposed on the parameters, providing novel goodnessoffit tests. The methodology’s practical utility is illustrated in a data analysis.