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BOUNDARY CROSSING PROBABILITIES FOR THE CUMULATIVE SAMPLE MEAN

Published online by Cambridge University Press:  08 March 2017

Dashi I. Singham Open the ORCID record for Dashi I. Singham [Opens in a new window]  and
Michael P. Atkinson
Dashi I. Singham
Affiliation:
Operations Research Department, Naval Postgraduate School, Monterey, CA 93943, USA E-mail: dsingham@nps.edu
Michael P. Atkinson
Affiliation:
Operations Research Department, Naval Postgraduate School, Monterey, CA 93943, USA E-mail: mpatkins@nps.edu
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Abstract

We develop a new measure of reliability for the mean behavior of a process by calculating the probability the cumulative sample mean will stay within a given distance from the true mean over a period of time. This probability is derived using boundary-crossing properties of Brownian bridges. We derive finite sample results for independent and identically distributed normal data, limiting results for data meeting a functional central limit theorem, and draw parallels to standard normal confidence intervals. We deliver numerical results for i.i.d., dependent, and queueing processes.

Keywords

applied probabilityreliability theorysimulationBrownian bridge

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 32 , Issue 2 , April 2018 , pp. 275 - 295
Copyright
Copyright © Cambridge University Press 2017 

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References

1. Abundo, M. (2002). Some conditional crossing results of Brownian motion over a piecewise-linear boundary. Statistics & Probability Letters 58(2): 131–145.CrossRefGoogle Scholar
2. Alexopoulos, C. (2006). A comprehensive review of methods for simulation output analysis. In Perrone, L.P., Wieland, F.P., Liu, J., Lawson, B.G., Nicol, D.M., & Fujimoto, R.M. (eds.), Proceedings of the 2006 Winter Simulation Conference. Piscataway, New Jersey: Institute of Electrical and Electronics Engineers, Inc., pp. 168–178.CrossRefGoogle Scholar
3. Atkinson, M.P. & Singham, D.I. (2015). Multidimensional hitting time results for Brownian bridges with moving hyperplanar boundaries. Statistics and Probability Letters 100: 85–92.CrossRefGoogle Scholar
4. Batur, D., Goldsman, D., & Kim, S.-H. (2009). An improved standardized time series Durbin–Watson variance estimator for steady-state simulation. Operations Research Letters 37(4): 285–289.CrossRefGoogle Scholar
5. Billingsley, P. (1968). Convergence of probability measures. New York: Wiley.Google Scholar
6. Doob, J.L. (1949). Heuristic approach to the Kolmogorov–Smirnov theorems. The Annals of Mathematical Statistics 20(3): 393–403.CrossRefGoogle Scholar
7. Fu, J.C. & Wu, T.-L. (2010). Linear and nonlinear boundary crossing probabilities for Brownian motion and related processes. Journal of Applied Probability 47(4): 1058–1071.CrossRefGoogle Scholar
8. Glynn, P.W. & Iglehart, D.L. (1990). Simulation output analysis using standardized time series. Mathematics of Operations Research 15(1): 1–16.CrossRefGoogle Scholar
9. Goldsman, D., Meketon, M., & Schruben, L. (1990). Properties of standardized time series weighted area variance estimators. Management Science 36(5): 602–612.CrossRefGoogle Scholar
10. Karatzas, I. & Shreve, S. (1998). Brownian motion and stochastic calculus. New York, NY: Springer.CrossRefGoogle Scholar
11. Klenke, A. (2013). Probability theory: a comprehensive course. London: Springer Science & Business Media.Google Scholar
12. Lerche, H.R. & Siegmund, D. (1987). Approximate Exit Probabilities for a Brownian Bridge on a Short Time Interval, and Applications. Technical report, Stanford University.Google Scholar
13. Nagaev, S.V. (1970). On the speed of convergence in a boundary problem. I. Theory of Probability & Its Applications 15(2): 163–186.CrossRefGoogle Scholar
14. Scheike, T.H. (1992). A boundary-crossing result for Brownian motion. Journal of Applied Probability 29(2): 448–453.Google Scholar
15. Schruben, L. (1983). Confidence interval estimation using standardized time series. Operations Research 31(6): 1090–1108.CrossRefGoogle Scholar
16. Singham, D.I. & Atkinson, M.P. (2015). Construction of Cumulative Mean Bounds for Simulation Output Analysis. Technical Report NPS-OR-15-004, Naval Postgraduate School.Google Scholar
17. Tafazzoli, A. & Wilson, J.R. (2010). Skart: a skewness-and autoregression-adjusted batch-means procedure for simulation analysis. IIE Transactions 43(2): 110–128.Google Scholar
18. Whitt, W. (2002). Stochastic-process limits: an introduction to stochastic-process limits and their application to queues. New York, NY: Springer Science & Business Media.Google Scholar