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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 114 (32 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 36 (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 MultiLevel CrossClassified Model for Discrete Response Variables
 Transportation Research Part B
, 2000
"... In many spatial analysis contexts, the variable of interest is discrete and there is spatial clustering of observations. This paper formulates a model that accommodates clustering along more than one dimension in the context of a discrete response variable. For example, in a travel mode choice conte ..."
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Cited by 32 (11 self)
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In many spatial analysis contexts, the variable of interest is discrete and there is spatial clustering of observations. This paper formulates a model that accommodates clustering along more than one dimension in the context of a discrete response variable. For example, in a travel mode choice context, individuals are clustered by both the home zone in which they live as well as by their work locations. The model formulation takes the form of a mixed logit structure and is estimated by maximum likelihood using a combination of Gaussian quadrature and quasiMonte Carlo simulation techniques. An application to travel mode choice suggests that ignoring the spatial context in which individuals make mode choice decisions can lead to an inferior data fit as well as provide inconsistent evaluations of transportation policy measures.
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 26 (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 20 (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...
SPLITTING FOR RAREEVENT SIMULATION
, 2006
"... Splitting and importance sampling are the two primary techniques to make important rare events happen more frequently in a simulation, and obtain an unbiased estimator with much smaller variance than the standard Monte Carlo estimator. Importance sampling has been discussed and studied in several ar ..."
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Cited by 17 (1 self)
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Splitting and importance sampling are the two primary techniques to make important rare events happen more frequently in a simulation, and obtain an unbiased estimator with much smaller variance than the standard Monte Carlo estimator. Importance sampling has been discussed and studied in several articles presented at the Winter Simulation Conference in the past. A smaller number of WSC articles have examined splitting. In this paper, we review the splitting technique and discuss some of its strengths and limitations from the practical viewpoint. We also introduce improvements in the implementation of the multilevel splitting technique. This is done in a setting where we want to estimate the probability of reaching B before reaching (or returning to) A when starting from a fixed state x0 ∈ B, where A and B are two disjoint subsets of the state space and B is very rarely attained. This problem has several practical applications.
Econometric Choice Formulations: Alternative Model Structures, Estimation Techniques, and Emerging Directions
 ECONOMETRIC MODELS OF CHOICE: FORMULATION AND ESTIMATION
, 2003
"... The last six years since the Austin IATBR conference has been a very fertile period for the germination of new conceptual, theoretical, and computational developments in the field of econometric choice models. There is a sense today of absolute control over the kind of choice behavior structures one ..."
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Cited by 16 (5 self)
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The last six years since the Austin IATBR conference has been a very fertile period for the germination of new conceptual, theoretical, and computational developments in the field of econometric choice models. There is a sense today of absolute control over the kind of choice behavior structures one wants to specify in empirical contexts and a renewed excitement in the field. This paper reviews these recent developments and assembles a list of recent applications of advanced discrete choice models.
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 16 (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.