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On the theoretical comparison of low-bias steady-state estimators

Published: 01 January 2007 Publication History

Abstract

The time-average estimator is typically biased in the context of steady-state simulation, and its bias is of order 1/t, where t represents simulated time. Several “low-bias” estimators have been developed that have a lower order bias, and, to first-order, the same variance of the time-average. We argue that this kind of first-order comparison is insufficient, and that a second-order asymptotic expansion of the mean square error (MSE) of the estimators is needed. We provide such an expansion for the time-average estimator in both the Markov and regenerative settings. Additionally, we provide a full bias expansion and a second-order MSE expansion for the Meketon--Heidelberger low-bias estimator, and show that its MSE can be asymptotically higher or lower than that of the time-average depending on the problem. The situation is different in the context of parallel steady-state simulation, where a reduction in bias that leaves the first-order variance unaffected is arguably an improvement in performance.

References

[1]
Beale, E. M. L. 1962. Some use of computers in operational research. Indust. Org. 31, 27--28.
[2]
Breiman, L. 1968. Probability. Addison-Wesley series in Statistics. Addison-Wesley, Reading, MA.
[3]
Brémaud, P. 1999. Markov Chains, Gibbs Fields, Monte Carlo Simulation and Queues. Springer-Verlag, New York.
[4]
Down, D., Meyn, S., and Tweedie, R. 1995. Exponential and uniform ergodicity of Markov processes. Ann. Prob. 23, 4, 1671--1691.
[5]
Durbin, J. 1959. A note on the application of Quenouille's method of bias reduction to the estimation of ratios. Biometrika 46, 477--480.
[6]
Durrett, R. 1995. Probability: Theory and Examples, 2nd ed. Duxbury Press, Belmont, CA.
[7]
Fieller, E. C. 1940. The biological standarization of insulin. J. Roy. Stat. Soc. Ser. B 7, 1--64.
[8]
Glynn, P. W. 1987. Limit theorems for the method of replications. Commun. Statist. Stoch. Models 3, 3, 343--355.
[9]
Glynn, P. W. 1989. A GSMP formalism for discrete-event systems. Proc. IEEE 77, 1, 14--23.
[10]
Glynn, P. W. 1994. Some topics in regenerative steady-state simulation. Acta Appl. Math. 34, 1-2, 225--236.
[11]
Glynn, P. W. and Heidelberger, P. 1990. Bias properties of budget constrained simulation. Oper. Res. 38, 5, 801--814.
[12]
Glynn, P. W. and Heidelberger, P. 1991. Analysis of initial transient deletion for replicated steady-state simulations. Oper. Res. Lett. 10, 8, 437--443.
[13]
Glynn, P. W. and Heidelberger, P. 1992a. Analysis of initial transient deletion for parallel steady-state simulations. SIAM J. Sci. Stat. Comput. 13, 4, 904--922.
[14]
Glynn, P. W. and Heidelberger, P. 1992b. Jackknifing under a budget constraint. ORSA J. Comput. 4, 3, 226--234.
[15]
Glynn, P. W. and Whitt, W. 1992. The asymptotic efficiency of simulation estimators. Oper. Res. 40, 3, 505--520.
[16]
Heidelberger, P. 1988. Discrete event simulation and parallel processing: Statistical properties. SIAM J. Sci. Stat. Comput. 9, 1114--1132.
[17]
Henderson, S. G. and Glynn, P. W. 2001. Regenerative steady-state simulation of discrete-event systems. ACM Trans. Model. Comput. Simul. 11, 4, 313--345.
[18]
Henderson, S. G. and Glynn, P. W. 2003. Nonexistence of a class of variate generation schemes. Oper. Res. Lett. 31, 2, 83--89.
[19]
Hsieh, M.-H., Iglehart, D. L., and Glynn, P. W. 2004. Empirical performance of bias-reducing estimators for regenerative steady-state simulations. ACM Trans. Model. Comput. Simul. 14, 4, 325--343.
[20]
Iglehart, D. L. 1975. Simulating stable stochastic systems V. Comparison of ratio estimators. Naval Res. Logist. 22, 3, 553--565.
[21]
Karlin, S. and Taylor, H. M. 1975. A First Course in Stochastic Processes, 2nd ed. Academic Press, San Diego, CA.
[22]
Meketon, M. S. and Heidelberger, P. 1982. A renewal theoretic approach to bias reduction in regenerative simulations. Manage. Sci. 28, 2, 173--181.
[23]
Meyn, S. and Kontoyiannis, I. 2003. Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Prob. 13, 1, 304--362.
[24]
Meyn, S. and Tweedie, R. 1993. Markov Chains and Stochastic Stability. Communications and Control Engineering series. Springer-Verlag, New York.
[25]
Nummelin, E. and Tuominen, P. 1983. The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory. Stochas. Proc. Their Appl. 15, 295--311.
[26]
Quenouille, M. 1956. Notes on bias in estimation. Biometrika 43, 3-4, 353--360.
[27]
Resnick, S. I. 1992. Adventures in Stochastic Processes. Birkhäuser, Boston, MA.
[28]
Rogers, L. and Williams, D. 1994. Diffusions, Markov Processes, and Martingales, 2nd ed. Vol. 1. Wiley, New York.
[29]
Stroock, D. W. 2005. An Introduction to Markov Processes. Springer-Verlag, Berlin, Germany.
[30]
Thorisson, H. 2000. Coupling, Stationarity, and Regeneration. Springer-Verlag, New York.
[31]
Tin, M. 1965. Comparison of some ratio estimators. J. ASA 60, 309, 294--307.

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Published In

cover image ACM Transactions on Modeling and Computer Simulation
ACM Transactions on Modeling and Computer Simulation  Volume 17, Issue 1
January 2007
84 pages
ISSN:1049-3301
EISSN:1558-1195
DOI:10.1145/1189756
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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 January 2007
Published in TOMACS Volume 17, Issue 1

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Author Tags

  1. Low-bias estimators
  2. mean-square error expansion
  3. steady-state simulation

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