### 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. 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 language | English (US) |
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Pages (from-to) | 807-830 |

Number of pages | 24 |

Journal | Advances in Applied Probability |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1998 |

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Applied Mathematics