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A boundary crossing probability for the Bessel process

Part of: Stochastic processes Sequential methods

Published online by Cambridge University Press:  01 July 2016

Rebecca A. Betensky
Rebecca A. Betensky*
Affiliation:
Harvard School of Public Health
*
∗ Postal address: Department of Biostatistics, Harvard School of Public Health, 677 Huntington Avenue, Boston, MA 02115, USA. Email address: [email protected]
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Abstract

Analytic approximations are derived for the distribution of the first crossing time of a straight-line boundary by a d-dimensional Bessel process and its discrete time analogue. The main ingredient for the approximations is the conditional probability that the process crossed the boundary before time m, given its location beneath the boundary at time m. The boundary crossing probability is of interest as the significance level and power of a sequential test comparing d+1 treatments using an O'Brien-Fleming (1979) stopping boundary (see Betensky 1996). Also, it is shown by DeLong (1980) to be the limiting distribution of a nonparametric test statistic for multiple regression. The approximations are compared with exact values from the literature and with values from a Monte Carlo simulation.

Keywords

O'Brien-Fleming testsequential testfirst passage time

MSC classification

Secondary: 60G99: None of the above, but in this section 62L10: Sequential analysis
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 807 - 830
Copyright
Copyright © Applied Probability Trust 1998 

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References

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