A boundary crossing probability for the bessel process

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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. Themain 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.

Original languageEnglish (US)
Pages (from-to)807-830
Number of pages24
JournalAdvances in Applied Probability
Issue number3
StatePublished - 1998


  • First passage time
  • O’Brien–Fleming test
  • Sequential test

ASJC Scopus subject areas

  • Statistics and Probability
  • Applied Mathematics


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