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Monte Carlo pp 493-586 | Cite as

Designing and Analyzing Sample Paths

  • George S. Fishman
Part of the Springer Series in Operations Research book series (ORFE)

Abstract

As in Ch. 5, we take as our objective the estimation of
$$\mu = \mu \left( g \right) = \int_X {g\left( {\text{x}} \right){\text{d}}F\left( {\text{x}} \right)} $$
(1)
, where F denotes an m-dimensional d.f. on \(X \subseteq {\mathbb{R}^m}\). Consider a Monte Carlo Markov sampling experiment composed of n independent replications, each of which begins in a state drawn from an initializing nonequilibrium distribution π0. After a warm-up interval of k — 1 steps on each replication, sampling continues for t additional steps and one uses the n independent truncated sample paths or realizations, each of length t, to estimate µ. Whereas Ch. 5 concentrates on sample path generating algorithms and a conceptual understanding of convergence to an equilibrium state, this chapter focuses on sampling plan design and statistical inference. With regard to design, the chapter shows how the choices of k, n, π0, and t affect computational and statistical efficiency. With regard to statistical inference, it describes methods for estimating the warm-up interval k that significantly mitigate the influence of the initial states drawn from the nonequilibrium distribution π0. Also, it describes methods for computing asymptotically valid confidence intervals for µ in expression (1) as n → ∞ for fixed t, as t → ∞ for fixed n and as both n → ∞ and t → ∞. Because confidence intervals inevitably depend on variance estimates, we need to impose a moderately stronger restriction on g. Whereas the assumption \(\int_X {{g^2}\left( {\text{x}} \right){\text{d}}F\left( {\text{x}} \right)} < \infty \) in Ch. 5 guarantees a finite variance for a single observation on a sample path, the assumption \(\int_X {{g^4}\left( {\text{x}} \right){\text{d}}F\left( {\text{x}} \right)} < \infty \) is necessary for us to obtain consistent estimators of that variance and of other variances that play essential roles in the derivation of asymptotically valid confidence intervals for µ.

Keywords

Sample Path Batch Size Percent Confidence Interval Independent Replication Autocovariance Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • George S. Fishman
    • 1
  1. 1.Department of Operations ResearchUniversity of North CarolinaChapel HillUSA

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