Kendall, Wilfrid S., Marin, Jean-Michel and Robert, Christian P., 1961- (2005) Brownian confidence bands on Monte Carlo output. Working Paper. University of Warwick. Centre for Research in Statistical Methodology, Coventry.

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Abstract

When considering a Monte Carlo estimation procedure, the path produced by successive partial estimates is often used as a guide for informal convergence diagnostics. However the confidence region associated with that path cannot be derived simplistically from the confidence interval for the estimate itself. An asymptotically correct approach can be based on the Brownian motion approximation of the path, but no exact formula for the corresponding area-minimizing confidence region is yet known. We construct proxy regions based on local time arguments and consider numerical approximations. These are then available for a more incisive assessment of the Monte Carlo procedure and thence of the estimate itself.

Item Type:	Working or Discussion Paper (Working Paper)
Alternative Title:	Confidence bands for Brownian motion and applications to Monte Carlo simulation
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Monte Carlo method, Brownian motion processes
Series Name:	Working papers
Publisher:	University of Warwick. Centre for Research in Statistical Methodology
Place of Publication:	Coventry
Date:	2005
Volume:	Vol.2005
Number:	No.2
Number of Pages:	18
Status:	Not Peer Reviewed
Access rights to Published version:	Open Access
References:	Anderson, T. (1960). A Modification of the Sequential Probability Ratio Test to Reduce the Sample Size. Ann. Math. Statist., 31(1):165–197. Billingsley, P. (1995). Probability and Measure. John Wiley, New York, third edition. Borodin, A. and Salminen, P. (2002). Handbook of Brownian motion - facts and formulae. Birkhauser Verlag, 2 edition. Corless, R., Gonnet, G., Hare, E., Jeffrey, D., and Knuth, D. (1996). On the Lambert W function. Adv. Computational Math., 5:329–359. Daniels, H. (1996). Approximating the first crossing-time density for a curved boundary. Bernoulli, 2:133–143. Durbin, J. (1971). Boundary crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the kolmogorov-smirnov test. J. Appl. Prob., 8:431– 453. Durbin, J. (1992). The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob., 29:291–304. Feller, W. (1971). An Introduction to Probability Theory and its Applications, volume 2. John Wiley, 2 edition. Hall, W. (1997). The distribution of Brownian motion on linear stopping boundaries. Sequential Anal., 4:345–352. Lerche, H. R. (1986). Boundary Crossing of Brownian Motion. Lecture Notes in Statistics. Springer- Verlag. Li, W. V. (2003). The first exit time of a Brownian motion from an unbounded convex domain. Ann. Prob., 31(2):1078–1096. Loader, C. and Deely, J. (1987). Computations of Boundary Crossing Probabilities for the Wiener Process. J. Stat. Comp. Simul., 27:96–105. Novikov, A., Frishling, V., and Kordzakhia, N. (1999). Approximation of boundary crossing probabilities for a Brownian motion. J. Appl. Prob., 36:1019–1030. Park, C. and Schuurmann, F. (1976). Evaluation of barrier crossing probabilities for Wiener path. J. Appl. Prob., 17:197–201. Potzelberger, K. and Wang, L. (2001). Boundary crossing probability for Brownian motion. J. Appl. Prob., 38:152–164. Revuz, D. and Yor, M. (1999). Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition. Robbins, H. and Siegmund, D. (1970). Boundary crossing probabilities for the Wiener process and partials sums. Ann. Math. Statist., 38:152–164. Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods. Springer-Verlag, second edition. Wang, L. and Potzelberger, K. (1997). Boundary crossing probability for Brownian motion and general boundaries. J. Appl. Prob., 34:54–65.
URI:	http://wrap.warwick.ac.uk/id/eprint/35579

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