nonparametric least squares estimation of a multivariate convex regression function this paper deals with the consistency of the least squares estimator of a convex regression function when the predictor is multidimensional we characterize and discuss the computation of such an estimator via the solution of certain quadratic and linear programs mild sufficient conditions for the consistency of this estimator and its subdifferentials in fixed and stochastic design regression settings are provided we also consider a regression function which is known to be convex and componentwise nonincreasing and discuss the characterization computation and consistency of its least squares estimator

revealed preference tests for weak separability an integer programming approach we present the revealed preference conditions that characterize the data sets that are consistent with the maximization of a weakly separable utility function we show that verifying these revealed preference conditions is np hard we also present an integer programming approach which is particularly attractive in view of empirical analysis we demonstrate the versatility of this integer programming approach by showing that it allows for testing homothetic separability and weak separability of the indirect utility function we illustrate the practical usefulness of the approach by an empirical application to spanish household consumption data

non parametric hypothesis testing procedures and applications to demand analysis abstract this paper proposes a test of the hypothesis that a possibly vector valued regression function g lies in a particular compact family of functions f where f is not necessarily a finite dimensional parametric family under the assumption of a known covariance matrix of the errors the test is shown to be consistent for a broad range of alternatives and an upper bound on the significance level is established it is shown how the test may be used to test the utility maximization hypothesis and homotheticity and the paper indicates how to apply the test to a broad range of other non parametric hypotheses

identification and nonparametric estimation of a transformed additively separable model let r x z be a function that along with its derivatives can be consistently estimated nonparametrically this paper discusses the identification and consistent estimation of the unknown functions h m g and f where r x z h m x z m x z g x f z and h is strictly monotonic an estimation algorithm is proposed for each of the model s unknown components when r x z represents a conditional mean function the resulting estimators use marginal integration to separate the components g and f our estimators are shown to have a limiting normal distribution with a faster rate of convergence than unrestricted nonparametric alternatives their small sample performance is studied in a monte carlo experiment we apply our results to estimate generalized homothetic production functions for four industries in the chinese economy

singularly perturbed finite element methods this paper introduces a new piecewise linear finite element which is designed to handle singularly perturbed ordinary differential equations both pointwise and global estimates which are independent of the perturbation parameter are obtained

uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems convection diffusion type in this paper we consider the standard bilinear finite element method fem and the corresponding streamline diffusion fem for the singularly perturbed elliptic boundary value problem e 2u x2 2u y2 b x y u a x y u f x y in the two space dimensions by using the asymptotic expansion method of vishik and lyusternik 36 and the technique we used in 21 22 we prove that the standard bilinear fem on a shishkin type mesh achieves first order uniform convergence rate globally in l2 norm for both the ordinary exponential boundary layer case and the parabolic boundary layer case extensive numerical results are carried out for both cases the results show that our methods perform much better than either the classical standard or streamline diffusion fem

deformations of unbounded convex bodies and hypersurfaces we study the topology of the space k of complete convex hypersurfaces of r which are homeomorphic to rn 1 in particular using minkowski sums we construct a deformation retraction of k onto the grassmannian space of hyperplanes so every hypersurface in k may be flattened in a canonical way further the total curvature of each hypersurface evolves continuously and monotonically under this deformation we also show that modulo proper rotations the subspaces of k consisting of smooth strictly convex or positively curved hypersurfaces are each contractible which settles a question of h rosenberg

generalized additive and index models with shape constraints summary we study generalized additive models with shape restrictions e g monotonicity convexity and concavity imposed on each component of the additive prediction function we show that this framework facilitates a non parametric estimator of each additive component obtained by maximizing the likelihood the procedure is free of tuning parameters and under mild conditions is proved to be uniformly consistent on compact intervals more generally our methodology can be applied to generalized additive index models here again the procedure can be justified on theoretical grounds and like the original algorithm has highly competitive finite sample performance practical utility is illustrated through the use of these methods in the analysis of two real data sets our algorithms are publicly available in the r package scar short for shape constrained additive regression

a superlative indicator for the luenberger hicks moorsteen productivity indicator theory and application abstract consisting of the difference between an output indicator and an input indicator the luenberger hicks moorsteen lhm productivity indicator allows straightforward interpretation however its computation requires estimating distance functions that are inherently unknown this paper shows that a computationally simple bennet indicator is a superlative indicator for the lhm indicator when one can assume profit maximizing behavior and the input and output directional distance functions can be represented up to the second order by a quadratic functional form we also show that the luenberger and lhm approximating bennet indicators coincide for an appropriate choice of directional vectors focusing on a large sample of italian food and beverages companies for the years 1995 2007 we empirically investigate the extent to which this theoretical equivalence translates into similar estimates we find that the bennet indicator is a close empirical alternative to the lhm indicator for the sample

accurate discretization for singular perturbations the one dimensional case this paper develops a discretization method for one dimensional singular perturbation problems based on a petrov galerkin finite element or an equivalent finite volume scheme the method is unique in that its discretization error has a bound that is second order in the mesh size and uniform in the perturbation parameter it satisfies a local discrete conservation law and it exhibits a discrete maximum principle numerical results are included for comparison with the theory

the influence curve approach in data envelopment analysis the behavior of efficiency in the stochastic dea model is examined here in terms of the influence curve approach which quantifies the influence of observed data on the empirical fit of the production frontier data influence is analyzed here through robustness two types of robustness and sensitivity issues are analyzed here in terms of a a new class of minimax measures and b a set nonlinear efficiency measures and it is shown through empirical applications that in suitable cases these measures outperform the conventional ones

a uniformly accurate finite element method for a singularly perturbed one dimensional reaction diffusion application d une methode a elements finis avec des elements de base exponentiels a un probleme autoadjoint a perturbation singuliere et a deux points limites generation d un schema aux differences tridiagonal uniformement precis du second ordre

a computational framework for multivariate convex regression and its variants we study the nonparametric least squares estimator lse of a multivariate convex regression function the lse given as the solution to a quadratic program with o n2 linear constraints n being t

multivariate convex regression with adaptive partitioning we propose a new nonparametric method for multivariate regression subject to convexity or concavity constraints on the response function convexity constraints are common in economics statistics operations research financial engineering and optimization but there is currently no multivariate method that is stable and computationally feasible for more than a few thousand observations we introduce convex adaptive partitioning cap which creates a globally convex regression model from locally linear estimates fit on adaptively selected covariate partitions cap is a computationally efficient consistent method for convex regression we demonstrate empirical performance by comparing the performance of cap to other shape constrained and unconstrained regression methods for predicting weekly wages and value function approximation for pricing american basket options

uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids the lowest order raviart thomas rectangular element is considered for solving the singular perturbation problem mbox div a nabla p bp f where the diagonal tensor a varepsilon 2 1 or a varepsilon 2 varepsilon 2 global uniform convergence rates of o n 1 for both p and a1 2 nabla p in the l2 norm are obtained in both cases where n is the number of intervals in either direction the pointwise interior away from the boundary layers convergence rates of o n 1 for p are also proved superconvergence i e o n 2 at special points and o n 2 global l2 estimate for both p and a 1 2 nabla p are obtained by a local postprocessing numerical results support our theoretical analysis moreover numerical experiments show that an anisotropic mesh gives more accurate results than the standard global uniform mesh

method of reduction of order for solving singularly perturbed two point boundary value problems in this paper a method of reduction of order is proposed for solving singularly perturbed two point boundary value problems with a boundary layer at one end point it is distinguished by the following fact the original singularly perturbed boundary value problem is replaced by a pair of initial value problems classical fourth order runge kutta method is used to solve these initial value problems several linear and non linear singular perturbation problems have been solved and the numerical results are presented to support the theory

the convex hull property and topology of hypersurfaces with nonnegative curvature abstract we prove that in euclidean space any nonnegatively curved compact smoothly immersed hypersurface lies outside the convex hull of its boundary provided the boundary satisfies certain required conditions this gives a convex hull property dual to the classical one for surfaces with nonpositive curvature a version of this result in the nonsmooth category is obtained as well we show that our boundary conditions determine the topology of the surface up to at most two choices the proof is based on uniform estimates for radii of convexity of these surfaces under a clipping procedure a noncollapsing convergence theorem and a gluing procedure

total positive curvature of hypersurfaces with convex boundary we prove that if is a compact hypersurface in euclidean space rn its boundary lies on the boundary of a convex body c and meets c orthogonally from the outside then the total positive curvature of is bigger than or equal to half the area of the sphere sn 1 also we obtain necessary and sufficient conditions for the equality to hold

a finite element method for a singularly perturbed boundary value problem we examine the problem eu a x u b x u f x for 0 0 b x 2 4e 0 a b andf inc 2 0 1 e in 0 1 u 0 andu 1 given using finite elements and a discretized green s function we show that the el mistikawy and werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem i e the nodal errors are bounded bych 2 wherec is independent ofh and e with a natural choice of trial functions uniform first order accuracy is obtained in thel 0 1 norm on choosing piecewise linear trial functions hat functions uniform first order accuracy is obtained in thel 1 0 1 norm

estimating nonparametric convex functions using gradient data abstract the estimation and use of nonparametric convex functions has been discussed in the recent literature where it is assumed that the data used in the estimation consists of a sample of observations of the convex function however an important additional or alternative source of data consists of observations on the sub gradient of the function in the present paper we discuss how to use such sub gradient data to estimate nonparametric functions this a allows us to estimate nonparametric convex functions when other point observation data is unavailable or unobservable or unreliable or costly to obtain b allows us an alternative even when other point observation data is available and c allows us to improve our estimates by using both types of data together the resultant function estimators may not be unique hence we compute bounds on the convex set of estimators we also consider introducing other data in addition to the gradient data the discussion is illustrated by reference to estimating production functions benefit functions and demand functions

nonparametric option pricing under shape restrictions frequently economic theory places shape restrictions on functional relationships between economic variables this paper develops a method to constrain the values of the first and second derivatives of nonparametric locally polynomial estimators we apply this technique to estimate the state price density spd or risk neutral density implicit in the market prices of options the option pricing function must be monotonic and convex simulations demonstrate that nonparametric estimates can be quite feasible in the small samples relevant for day to day option pricing once appropriate theory motivated shape restrictions are imposed using s p 500 option prices we show that unconstrained nonparametric estimators violate the constraints during more than half the trading days in 1999 unlike the constrained estimator we propose

a global uniformly convergent finite element method for a quasi linear singularly perturbed elliptic problem abstract in this paper we construct a bilinear finite element method based on a special piecewise uniform mesh for solving a quasi linear singularly perturbed elliptic problem in two space dimensions a quasi optimal global uniform convergence rate o n 2x ln2 nx n 2y ln2 ny was obtained which is independent of the perturbation parameter here nx and ny are the number of elements in the x and y directions respectively

nonparametric tests of regularity farrell efficiency and goodness of fit abstract the goal of this paper is to reveal the links among several seemingly unrelated literatures 1 1 the nonparametric regularity test literature see afriat 1972 diewert and parkan 1983 hanoch and rothschild 1972 varian 1984 2 2 the efficiency measurement literature farrell 1957 is the classic reference but also included is what has been dubbed data envelopment analysis and 3 3 the goodness of fit approach associated with varian 1990 the unifying threads are the shephard type distance function and the mahler inequality mahler 1939 our synthesis yields regularity tests of cost minimization and technical change which are nonparametric yet provide information as to goodness of fit based on farrell type efficiency measures

uniform high order difference schemes for a singularly perturbed two point boundary value problem on construit une famille de schemas aux differences finies uniformement precis pour le probleme du type eu a x u b x u f x on donne une analyse complete de l erreur de discretisation basee sur les resultats de stabilite on presente des essais numeriques

multigrid convergence for discretizations of singular perturbation problems with grid aligned flow we use fourier analysis to examine multigrid convergence for a variety of discretizations of the two dimensional convection diffusion equation with grid aligned flow velocity emphasis is placed on the problem with a small diffusion coefficient we consider three distinct discretizations of the problem centered differencing standard upwind differencing and a generalization to two dimensions of the el mistikawy and werle discretization emw the analysis covers the effectiveness of block jacobi relaxation for the problem and stability as the diffusion coefficient epsilon to 0 of the discretizations in stark contrast to the problem with general non grid aligned flow we obtain v cycle rates independent of meshwidth h for central differencing as well as for the other discretizations a striking similarity is found between tandard upwinding and the second order accurate generalization of emw the latter is also found to be superior to central differencing in terms of stability superior multigr

linear production functions and the data envelopment analysis abstract this paper shows that the data envelopment analysis or farrell method is not nonparametric in the sense that the method implicitly employs linear production functions frontiers to measure efficiency deriving positive multiplier linear production frontiers and grouping the dmus by their corresponding frontiers can simplify the estimation procedures when there are new dmus included and can easily determine how many inputs should be reduced or and how many outputs should be increased to make an inefficient dmu efficiently this procedure can also circumvent the problems of the non archimedean infinitesmal in the ccr literature it is also shown that because of the linear production functions both banker s and the banker charnes cooper methods fail in testing for returns to scale

an analysis of a uniformly convergent finite difference finite element scheme for a model singular perturbation problem on etablit la convergence uniforme de la discretisation d un probleme a perturbation singuliere on utilise quelques resultats de stabilite et un theoreme de representation des solutions

dea with efficiency classification preserving conditional convexity abstract we propose to relax the standard convexity property used in data envelopment analysis dea by imposing additional qualifications for feasibility of convex combinations we specifically focus on a condition that preserves the koopmans efficiency classification this yields an efficiency classification preserving conditional convexity property which is implied by both monotonicity and convexity but not conversely substituting convexity by conditional convexity we construct various empirical dea approximations as the minimal sets that contain all dmus and are consistent with the imposed production assumptions imposing an additional disjunctive constraint to standard convex dea formulations can enforce conditional convexity computation of efficiency measures relative to conditionally convex production set can be performed through disjunctive programming dp

consistency of multidimensional convex regression convex regression is concerned with computing the best fit of a convex function to a data set of n observations in which the independent variable is possibly multidimensional such regression problems arise in operations research economics and other disciplines in which imposing a convexity constraint on the regression function is natural this paper studies a least squares estimator that is computable as the solution of a quadratic program and establishes that it converges almost surely to the true function as n under modest technical assumptions in addition to this multidimensional consistency result we identify the behavior of the estimator when the model is misspecified so that the true function is nonconvex and we extend the consistency result to settings in which the function must be both convex and nondecreasing as is needed for consumer preference utility functions

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

an analysis of a superconvergence result for a singularly perturbed boundary value problem we give a new proof that the el mistikawy and werle finite difference scheme is uniformly second order accurate for a nonselfadjoint singularly perturbed boundary value problem to do this we use exponential finite elements and a discretized green s function the proof is direct gives the nodal errors explicitly in integral form and involves much less computation than in previous proofs of the result

isoperimetric inequalities in euclidean convex bodies in this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body i e a compact convex set in euclidean space with interior points we shall not impose any regularity assumption on the boundary of the convex set amongst other results we shall prove the equivalence between hausdorff and lipschitz convergence the continuity of the isoperimetric profile with respect to the hausdorff distance and the convergence in hausdorff distance of sequences of isoperimetric regions and their free boundaries we shall also describe the behavior of the isoperimetric profile for small volume and the behavior of isoperimetric regions for small volume

adaptivity and error estimation for the finite element method applied to convection diffusion problems a detailed analysis is performed for a finite element method applied to the general one dimensional convection diffusion problem piecewise polynomials are used for the trial space the test space is formed by locally projecting l spline basis functions onto upwinded polynomials the error is measured in the l p mesh dependent norm the method is shown to be quasi optimal provided that the input data is piecewise smooth a reasonable assumption in practice a posteriors error estimates are derived having the property that the effectivity index theta error estimate true error converges to one as the maximum mesh size goes to zero these error estimates are composed of locally computable error indicators providing for an adaptive mesh refinement strategy numerical results show that theta is nearly one even on coarse meshes and optimal rates of convergence are attained by the adaptive procedure the robustness of the algorithm is tested on a nonlinear turning point problem modeling flow th

a uniform finite element method for a conservative singularly perturbed problem abstract we examine the problem p x u q x u r x u f x for 0 x p 0 q 0 r 0 p q r and f in c 2 0 1 in 0 1 u 0 and u 1 given existence of a unique solution u and bounds on u and its derivatives are obtained using finite elements on an equidistant mesh of width h we generate a tridiagonal difference scheme which is shown to be uniformly second order accurate for this problem i e the nodal errors are bounded by ch 2 where c is independent of h and with a natural choice of trial functions uniform first order accuracy is obtained in the l 0 1 norm using trial functions which interpolate linearly between the nodal values generated by the difference scheme gives uniform first order accuracy in the l 1 0 1 norm

estimation and confidence regions for parameter sets in econometric models this paper develops a framework for performing estimation and inference in econometric models with partial identification focusing particularly on models characterized by moment inequalities and equalities applications of this framework include the analysis of game theoretic models revealed preference restrictions regressions with missing and corrupted data auction models structural quantile regressions and asset pricing models specifically we provide estimators and confidence regions for the set of minimizers i of an econometric criterion function q in applications the criterion function embodies testable restrictions on economic models a parameter value that describes an economic model satisfies these restrictions if q attains its minimum at this value interest therefore focuses on the set of minimizers called the identified set we use the inversion of the sample analog q n of the population criterion q to construct estimators and confidence regions for the identified set and develop consistency rates of convergence and inference results for these estimators and regions to derive these results we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification

an operator compact implicit method of exponential type a fourth order accurate tridiagonal operator compact implicit finite difference method for diffusion convection equations is developed the coefficients of the difference operator contain exponentials of the coefficients of the differential operator in this way the method avoids spurious oscillations when the cell reynolds number is large numerical results and comparisons to other methods are presented

die gleichmaβige stabilitat singular gestorter diskreter und kontinuierlicher randwertprobleme in this paper we consider singular perturbation problems for ordinary differential operators of ordern and their discrete counterparts on arbitrary nonuniform grids we prove that the singularly perturbed initial value problem is stable uniformly in the perturbation parameter in both the continuous and the discrete case we use this result to characterize the stability of the corresponding continuous and discrete boundary value problems if the continuous problem is stable and if the consistency error is smaller than a certain constant the discrete problem is also stable

non parametric tests of productive efficiency with errors in variables we develop a non parametric test of productive efficiency that accounts for errors in variables following the approach of varian 1985 nonparametric analysis of optimizing behavior with measurement error journal of econometrics 30 1 2 445 458 the test is based on the general pareto koopmans notion of efficiency and does not require price data statistical inference is based on the sampling distribution of the l norm of errors the test statistic can be computed using a simple enumeration algorithm the finite sample properties of the test are analyzed by means of a monte carlo simulation using real world data of large eu commercial banks

an analysis of a uniformly accurate difference method for a singular perturbation problem it will be proven that an exponential tridiagonal difference scheme when applied with a uniform mesh of size h to euxx b x ux f x for o o b and f smooth e in 0 1 and u o and u i given is uniformly second order accurate i e the maximum of the errors at the grid points is bounded by ch2 with the constant c independent of h and e this scheme was derived by el mistikawy and werle by a c patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point the behavior of the approximate solution in between the grid points will be analyzed and some numerical results will also be given

a penalized method for multivariate concave least squares with application to productivity analysis abstract we propose a penalized method for the least squares estimator of a multivariate concave regression function this estimator is formulated as a quadratic programing qp problem with o n2 constraints where n is the number of observations computing such an estimator is a very time consuming task and the computational burden rises dramatically as the number of observations increases by introducing a quadratic penalty function we reformulate the concave least squares estimator as a qp with only non negativity constraints this reformulation can be adapted for estimating variants of shape restricted least squares i e the monotonic concave convex least squares the experimental results and an empirical study show that the reformulated problem and its dual are solved significantly faster than the original problem the matlab and r codes for implementing the penalized problems are provided in the paper

direction selection in stochastic directional distance functions abstract researchers rely on the distance function to model multiple product production using multiple inputs a stochastic directional distance function sddf allows for noise in potentially all input and output variables yet when estimated the direction selected will affect the functional estimates because deviations from the estimated function are minimized in the specified direction specifically the parameters of the parametric sddf are point identified when the direction is specified we show that the parameters of the parametric sddf are set identified when multiple directions are considered further the set of identified parameters can be narrowed via data driven approaches to restrict the directions considered we demonstrate a similar narrowing of the identified parameter set for a shape constrained nonparametric method where the shape constraints impose standard features of a cost function such as monotonicity and convexity our monte carlo simulation studies reveal significant improvements as measured by out of sample radial mean squared error in functional estimates when we use a directional distance function with an appropriately selected direction and the errors are uncorrelated across variables we show that these benefits increase as the correlation in error terms across variables increase this correlation is a type of endogeneity that is common in production settings from our monte carlo simulations we conclude that selecting a direction that is approximately orthogonal to the estimated function in the central region of the data gives significantly better estimates relative to the directions commonly used in the literature for practitioners our results imply that selecting a direction vector that has non zero components for all variables that may have measurement error provides a significant improvement in the estimator s performance we illustrate these results using cost and production data from samples of approximately 500 us hospitals per year operating in 2007 2008 and 2009 respectively and find that the shape constrained nonparametric methods provide a significant increase in flexibility over second order local approximation parametric methods

the convex hull property of noncompact hypersurfaces with positive curvature we prove that in euclidean space r n 1 every metrically complete positively curved immersed hypersurface m with compact boundary m lies outside the convex hull of m provided that m is embedded on the boundary of a convex body and n 2 for n 2 on the other hand we construct examples which contradict this property