efficient high dimensional importance sampling the paper describes a simple generic and yet highly accurate efficient importance sampling eis monte carlo mc procedure for the evaluation of high dimensional numerical integrals eis is based upon a sequence of auxiliary weighted regressions which actually are linear under appropriate conditions it can be used to evaluate likelihood functions and byproducts thereof such as ml estimators for models which depend upon unobservable variables a dynamic stochastic volatility model and a logit panel data model with unobserved heterogeneity random effects in both dimensions are used to provide illustrations of eis high numerical accuracy even under small number of mc draws mc simulations are used to characterize the finite sample numerical and statistical properties of eis based ml estimators

improving the performance of random coefficients demand models the role of optimal instruments abstract we shed new light on the performance of berry levinsohn and pakes 1995 gmm estimator of the aggregate random coefficient logit model based on an extensive monte carlo study we show that the use of chamberlain s 1987 optimal instruments overcomes many problems that have recently been documented with standard non optimal instruments optimal instruments reduce small sample bias but they prove even more powerful in increasing the estimator s efficiency and stability we consider a wide variety of data generating processes and an empirical application to the automobile market we also consider the gains of other recent methodological advances when combined with optimal instruments

an efficient decomposition of the expectation of the maximum for the multivariate normal and related distributions abstract in structural dynamic discrete choice models monte carlo integration has been the only way to evaluate the expectation of the maximum when errors are normally distributed in this paper however i show that the expectation of the maximum can be decomposed as a linear combination of multivariate normal cdfs for related distributions such as the multivariate t distribution this expectation has a similar decomposition my computational results show speed benefits of my proposed method for models with a low number of choices although the speed gains are contingent on the use of analytical derivatives as opposed to numerical derivatives

a computationally efficient quadrature procedure for the one factor multinomial probit model a problem of estimation that has long confronted many economists is the difficulty of estimating the parameters of equations with limited dependent variables on cross section time series i e panel data while there are widely available packaged computer programs for estimating either a cross section probit and tobit models or b simple permanent transitory random effects panel models with continuous dependent variables there are no available computationally feasible methods of combining these two models this is because the likelihood function that arises in such a combined model contains multivariate normal integrals whose evaluation is quite difficult if not impossible with conventional approximation methods there is a widespread feeling among those working in the area that one possible method of evaluation the use of quadrature techniques is in principle possible but is in practice computationally too burdensome to consider e g albright et al 2 p 13 hausman and wise 6 p 12 in this note we point out that this is true only of standard quadrature techniques such as trapezoidal integration or its improved variants gaussian quadrature on the other hand is extremely efficient and is well within the bounds of computational feasibility on modern computers in what follows we state the nature of the integrals that need to be evaluated provide a brief exposition of gaussian quadrature and provide a numerical illustration of its use in

the eigenfunction solution for scattered fields and surface currents of a vertex the surface current and the scattered field of a vertex are studied by employing the exact eigenfunction solution the first few contributing eigenvalues are presented in easy to use graphical form results will be shown for both the current on a plane angular sector and also for the scattered field

sparse grid quadrature nonlinear filtering in this paper a novel nonlinear filter named sparse grid quadrature filter sgqf is proposed the filter utilizes weighted sparse grid quadrature points to approximate the multi dimensional integrals in the nonlinear bayesian estimation algorithm the locations and weights of the univariate quadrature points with a range of accuracy levels are determined by the moment matching method then the univariate quadrature point sets are extended to form a multi dimensional grid using the sparse grid theory compared with the conventional point based methods the estimation accuracy level of the sgqf can be flexibly controlled and the number of sparse grid quadrature points for the sgqf is a polynomial of the dimension of the system which alleviates the curse of dimensionality for high dimensional problems the unscented kalman filter ukf is proven to be a subset of the sgqf at the level 2 accuracy the performance of this filter is demonstrated by an orbit estimation problem the simulation results show that the sgqf achieves higher accuracy than the extended kalman filter ekf the ukf and the cubature kalman filter ckf in addition the sgqf is computationally much more efficient than the multi dimensional gauss hermite quadrature filter ghqf with the same performance

numerical calculation of the diffraction coefficients for an arbitrary shaped perfectly conducting cone a method for numerical calculation of the diffraction coefficients for electromagnetic diffraction by arbitrarily shaped perfectly conducting cones is proposed the approach makes an extensive use of the analytic formulas of smyshlyaev in combination with further developments including a use of the potential theory adapted to the laplace beltrami operator on a subdomain of unit sphere this reduces the problem to a fredholm integral equation on the closed curve of the unit sphere defining the cone s geometry which can be solved numerically this strategy permits us to implement a numerical code for calculation of the diffraction coefficients for cones of rather general cross sections results of sample calculations for the circular and elliptic cones are given

chebyshev polynomials moment matching and optimal estimation of the unseen we consider the problem of estimating the support size of a discrete distribution whose minimum non zero mass is at least 1 k using independent samples we show that the minimax sample complexity to achieve an additive error of k with probability at least 0 5 is within universal constant factors of k log k log 2 1 which improves the state of the art result k 2 log k due to valiant and valiant the optimal procedure is a linear estimator based on the chebyshev polynomial and its approximation theoretic properties we also study the closely related species problem where the goal is to estimate the number of distinct colors in an urn containing k balls from repeated draws while achieving an additive error proportional to k still requires k log k samples we show that with k samples one can strictly outperform a general support size estimator using interpolating polynomials

a decomposition strategy for the variational inference of complex systems markov chain monte carlo approaches have been widely used for bayesian inference the drawback of these methods is that they can be computationally prohibitive especially when complex models are analyzed in such cases variational methods may provide an efficient and attractive alternative however the variational methods reported to date are applicable to relatively simple models and most are based on a factorized approximation to the posterior distribution here we propose a variational approach that is capable of handling models that consist of a system of differential algebraic equations and whose posterior approximation can be represented by a multivariate distribution under the proposed approach the solution of the variational inference problem is decomposed into three steps a maximum a posteriori optimization which is facilitated by using an orthogonal collocation approach a preprocessing step which is based on the estimation of the eigenvectors of the posterior covariance matrix and an e

using penalized likelihood to select parameters in a random coefficients multinomial logit model abstract this paper is about estimating a random coefficients logit model in which the distribution of each coefficient is characterized by finitely many parameters some of which may be zero the paper gives conditions under which with probability approaching 1 as the sample size increases penalized maximum likelihood pml estimation with the adaptive lasso al penalty distinguishes correctly between zero and non zero parameters the paper also gives conditions under which pml reduces the asymptotic mean square estimation error of any continuously differentiable function of the model s parameters the paper describes a method for computing pml estimates and presents the results of monte carlo experiments that illustrate their performance it also presents the results of pml estimation of a random coefficients logit model of choice among brands of butter and margarine in the british groceries market

uncertainty quantification for a 1d thermo hyperelastic coupled problem using polynomial chaos projection and p fems numerical solutions of non linear stochastic thermo hyperelastic problems at finite strains are addressed these belong to a category of non linear coupled problems that impose challenges on their numerical treatment both in the physical and stochastic spaces combining the high order finite element methods fems for discretizing the physical space and the polynomial chaos projection pcp method for discretizing the stochastic space a non intrusive scheme is obtained manifesting an exponential convergence rate the method is applied to a 1 d coupled stationary thermo hyperelastic system with stochastic material properties we derive exact stochastic solutions that serve for comparison to numerical results allowing their verification these demonstrate that stochastic coupled problems intractable by standard monte carlo mc methods may be easily computed by combining high order fems with the pcp method controlling discretization errors

a note on gauss hermite quadrature summary for gauss hermite quadrature we consider a systematic method for transforming the variable of integration so that the integrand is sampled in an appropriate region the effectiveness of the quadrature then depends on the ratio of the integrand to some gaussian density being a smooth function well approximated by a low order polynomial it is pointed out that in this approach order one gauss hermite quadrature becomes the laplace approximation thus the quadrature as implemented here can be thought of as a higher order laplace approximation

simulated maximum likelihood estimation of dynamic discrete choice statistical models some monte carlo results this article reports monte carlo results on the simulated maximum likelihood estimation of discrete dynamic panel models introduced by james heckman 1981a the simulated maximum likelihood method is numerically stable even for long panels regression models and polya and renewal models can be better estimated than markov models with a moderate number of simulation draws most of these complex models can be adequately estimated for panels with length up to 30 polya and renewal models can be accurately estimated for panels up to 50 periods estimates of markov models can be sensitive to misspecified initial states but polya renewal and habit persistence models may not

econometric illustrations of novel numerical integration strategies for bayesian inference abstract recent progress with numerical techniques for implementing bayesian methods for irregular multi parameter likelihoods is reviewed and illustrated with two econometric examples the numerical integration strategies involve novel iterative adaptive uses of cartesian product and spherical rule quadrature formulae together with importance sampling techniques

likelihood approximation by numerical integration on sparse grids the calculation of likelihood functions of many econometric models requires the evaluation of integrals without analytical solutions approaches for extending gaussian quadrature to multiple dimensions discussed in the literature are either very specific or suffer from exponentially rising computational costs in the number of dimensions we propose an extension that is very general and easily implemented and does not suffer from the curse of dimensionality monte carlo experiments for the mixed logit model indicate the superior performance of the proposed method over simulation techniques

a simple nonparametric approach to estimating the distribution of random coefficients in structural models abstract we explore least squares and likelihood nonparametric mixtures estimators of the joint distribution of random coefficients in structural models the estimators fix a grid of heterogeneous parameters and estimate only the weights on the grid points an approach that is computationally attractive compared to alternative nonparametric estimators we provide conditions under which the estimated distribution function converges to the true distribution in the weak topology on the space of distributions we verify most of the consistency conditions for three discrete choice models we also derive the convergence rates of the least squares nonparametric mixtures estimator under additional restrictions we perform a monte carlo study on a dynamic programming model

fully symmetric interpolatory rules for multiple integrals over infinite regions with gaussian weight fully symmetric interpolatory integration rules are constructed for multidimensional integrals over infinite integration regions with a gaussian weight function the points for these rules are determined by successive extensions of the one dimensional three point gauss hermite rule the new rules are shown to be efficient and only moderately unstable

smooth unbiased multivariate probability simulators for maximum likelihood estimation of limited dependent variable models abstract we apply a new simulation method that solves the multidimensional probability integrals that arise in maximum likelihood estimation of a broad class of limited dependent variable models the simulation method has four key features the simulated choice probabilities are unbiased they are a continuous and differentiable function of the parameters of the model they are bounded between 0 and 1 and their computation takes an effort that is nearly linear in the dimension of the probability integral independent of the magnitudes of the true probabilities we also show that the new simulation method produces probability estimates with substantially smaller variance than those generated by acceptance rejection methods or by stern s 1992 method the simulated probabilities can therefore be used to revive the lerman and manski 1981 procedure of approximating the likelihood function using simulated choice probabilities by overcoming its computational disadvantages

non intrusive reduced order modelling of the navier stokes equations abstract this article presents two new non intrusive reduced order models based upon proper orthogonal decomposition pod for solving the navier stokes equations the novelty of these methods resides in how the reduced order models are formed that is how the coefficients of the pod expansions are calculated rather than taking a standard approach of projecting the underlying equations onto the reduced space through a galerkin projection here two different techniques are employed the first method applies a second order taylor series to calculate the pod coefficients at each time step from the pod coefficients at earlier time steps the second method uses a smolyak sparse grid collocation method to calculate the pod coefficients where again the coefficients at earlier time steps are used as the inputs the advantage of both approaches are that they are non intrusive and so do not require modifications to a system code they are therefore very easy to implement they also provide accurate solutions for modelling flow problems and this has been demonstrated by the simulation of flows past a cylinder and within a gyre it is demonstrated that accuracy relative to the high fidelity model is maintained whilst cpu times are reduced by several orders of magnitude in comparison to high fidelity models

electric field behavior near metallic wedges electric field behavior is determined on perfect metallic wedges of bidimensional conical and three dimensional shape the analytical treatment is based on solving laplace s equation in spherical and conical coordinate systems for both cases the components of the electric field behave like r sup alpha presenting a singularity when alpha

alternative sampling methods for estimating multivariate normal probabilities we study the performance of alternative sampling methods for estimating multivariate normal probabilities through the ghk simulator the sampling methods are randomized versions of some quasi monte carlo samples halton niederreiter niederreiter xing sequences and lattice points and some samples based on orthogonal arrays latin hypercube orthogonal array and orthogonal array based latin hypercube samples in general these samples turn out to have a better performance than monte carlo and antithetic monte carlo samples improvements over these are large for low dimensional 4 and 10 cases and still significant for dimensions as large as 50

a method for smoothing simulated moments of discrete probabilities in multinomial probit models there has been much interest over the years in estimating discrete choice models however with the exception of multinomial logit models agents have been restricted to few choices so that multivariate integrals could be feasibly integrated numerically 2 pakes and pollard 1989 and mcfadden 1989 have independently developed the method of simulated moments msm to deal with estimating a wide class of models of which high order discrete choice models are a subset one of the problems a researcher must handle when using msm for a discrete choice model is how to smooth the discrete simulated random variables this is necessary to keep small the number of draws required to simulate derivatives and to simulate variation in the data geweke 1989 and mcfadden 1989 have suggested importance sampling methods and other methods to smooth the simulated variables for general error structures in this paper i present a factor analytic smoothing method that can be applied to probit problems although it is not as general as the methods geweke and mcfadden suggest it is easy to use and has clear intuition furthermore simulated probabilities will be unbiased will be bounded between zero and unity and will have smaller variances than unsmoothed probabilities always and smaller variances than importance sampling estimates for a large class of probabilities the second section of this paper develops notation defines the problem and presents the smoothing method and an algorithm to employ it the last section presents monte carlo comparisons of the smoothing method to the importance sampling method

topology optimization under uncertainty via non intrusive polynomial chaos expansion abstract this paper presents a systematic approach for topology optimization under uncertainty that integrates non intrusive polynomial chaos expansion with design sensitivity analysis for reliability based and robust topology optimization uncertainty is introduced in loading and in geometry to address the manufacturing variability the manufacturing variability is modeled via a thresholding technique in which the threshold field is represented by a reduced dimensional random field response metrics such as compliance and volume are characterized as polynomial chaos expansions of the underlying uncertain parameters thus allowing accurate and efficient estimation of statistical moments failure probabilities and their sensitivities the number of simulations is reduced for linear structures under loading uncertainty by means of superposition efficiency of the non intrusive polynomial chaos approach is highlighted by comparison with the monte carlo method in terms of the number of simulations to demonstrate the effect of uncertainty optimized designs that consider uncertainty are compared to those that do not comparisons of polynomial chaos expansion to existing analytical methods on a benchmark numerical example are also provided for reliability based and worst case designs

smolyak cubature of given polynomial degree with few nodes for increasing dimension some recent investigations see e g gerstner and griebel 5 novak and ritter 9 and 10 novak ritter and steinbauer 11 wasilkowski and wo niakowski 18 or petras 13 show that the so called smolyak algorithm applied to a cubature problem on the d dimensional cube seems to be particularly useful for smooth integrands the problem is still that the numbers of nodes grow polynomially but fast for increasing dimensions we therefore investigate how to obtain smolyak cubature formulae with a given degree of polynomial exactness and the asymptotically minimal number of nodes for increasing dimension d and obtain their characterization for a subset of smolyak formulae error bounds and numerical examples show their good behaviour for smooth integrands a modification can be applied successfully to problems of mathematical finance as indicated by a further numerical example

a method of simulated moments for estimation of discrete response models without numerical integration this paper proposes a simple modification of a conventional method of moments estimator for a discrete response model replacing response probabilities that require numerical integration with estimators obtained by monte carlo simulation this method of simulated moments msm does not require precise estimates of these probabilities for consistency and asymptotic normality relying instead on the law of large numbers operating across observations to control simulation error and hence can use simulations of practical size the method is useful for models such as high dimensional multinomial probit mnp where computation has restricted applications

a numerically stable quadrature procedure for the one factor random component discrete choice model abstract the gaussian quadrature formula had been popularized by butler and moffitt 1982 econometrika 50 761 764 for the estimation of the error component probit panel model borjas and sueyoshi 1994 journal of econometrics 64 164 182 pointed out some numerical and statistical difficulties of applying it to models with group effects with a moderate or large number of individuals in a group the likelihood function of the model evaluated by the gaussian quadrature formula can be numerically unstable and at worst impossible to evaluate statistical inference may also be inaccurate we point out that some of these difficulties can be overcome with a carefully designed algorithm and the proper selection of the number of quadrature points however with a very large number of individuals in a group the gaussian quadrature formulation of integral may have large numerical approximation errors

spectral domain solution and asymptotics for the diffraction by an impedance cone the problem of diffraction of a plane wave by a cone with impedance condition on its surface is studied for a scalar wave field satisfying the helmholtz equation the integral kontorovich lebedev sommerfeld maliuzhinets and fourier transformations are exploited to investigate the problem and to separate the radial variable the problem in question is reduced to that for a spectral function satisfying a laplace beltrami type equation on the unit sphere with a hole cut out by the conical surface an impedance type boundary condition with a nonlocal impedance operator is determined on the boundary of the hole then the problem for the spectral function can be transformed to a second kind integral equation with a nonoscillatory kernel in the particular case of a circular impedance cone the problem is simplified as an application a closed form expression for the scattering diagram or pattern is deduced in the narrow cone approximation the leading and first correction terms are represented

estimating marginal properties of quantitative real time pcr data using nonlinear mixed models summary a unified modeling framework based on a set of nonlinear mixed models is proposed for flexible modeling of gene expression in real time pcr experiments focus is on estimating the marginal or population based derived parameters cycle thresholds and but retaining the conditional mixed model structure to adequately reflect the experimental design additionally the calculation of model average estimates allows incorporation of the model selection uncertainty the methodology is applied for estimating the differential expression of a phosphate transporter gene ospt6 in rice in comparison to a reference gene at several states after phosphate resupply in a small simulation study the performance of the proposed method is evaluated and compared to a standard method

statistical inference in the multinomial multiperiod probit model abstract statistical inference in multinomial multiperiod probit models has been hindered in the past by the high dimensional numerical integrations necessary to form the likelihood functions posterior distributions or moment conditions in these models we describe three alternative estimators implemented using simulation based approaches to inference that circumvent the integration problem posterior means computed using gibbs sampling and data augmentation gibbs simulated maximum likelihood sml estimation using the ghk probability simulator and method of simulated moment msm estimation using ghk we perform a set of monte carlo experiments to compare the sampling distributions of these estimators although all three estimators perform reasonably well some important differences emerge our most important finding is that holding simulation size fixed the relative and absolute performance of the classical methods especially sml gets worse when serial correlation in disturbances is strong in data sets with an ar 1 parameter of 0 50 the rmses for sml and msm based on ghk with 20 draws exceed those of gibbs by 9 and 0 respectively but when the ar 1 parameter is 0 80 the rmses for sml and msm based on 20 draws exceed those of gibbs by 79 and 37 respectively and the number of draws needed to reduce the rmses to within 10 of gibbs are 160 and 80 respectively also the sml estimates of serial correlation parameters exhibit significant downward bias thus while conventional wisdom suggests that 20 draws of ghk is enough to render the bias and noise induced by simulation negligible our results suggest that much larger simulation sizes are needed when serial correlation in disturbances is strong

electromagnetic excitation of pec slotted cones by elementary radial dipoles a semi inversion analysis a novel method is presented that is based on the use of the kontorovich lebedev integral transform and the semi inversion technique developed for building the efficient and accurate numerical solutions to the boundary value problems of electromagnetic wave scattering by the 3d coaxial slotted cones a generic structure under consideration consists of two semi infinite coaxial circular perfectly electrical conducting and zero thickness cones with periodic longitudinal slots excited by a radial dipole the considered problem is reduced to an infinite set of linear algebraic equations of the fredholm second kind that is truncated and solved numerically a detail analysis of the accuracy and convergence of the method is presented the basic electromagnetic characteristics such as the field behavior near the structure singularities field patterns in the wave zone and field polarization for various problem parameters are investigated for the cones excited by on axis elementary dipoles the slotted cones allow obtaining more directional patterns in wider band than the solid ones

application of polynomial chaos to quantify uncertainty in deterministic channel models a non intrusive formulation of the polynomial chaos method is applied to quantify the uncertainties in deterministic models of the indoor radio channel deterministic models based on the finite difference time domain fdtd method and ray tracing are examined various sources of parameter uncertainty are considered including randomness in the material properties building geometry and the spatial location of transmitting and receiving antennas the polynomial chaos results are confirmed against monte carlo simulations and experimental measurements the analysis shows the expected variation in the sector averaged path loss can be considerable for relatively small input parameter uncertainties leading to the conclusion that a single simulation run using nominal values may be insufficient to adequately characterize the indoor radio channel

maximum likelihood estimation of limited and discrete dependent variable models with nested random effects abstract gauss hermite quadrature is often used to evaluate and maximize the likelihood for random component probit models unfortunately the estimates are biased for large cluster sizes and or intraclass correlations we show that adaptive quadrature largely overcomes these problems we then extend the adaptive quadrature approach to general random coefficient models with limited and discrete dependent variables the models can include several nested random effects intercepts and coefficients representing unobserved heterogeneity at different levels of a hierarchical dataset the required multivariate integrals are evaluated efficiently using spherical quadrature rules simulations show that adaptive quadrature performs well in a wide range of situations