adaptive ofdm synchronization algorithms based on discrete stochastic approximation this paper presents discrete stochastic approximation algorithms dsa for time synchronization in orthogonal frequency division multiplexing ofdm systems it is shown that the discrete stochastic approximation algorithms can be effectively used to achieve a significant reduction in computational complexity compared to brute force maximum likelihood ml methods for ofdm synchronization the most important property of the proposed algorithms is their recursive self learning capability most of the computational effort is spent at the global or a local optimizer of the objective function the convergence of the algorithms is analyzed an adaptive version of the discrete stochastic approximation algorithm is also presented for tracking time varying time delays and frequency offsets in time selective fading channels detailed numerical examples illustrate the performance gains of these dsa based synchronization algorithms

a chebyshev method for the solution of boundary value problems abstract an expansion procedure using the chebyshev polynomials as base functions is proposed the method yields more accurate results than either of the galerkin or tau methods as indicated from solving the orr sommerfeld equation for both the plane poiseuille flow and the blasius velocity profile the chebyshev approximation is also applied to resolve the radial dependence of the flow field for a circular cylinder or a sphere in a uniform flow

unbiased estimation with square root convergence for sde models in many settings in which monte carlo methods are applied there may be no known algorithm for exactly generating the random object for which an expectation is to be computed frequently however one can generate arbitrarily close approximations to the random object we introduce a simple randomization idea for creating unbiased estimators in such a setting based on a sequence of approximations applying this idea to computing expectations of path functionals associated with stochastic differential equations sdes we construct finite variance unbiased estimators with a square root convergence rate for a general class of multidimensional sdes we then identify the optimal randomization distribution numerical experiments with various path functionals of continuous time processes that often arise in finance illustrate the effectiveness of our new approach

sample path properties of self similar processes with stationary increments a real valued process x x t u is self similar with exponent h h ss if x a d ahx for all a 0 sample path properties of h ss processes with stationary increments are investigated the main result is that the sample paths have nowhere bounded variation if 0 1 in which case they are singular however nowhere bounded variation may occur also for h 1 examples exhibiting this combination of properties are constructed as well as many others most are obtained by subordination of random measures to point processes in ir2 that are poincar6 i e invariant in distribution for the transformations t x at b ax of r in a final section it is shown that the self similarity and stationary increment properties are preserved under composition of independent processes x1 a x2 x1 x2 t l some interesting examples are obtained this way

scrambled net variance for integrals of smooth functions hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter one version randomized t m s nets has the property that the integral estimates are unbiased and that the variance is o 1 n for any square integrable integrand stronger assumptions on the integrand allow one to find rates of convergence this paper shows that for smooth integrands over s dimensions the variance is of order n 3 log n 3 1 compared to n 1 for ordinary monte carlo thus the integration errors are of order n 3 2 log n s 1 2 in probability this compares favorably with the rate n 1 log n s 1 for unrandomized t m s nets

asymptotics of likelihood ratio derivative estimators in simulations of highly reliable markovian systems we discuss the estimation of derivatives of a performance measure using the likelihood ratio method in simulations of highly reliable markovian systems we compare the difficulties of estimating the performance measure and of estimating its partial derivatives with respect to component failure rates as the component failure rates tend to 0 and the component repair rates remain fixed we first consider the case when the quantities are estimated using naive simulation i e when no variance reduction technique is used in particular we prove that in the limit some of the partial derivatives can be estimated as accurately as the performance measure itself this result is of particular interest in light of the somewhat pessimistic empirical results others have obtained when applying the likelihood ratio method to other types of systems however the result only holds for certain partial derivatives of the performance measure when using naive simulation more specifically we can estimate a certain partial

sensitivity estimation for gaussian systems in this paper we address the construction of efficient algorithms for the estimation of gradients of general performance measures of gaussian systems exploiting a clever coupling between the normal and the maxwell distribution we present a new gradient estimator and we show that it outperforms both the single run based infinitesimal perturbation analysis ipa estimator and the score function sf estimator in the one dimensional case for a dense class of test functions next we present an example of the multi dimensional case with a system from the area of stochastic activity networks our numerical experiments show that this new estimator also has significantly smaller sample variance than ipa and sf to increase efficiency in addition to variance reduction we present an optimized method for generating the maxwell distribution which minimizes the cpu time

unstable asymptotics for nonstationary queues we relate laws of large numbers and central limit theorems for nonstationary counting processes to corresponding limits for their inverse processes we apply these results to develop approximations for queues that are unstable in a nonstationary manner we obtain unstable nonstationary analogs of the queueing relation l w and associated central limit theorem versions for modeling and to obtain the first limits we can construct nonstationary point processes as random time transformations of familiar point processes such as renewal processes and stationary point processes we deduce the asymptotic behavior of the nonstationary point process from the asymptotic behavior of the familiar point process and the time transformation

the asymptotic efficiency of simulation estimators a decision theoretic framework is proposed for evaluating the efficiency of simulation estimators the framework includes the cost of obtaining the estimate as well as the cost of acting based on the estimate the cost of obtaining the estimate and the estimate itself are represented as realizations of jointly distributed stochastic processes in this context the efficiency of a simulation estimator based on a given computational budget is defined as the reciprocal of the risk the overall expected cost this framework is appealing philosophically but it is often difficult to apply in practice e g to compare the efficiency of two different estimators because only rarely can the efficiency associated with a given computational budget be calculated however a useful practical framework emerges in a large sample context when we consider the limiting behavior as the computational budget increases a limit theorem established for this model supports and extends a fairly well known efficiency principle

a large deviations perspective on the efficiency of multilevel splitting we analyze the performance of a multilevel splitting method for rare event simulation related to one recently proposed in the telecommunications literature this method splits promising paths into subpaths at intermediate levels to increase the number of observations of a rare event in our previous paper 1997 we gave sufficient conditions in specific classes of models for this method to be asymptotically optimal here we focus on necessary conditions in a general setting we show through a variety of results the importance of choosing the intermediate thresholds in a way consistent with the most likely path to a rare set both when the number of levels is fixed and when it increases with the rarity of the event in the latter case we give very general necessary conditions based on large deviations rate functions these indicate that even when the intermediate levels are chosen appropriately the method will frequently fail to be asymptotically optimal we illustrate the conditions with examples

indirect estimation via l w for a large class of queueing systems little s law l w helps provide a variety of statistical estimators for the long run time average queue length l and the long run customer average waiting time w we apply central limit theorem versions of little s law to investigate the asymptotic efficiency of these estimators we show that an indirect estimator for l using the natural estimator for w plus the known arrival rate is more efficient than a direct estimator for l provided that the interarrival and waiting times are negatively correlated thus extending a variance reduction principle for the gi g s model due to a m law and j s carson we also introduce a general framework for indirect estimation which can be applied to other problems besides l w we show that the issue of indirect versus direct estimation is related to estimation using nonlinear control variables we also show under mild regularity conditions that any nonlinear control variable scheme is equivalent to a linear control va

an automatic orthonormalization method for solving stiff boundary value problems abstract a new initial value method is described based on a remark by drury for solving stiff linear differential two point cigenvalue and boundary value problems the method is extremely reliable it is especially suitable for high order differential systems and it is capable of accommodating realms of stiffness which other methods cannot reach the key idea behind the method is to decompose the stiff differential operator into two non stiff operators one of which is nonlinear the nonlinear one is specially chosen so that it advances an orthonormal frame indeed the method is essentially a kind of automatic orthonormalization the second is auxiliary but it is needed to determine the required function the usefulness of the method is demonstrated by calculating some eigenfunctions for an orr sommerfeld problem when the reynolds number is as large as 10

antithetic acceleration of monte carlo integration in bayesian inference abstract it is proposed to sample antithetically rather than randomly from the posterior density in bayesian inference using monte carlo integration conditions are established under which the number of replications required with antithetic sampling relative to the number required with random sampling is inversely proportional to sample size as sample size increases the result is illustrated in an experiment using a bivariate vector autoregression

a quasirandom approach to integration in bayesian statistics on passe en revue la theorie existante et on introduit de nouveaux criteres d identification de suites quasi aleatoires on suggere des extensions des regles d integration de base

multilevel splitting for estimating rare event probabilities we analyze the performance of a splitting technique for the estimation of rare event probabilities by simulation a straightforward estimator of the probability of an event evaluates the proportion of simulated paths on which the event occurs if the event is rare even a large number of paths may produce little information about its probability using this approach the method we study reinforces promising paths at intermediate thresholds by splitting them into subpaths which then evolve independently if implemented appropriately this has the effect of dedicating a greater fraction of the computational effort to informative runs we analyze the method for a class of models in which roughly speaking the number of states through which each threshold can be crossed is bounded under additional assumptions we identify the optimal degree of splitting at each threshold as the rarity of the event increases it should be set so that the expected number of subpaths reaching each threshold remains roughly constant thus implemented the method is provably effective in a sense appropriate to rare event simulations these results follow from a branching process analysis of the method we illustrate our theoretical results with some numerical examples for queueing models

asymptotic formulas for markov processes with applications to simulation the simulation run length required to achieve desired statistical precision for a sample mean in a steady state stochastic simulation experiment is largely determined by the asymptotic variance of the sample mean and to a lesser extent by the second order asymptotics of the variance and the asymptotic bias the asymptotic variance the second order asymptotics of the variance and the asymptotic bias of the sample mean of a function of an ergodic markov process can be expressed in terms of solutions of poisson s equation as indicated by positive recurrent potential theory we review this positive recurrent potential theory giving special attention to continuous time markov chains we provide explicit formulas for birth and death processes and diffusion processes and recursive computational procedures for skip free chains these results can be used to help design simulation experiments after approximating the stochastic process of interest by one of the elementary markov processes considered here

the compound matrix method for ordinary differential systems abstract a generalization of the compound matrix method is presented to deal with eigenvalue and boundary value problems involving unstable systems of ordinary differential equations details are given for fourth and sixth order problems next is shown that a simple equivalence relation exists between the compounds of the solution matrices of the eigenvalue problems and their adjoints and how this relation can be exploited to simplify the calculation of the adjoint eigenfunctions is discussed using the orr sommerfeld problem as an example it is also shown how the techniques used in the derivation of certain auxiliary systems which play a crucial role in the generalization of the compound matrix method can also provide an alternate method for the direct computation of certain quantities defined in terms of the eigenfunctions finally there is a brief discussion of the relationship between the compound matrix method anf the riccati method

quadrature error bounds with applications to lattice rules reproducing kernel hilbert spaces are used to derive error bounds and worst case integrands for a large family of quadrature rules in the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature however the theory developed here does not require periodicity and is not restricted to lattice rules an analysis of variance anova decomposition is employed in defining the inner product it is shown that imbedded rules are superior when integrating functions with large high order anova effects

an introduction to finite element methods for transient advection problems abstract an introduction is presented to time accurate finite element methods recently developed for solving unsteady problems governed by linear and nonlinear advection equations this includes methods making an explicit use of the characteristic curves methods based on the taylor series in time and least squares methods

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

the close relationships between methods for solving two point boundary value problems the well conditioning of a boundary value problem is shown to be related to bounding two quantities one involving the boundary conditions and the other involving the green s function in the case of a solution dichotomy they are related to known stability results these results easily explain why difficulties arise using superposition and reduced superposition it is shown how multiple shooting overcomes these difficulties by relating its matrix conditioning to the underlying boundary value problem some factorization methods for the multiple shooting method are considered these show the close relationships between multiple shooting the stabilized march and invariant imbedding and one factorization leads to an efficient new way to implement each of these three methods many of the results apply for collocation and finite difference methods too finally it is shown both theroetically and computationally when matrix compactification can lead to difficulty

general multilevel monte carlo methods for pricing discretely monitored asian options abstract we describe general multilevel monte carlo methods that estimate the price of an asian option monitored at m fixed dates for a variety of processes that can be simulated exactly we prove that for the same computational cost our method yields an unbiased estimator with variance lower than the variance of the standard monte carlo estimator by a factor of order m we show how to combine our approach with the milstein scheme for processes driven by scalar stochastic differential equations and with the euler scheme for processes driven by multidimensional stochastic differential equations numerical experiments confirm that our method outperforms the conventional monte carlo algorithm by a factor proportional to m

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

extensible lattice sequences for quasi monte carlo quadrature integration lattices are one of the main types of low discrepancy sets used in quasi monte carlo methods however they have the disadvantage of being of fixed size this article describes the construction of an infinite sequence of points the first bm of which forms a lattice for any nonnegative integer m thus if the quadrature error using an initial lattice is too large the lattice can be extended without discarding the original points generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice the spectral test used for finding pseudorandom number generators is one important example of such a discrepancy the performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

the efficiency of one long run versus independent replication in steady state simulation we evaluate the efficiency of one long run versus independent replications in steady state discrete event simulation assuming that an initial portion of each replication will be deleted to allow the process to approach steady state we provide supporting evidence in favor of one long run but we also show that multiple replications can be more efficient the advantage of one long run increases if the amount deleted increases or if the covariance function decreases more quickly assuming it is nonnegative and decreasing thus assuming that the amount deleted depends on the way the process approaches steady state one long run tends to be efficient when the covariance function decays rapidly compared to the rate the process approaches steady state we also discuss ways to determine the initial portion to delete we consider the case of an exponential covariance function in detail and use it as a basis for approximations we also consider the m g queueing model and reflected brownian motion the latter

a shooting method for nonlinear heat transfer using automatic differentiation the steady state temperature distribution in a cylinder of unit radius is modeled by a nonlinear two point boundary value problem with an endpoint singularity we present a simple shooting method for the solution of this problem it is well known that shooting methods solve initial value problems repeatedly until the boundary conditions are satisfied for the solution of the initial value problems the method of this paper uses a technique known as automatic differentiation and obtains a taylor series expansion for the solution automatic differentiation is the process of obtaining the exact values of derivatives needed in the taylor series expansion using recursive formulas derived from the governing differential equation itself thus the method does not face the need to deal with step size issues or the need to carry out lengthy algebraic manipulations for obtaining higher order derivatives the method successfully reproduces the solutions obtained by previous researchers

adaptive mimo antenna selection via discrete stochastic optimization recently it has been shown that it is possible to improve the performance of multiple input multiple output mimo systems by employing a larger number of antennas than actually used and selecting the optimal subset based on the channel state information existing antenna selection algorithms assume perfect channel knowledge and optimize criteria such as shannon capacity or various bounds on error rate this paper examines mimo antenna selection algorithms where the set of possible solutions is large and only a noisy estimate of the channel is available in the same spirit as traditional adaptive filtering algorithms we propose simulation based discrete stochastic optimization algorithms to adaptively select a better antenna subset using criteria such as maximum mutual information bounds on error rate etc these discrete stochastic approximation algorithms are ideally suited to minimize the error rate since computing a closed form expression for the error rate is intractable we also consider scenarios of time varying channels for which the antenna selection algorithms can track the time varying optimal antenna configuration we present several numerical examples to show the fast convergence of these algorithms under various performance criteria and also demonstrate their tracking capabilities

gradient estimation for multicomponent maintenance systems with age replacement policy we consider multicomponent maintenance systems with an f failure group age replacement policy it keeps failed components idling until f components are failed and then replaces all failed components together with the nonfailed components whose age has passed the critical threshold age n for components of type n with each maintenance action costs are associated we derive various unbiased gradient estimators based on the measure valued differentiation approach for the gradient of the average cost each estimator has its own domain of applicability we also compare the performance of our gradient estimators when applied to stochastic optimization with other general gradient free methods

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

efficient estimation and filtering for multivariate jump diffusions abstract this paper develops estimators of the transition density filters and parameters of multivariate jump diffusion models the drift volatility jump intensity and jump magnitude are allowed to be state dependent and non affine it is not necessary to diagonalize the volatility matrix our density and filter estimators converge at the canonical rate typically associated with exact monte carlo estimation our parameter estimators have the same asymptotic distribution as maximum likelihood estimators which are often intractable for the class of models we consider the results of this paper enable the empirical analysis of previously intractable models of asset prices and economic time series

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

ordinary clt and wlln versions of l λw the familiar queueing principle expressed by the formula l w little s law can be interpreted as a relation among strong laws of large numbers sllns here we prove central limit theorem clt and weak law of large numbers wlln versions of l w for example if the sequence of ordered pairs of interarrival times and waiting times is strictly stationary and satisfies a joint clt then the queue length process also obeys a clt with a related limiting distribution in a previous paper we proved a functional central limit theorem version of l w without stationarity by very different arguments the two papers highlight the differences between establishing ordinary limit theorems and their functional limit theorem counterparts

high order numerical method for two point boundary value problems in this paper the two point boundary value problem is transformed into general first order ordinary differential equation system through introduction of conditions of an integral character to supplement the simultaneous set of first order equations a new discrete approximation of a high order compact difference scheme is presented for the first order system it is a block bidiagonal profile and removes the limits of other high order discrete schemes at the interval ends the numerical tests of a seventh order compact difference scheme show that the proposed scheme is very convenient and efficient for linear and nonlinear two point boundary value problems

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

numerical solution of two point boundary value problems abstract this paper presents an original formulation of two point boundary value and eigenvalue problems expressed as a system of first order equations the fundamental difference between the new method and other methods based on a first order approach is the introduction of conditions of an integral character to supplement the simultaneous set of first order equations which are hence never regarded as an initial value problem the consideration of integral conditions leads to establish a class of linear multipoint schemes for the numerical solution of boundary value problems for ordinary differential equations furthermore the global character of the integral conditions nonlocality combined with the block structure of the system of algebraic equations allow dealing with stiff problems by means of the classical procedure of iterative refinement introduced by wilkinson the properties of the numerical schemes are illustrated by the solution of linear and nonlinear problems and by the accurate and efficient determination of some eigensolutions of a difficult problem of hydrodynamic stability the proposed method is conceptually simpler and numerically more convenient than existing initial value methods while still retaining all the advantages of a formulation based on a first order system

monte carlo variance of scrambled net quadrature hybrids of equidistribution and monte carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter this paper studies the variance of one such hybrid scrambled nets by applying a multidimensional multiresolution wavelet analysis to the integrand the integrand is assumed to be measurable and square integrable but not necessarily of bounded variation in simple monte carlo every nonconstant term of the multiresolution contributes to the variance of the estimated integral for scrambled nets certain low dimensional and coarse terms do not contribute to the variance for any integrand in l2 the sampling variance tends to zero faster under scrambled net quadrature than under monte carlo sampling as the number of function evaluations n tends to infinity some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the monte carlo variance uniformly over all integrands f latin hypercube sampling is a special case of scrambled net quadrature

dirichlet bridge sampling for the variance gamma process pricing path dependent options the authors develop a new monte carlo based method for pricing path dependent options under the variance gamma vg model the gamma bridge sampling method proposed by avramidis et al avramidis a n p l ecuyer p a tremblay 2003 efficient simulation of gamma and variance gamma processes proc 2003 winter simulation conf ieee press piscataway nj 319 326 and ribeiro and webber ribeiro c n webber 2004 valuing path dependent options in the variance gamma model by monte carlo with a gamma bridge j computational finance7 2 81 100 is generalized to a multivariate dirichlet construction bridging simultaneously over all time partition points of the trajectory of a gamma process the generation of the increments of the gamma process given its value at the terminal point is interpreted as a dirichlet partition of the unit interval the increments are generated in a decreasing stochastic order and under the kingman limit have a known distribution thus simulation of a trajectory from the gamma process requires generating only a small number of uniforms avoiding the expensive simulation of beta variates via numerical probability integral inversion the proposed method is then applied in simulating the trajectory of a vg process using its difference of gammas representation it has been implemented in both plain monte carlo and quasi monte carlo environments it is tested in pricing lookback barrier and asian options and is shown to provide consistent efficiency gains compared to the sequential method and the difference of gammas bridge sampling proposed by avramidis and l ecuyer avramidis a n p l ecuyer 2006 efficient monte carlo and quasi monte carlo option pricing under the variance gamma model management sci 52 12 1930 1944

spectral algorithms for vector elliptic equations in a spherical gap abstract this paper is devoted to the numerical solution of three dimensional elliptic equations of a vector field in the region contained between two concentric spheres the method of truncated series expansion in orthogonal functions is considered using chebyshev polynomials and vector spherical harmonics spectral algorithms are described for solving two point boundary value problems of the vector modes of the poisson helmholtz and biharmonic equations supplemented by boundary conditions typical of fluid dynamical applications the accuracy of the algorithms is illustrated by computing some analytical single mode examples the proposed algorithms when combined with efficient transform techniques can be used to solve multi mode nonlinear problems

randomized dimension reduction for monte carlo simulations we present a new unbiased algorithm that estimates the expected value of f u via monte carlo simulation where u is a vector of d independent random variables and f is a function of d variables

almost sure approximations to the robbins monro and kiefer wolfowitz processes with dependent noise we study a recursive algorithm which includes the multidimensional robbins monro and kiefer wolfowitz processes the assumptions on the disturbances are weaker than the usual assumption that they be a martingale difference sequence it is shown that the algorithm can be represented as a weighted average of the disturbances this representation can be used to prove asymptotic results for stochastic approximation procedures as an example we approximate the one dimensional kiefer wolfowitz process almost surely by brownian motion and as a byproduct obtain a law of the iterated logarithm

unbiased estimators and multilevel monte carlo multilevel monte carlo mlmc and recently proposed unbiased estimators are closely related this connection is elaborated by presenting a new general class of unbiased estimators which admits previous debiasing schemes as special cases new lower variance estimators are proposed which are stratified versions of earlier unbiased schemes under general conditions essentially when mlmc admits the canonical square root monte carlo error rate the proposed new schemes are shown to be asymptotically as efficient as mlmc both in terms of variance and cost the experiments demonstrate that the variance reduction provided by the new schemes can be substantial

bayesian inference in econometric models using monte carlo integration methods for the systematic application of monte carlo integration with importance sampling to bayesian inference in econometric models are developed conditions under which the numerical approximation of a posterior moment converges almost surely to the true value as the number of monte carlo replications increases and the numerical accuracy of this approximation may be assessed reliably are set forth methods for the analytical verification of these conditions are discussed

simulated likelihood estimators for discretely observed jump diffusions abstract this paper develops an unbiased monte carlo approximation to the transition density of a jump diffusion process with state dependent drift volatility jump intensity and jump magnitude the approximation is used to construct a likelihood estimator of the parameters of a jump diffusion observed at fixed time intervals that need not be short the estimator is asymptotically unbiased for any sample size it has the same large sample asymptotic properties as the true but uncomputable likelihood estimator numerical results illustrate its properties

continuous orthonormalization for boundary value problems abstract continuous orthonormalization describes an initial value method for linear 2 point boundary value problems which provides an orthogonal basis for the solution space at all points of the interval in this paper the equations of continuous orthonormalization are derived with elementary projection arguments to provide geometric insight and motivate some modifications of an earlier algorithm the method is then applied to some oscillatory and stiff boundary value problems to demonstrate that it is simple to use problem independent and as adaptive as the initial value code which is used to integrate the equations of continuous orthonormalization

orthogonal array based latin hypercubes abstract in this article we use orthogonal arrays oa s to construct latin hypercubes besides preserving the univariate stratification properties of latin hypercubes these strength r oa based latin hypercubes also stratify each r dimensional margin therefore such oa based latin hypercubes provide more suitable designs for computer experiments and numerical integration than do general latin hypercubes we prove that when used for integration the sampling scheme with oa based latin hypercubes offers a substantial improvement over latin hypercube sampling

monte carlo evaluation of multivariate normal probabilities abstract this paper extends research on the simulation of multivariate normal probabilities of high order dimension by developing a new family of simulators such simulators are useful for models with limited dependent variables including multinomial probit in panel studies spatial analysis and time series analysis the simulators are derived from a cholesky decomposition of the covariance matrix combined with a suitable choice of an importance sampling distribution the paper studies among others the impact of antithetical sampling the insights gained in this paper are of use in bayesian analysis as well in the evaluation of posterior densities

simulating discounted costs we numerically estimate via simulation the expected infinite horizon discounted cost d of running a stochastic system a naive strategy estimates a finite horizon approximation to d we propose alternatives all are ranked with respect to asymptotic variance as a function of computer time budget and discount rate when semi markov and or regenerative structure or neither is assumed in this setting the naive truncation estimator loses it may triumph however when the computer time budget is modest the discount rate is large and the process simulated is not regenerative or has long cycle lengths

on one dimensional stretching functions for finite difference calculations abstract the class of one dimensional stretching functions used in finite difference calculations is studied for solutions containing a highly localized region of rapid variation simple criteria for a stretching function are derived using a truncation error analysis these criteria are used to investigate two types of stretching functions one is an interior stretching function for which the location and slope of an interior clustering region are specified the simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine it was first employed by thomas et al aiaa j 10 1972 887 the other type of function is a two sided stretching function for which the arbitrary slopes at the two ends of the one dimensional interval are specified the simplest such general function is found to be one based on the inverse tangent the special case where the slopes were both equal and greater than one was first employed by roberts the general two sided function has many applications in the construction of finite difference grids examples of such applications are found in the listed references

filtered likelihood for point processes abstract point processes are widely used in finance and economics to model the timing of defaults market transactions unemployment spells births and a range of other events we develop and analyze likelihood estimators for the parameters of a marked point process and incompletely observed explanatory factors that influence the arrival intensity and mark distribution we establish an approximation to the likelihood and analyze the convergence and large sample properties of the associated estimators numerical results illustrate the behavior of our estimators

budget dependent convergence rate of stochastic approximation convergence rate results are derived for a stochastic optimization problem where a performance measure is minimized with respect to a vector parameter t assuming that a gradient estimator is available and that both the bias and the variance of the estimator are known functions of the budget devoted to its computation the gradient estimator is employed in conjunction with a stochastic approximation sa algorithm our interest is to figure out how to allocate the total available computational budget to the successive sa iterations the effort is devoted to solving the asymptotic version of this problem by finding the convergence rate of sa toward the optimizer first as a function of the number of iterations and then as a function of the total computational effort as a result the optimal rate of increase of the computational budget per iteration can be found explicit expressions for the case where the computational budget devoted to an iteration is a polynomial in the iteration number and where the bias and variance of the gradient estimator are polynomials of the computational budget are derived applications include the optimization of steady state simulation models with likelihood ratio perturbation analysis or finite difference gradient estimators optimization of infinite horizon models with discounting optimization of functions of several expectations and so on several examples are discussed our results readily generalize to general root finding problems

optimal unbiased estimation for expected cumulative discounted cost abstract we consider estimating an expected infinite horizon cumulative discounted cost reward contingent on an underlying stochastic process by monte carlo simulation an unbiased estimator based on truncating the cumulative cost at a random horizon is proposed explicit forms for the optimal distributions of the random horizon are given and explicit expressions for the optimal random truncation level are obtained leading to a full analysis of the bias variance tradeoff when comparing this new class of randomized estimators with traditional fixed truncation estimators moreover we characterize when the optimal randomized estimator is preferred over a fixed truncation estimator by considering the tradeoff between bias and variance this comparison provides guidance on when to choose randomized estimators over fixed truncation estimators in practice numerical experiments substantiate the theoretical results

estimating average production intervals using inventory measurements little s law for partially observable processes this paper proposes an indirect approach to estimate average production intervals the length of time between starting and finishing work on each product using work in process inventory measurements the idea is to apply a modified version of little s law l w from queueing theory to cope with stochastic processes that are not directly observable when the actual amount of completed product to be produced from the current work in process is not known we suggest working with an appropriate expected amount of completed product associated with current work in process taking care to properly account for such features as partial yields changing lot sizes and reconstituted lots this indirect estimation procedure can be applied to computer simulation as well as direct system measurement the approach also can be used to calculate expected values of steady state random variables in mathematical models

simulation of multivariate normal rectangle probabilities and their derivatives theoretical and computational results abstract an extensive literature in econometrics and in numerical analysis has considered the problem of evaluating the multiple integral p b a b n v dv e v 1 v b where v is a m dimensional normal vector with mean covariance matrix and density n v and 1 v b is an indicator for the event b v a a leading case of such an integral is the negative orthant probability where b v v the problem is computationally difficult except in very special cases the multinomial probit mnp model used in econometrics and biometrics has cell probabilities that are negative orthant probabilities with and depending on unknown parameters and in general on covariates estimation of this model requires for each trial parameter vector and each observation in a sample evaluation of p b and of its derivatives with respect to and this paper surveys monte carlo techniques that have been developed for approximations of p b and its linear and logarithmic derivatives that limit computation while possessing properties that facilitate their use in iterative calculations for statistical inference the crude frequency simulator cfs normal importance sampling nis a kernel smoothed frequency simulator kfs stern s decomposition simulator sds the geweke hajivassiliou keane simulator ghk a parabolic cylinder function simulator pcf deak s chi squared simulator dcs an acceptance rejection simulator ars the gibbs sampler simulator gss a sequentially unbiased simulator sus and an approximately unbiased simulator aus we also discuss gauss and fortran implementations of these algorithms and present our computational experience with them we find that ghk is overall the most reliable method

decoupling and stability of algorithms for boundary value problems the ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals it is shown that most methods like multiple shooting collocation invariant imbedding and difference methods derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later for instance when solving a linear system this decoupling is related to the dichotomy of the ordinary differential equations in fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed how this stability is related to the use of the boundary conditions is also investigated