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52
Random number generation
"... Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0, 1) random variables (i.e., continuous random variables dis ..."
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Cited by 136 (30 self)
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Random numbers are the nuts and bolts of simulation. Typically, all the randomness required by the model is simulated by a random number generator whose output is assumed to be a sequence of independent and identically distributed (IID) U(0, 1) random variables (i.e., continuous random variables distributed uniformly over the interval
Simulation estimation of mixed discrete choice models using randomized and scrambled halton sequences
 Transportation Research Part B
, 2002
"... The use of simulation techniques has been increasing in recent years in the transportation and related fields to accommodate flexible and behaviorally realistic structures for analysis of decision processes. This paper proposes a randomized and scrambled version of the Halton sequence for use in sim ..."
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Cited by 27 (3 self)
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The use of simulation techniques has been increasing in recent years in the transportation and related fields to accommodate flexible and behaviorally realistic structures for analysis of decision processes. This paper proposes a randomized and scrambled version of the Halton sequence for use in simulation estimation of discrete choice models. The scrambling of the Halton sequence is motivated by the rapid deterioration of the standard Halton sequence's coverage of the integration domain in high dimensions of integration. The randomization of the sequence is motivated from a need to statistically compute the simulation variance of model parameters. The resulting hybrid sequence combines the good coverage property of quasiMonte Carlo sequences with the ease of estimating simulation error using traditional Monte Carlo methods. The paper develops an evaluation framework for assessing the performance of the traditional pseudorandom sequence, the standard Halton sequence, and the scrambled Halton sequence. The results of computational experiments indicate that the scrambled Halton sequence performs better than the standard Halton sequence and the traditional pseudorandom sequence for simulation estimation of models with high dimensionality of integration.
The Dimension Distribution, and Quadrature Test Functions
"... This paper introduces the dimension distribution for a square integrable function f on [0; 1]^s. The dimension distribution is used to relate several definitions of the effective dimension of a function. Functions of low effective dimension can be easy to integrate numerically. ..."
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Cited by 24 (4 self)
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This paper introduces the dimension distribution for a square integrable function f on [0; 1]^s. The dimension distribution is used to relate several definitions of the effective dimension of a function. Functions of low effective dimension can be easy to integrate numerically.
A randomized quasiMonte Carlo simulation method for Markov chains
 Operations Research
, 2007
"... Abstract. We introduce and study a randomized quasiMonte Carlo method for estimating the state distribution at each step of a Markov chain. The number of steps in the chain can be random and unbounded. The method simulates n copies of the chain in parallel, using a (d + 1)dimensional highlyunifor ..."
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Cited by 20 (8 self)
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Abstract. We introduce and study a randomized quasiMonte Carlo method for estimating the state distribution at each step of a Markov chain. The number of steps in the chain can be random and unbounded. The method simulates n copies of the chain in parallel, using a (d + 1)dimensional highlyuniform point set of cardinality n, randomized independently at each step, where d is the number of uniform random numbers required at each transition of the Markov chain. This technique is effective in particular to obtain a lowvariance unbiased estimator of the expected total cost up to some random stopping time, when statedependent costs are paid at each step. It is generally more effective when the state space has a natural order related to the cost function. We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial. The variance can be reduced by factors of several thousands in some cases. We prove bounds on the convergence rate of the worstcase error and variance for special situations. In line with what is typically observed in randomized quasiMonte Carlo contexts, our empirical results indicate much better convergence than what these bounds guarantee.
Monte Carlo Extension Of QuasiMonte Carlo
 Proceedings of the 1998 Winter Simulation Conference
, 1998
"... This paper surveys recent research on using Monte Carlo techniques to improve quasiMonte Carlo techniques. Randomized quasiMonte Carlo methods provide a basis for error estimation. They have, in the special case of scrambled nets, also been observed to improve accuracy. Finally through Latin super ..."
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Cited by 15 (0 self)
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This paper surveys recent research on using Monte Carlo techniques to improve quasiMonte Carlo techniques. Randomized quasiMonte Carlo methods provide a basis for error estimation. They have, in the special case of scrambled nets, also been observed to improve accuracy. Finally through Latin supercube sampling it is possible to use Monte Carlo methods to extend quasiMonte Carlo methods to higher dimensional problems. 1 INTRODUCTION The problem we consider is the estimation of an integral I = Z [0;1] d f(x)dx: (1) Standard manipulations can be applied to express integrals over domains other than the unit cube or with respect to nonuniform measures in the form (1). Similarly, the integrand f in (1) subsumes weighting functions from importance sampling or periodization. We are especially interested in cases where the dimension d is large, and some of the methods considered here apply to the case d = 1. The focus of this article is on ways of combining Monte Carlo and quasiMo...
Randomized Polynomial Lattice Rules For Multivariate Integration And Simulation
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 2001
"... Lattice rules are among the best methods to estimate integrals in a large number of dimensions. They are part of the quasiMonte Carlo set of tools. A new class of lattice rules, defined in a space of polynomials with coefficients in a finite field, is introduced in this paper, and a theoretical fra ..."
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Cited by 12 (3 self)
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Lattice rules are among the best methods to estimate integrals in a large number of dimensions. They are part of the quasiMonte Carlo set of tools. A new class of lattice rules, defined in a space of polynomials with coefficients in a finite field, is introduced in this paper, and a theoretical framework for these polynomial lattice rules is developed. A randomized version is studied, implementations and criteria for selecting the parameters are discussed, and examples of its use as a variance reduction tool in stochastic simulation are provided. Certain types of digital net constructions, as well as point sets constructed by taking all vectors of successive output values produced by a Tausworthe random number generator, turn out to be special cases of this method.
Fast Generation of Randomized LowDiscrepancy Point Sets
, 2001
"... We introduce two novel techniques for speeding up the generation of digital (t,s)sequences. Based on these results a new algorithm for the construction of Owen's randomly permuted (t,s)sequences is developed and analyzed. An implementation is available at http://www.mcqmc.org/Software.html. ..."
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Cited by 12 (1 self)
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We introduce two novel techniques for speeding up the generation of digital (t,s)sequences. Based on these results a new algorithm for the construction of Owen's randomly permuted (t,s)sequences is developed and analyzed. An implementation is available at http://www.mcqmc.org/Software.html.
Probabilistic error bounds for simulation quantile estimators
 Management Science
, 2003
"... Quantile estimation has become increasingly important,particularly in the financial industry,where value at risk (VaR) has emerged as a standard measurement tool for controlling portfolio risk. In this paper,we analyze the probability that a simulationbased quantile estimator fails to lie in a pres ..."
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Cited by 11 (1 self)
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Quantile estimation has become increasingly important,particularly in the financial industry,where value at risk (VaR) has emerged as a standard measurement tool for controlling portfolio risk. In this paper,we analyze the probability that a simulationbased quantile estimator fails to lie in a prespecified neighborhood of the true quantile. First,we show that this error probability converges to zero exponentially fast with sample size for negatively dependent sampling. Then we consider stratified quantile estimators and show that the error probability for these estimators can be guaranteed to be 0 with sufficiently large,but finite,sample size. These estimators,however,require sample sizes that grow exponentially in the problem dimension. Numerical experiments on a simple VaR example illustrate the potential for variance reduction.
On rates of convergence for stochastic optimization problems under nonI.I.D. sampling
, 2006
"... In this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations. Our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size. This is a well studied problem in case the samples are ..."
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Cited by 11 (1 self)
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In this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations. Our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size. This is a well studied problem in case the samples are independent and identically distributed (i.e., when standard Monte Carlo is used); here, we study the case where that assumption is dropped. Broadly speaking, our results show that, under appropriate assumptions, the rates of convergence for pointwise estimators under a sampling scheme carry over to the optimization case, in the sense that convergence of approximating optimal solutions and optimal values to their true counterparts has the same rates as in pointwise estimation. Our motivation for the study arises from two types of sampling methods that have been widely used in the Statistics literature. One is Latin Hypercube Sampling (LHS), a stratified sampling method originally proposed in the seventies by McKay, Beckman, and Conover (1979). The other is the class of quasiMonte Carlo (QMC) methods, which have become popular especially after the work of Niederreiter (1992). The advantage of such methods is that they typically yield pointwise estimators which not only have lower variance than standard Monte Carlo but also possess better rates of convergence. Thus, it is important to study the use of these techniques in samplingbased optimization. The novelty of our work arises from the fact that, while there has been some work on the use of variance reduction techniques and QMC methods in stochastic optimization, none of the existing work — to the best of our knowledge — has provided a theoretical study on the effect of these techniques on rates of convergence for the optimization problem. We present numerical results for some twostage stochastic programs from the literature to illustrate the discussed ideas.
QuasiMonte Carlo Via Linear ShiftRegister Sequences
, 1999
"... Linear recurrences modulo 2 with long periods have been widely used for contructing (pseudo)random number generators. Here, we use them for quasiMonte Carlo integration over the unit hypercube. Any stochastic simulation fits this framework. The idea is to choose a recurrence with a short period leng ..."
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Cited by 10 (2 self)
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Linear recurrences modulo 2 with long periods have been widely used for contructing (pseudo)random number generators. Here, we use them for quasiMonte Carlo integration over the unit hypercube. Any stochastic simulation fits this framework. The idea is to choose a recurrence with a short period length and to estimate the integral by the average value of the integrand over all vectors of successive output values produced by the small generator. We examine randomizations of this scheme, discuss criteria for selecting the parameters, and provide examples. This approach can be viewed as a polynomial version of lattice rules.