### 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) |
---|---|

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 |

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### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Applied Mathematics

### Cite this

*Advances in Applied Probability*,

*30*(3), 807-830. https://doi.org/10.1239/aap/1035228130

**A boundary crossing probability for the bessel process.** / Betensky, Rebecca.

Research output: Contribution to journal › Article

*Advances in Applied Probability*, vol. 30, no. 3, pp. 807-830. https://doi.org/10.1239/aap/1035228130

}

TY - JOUR

T1 - A boundary crossing probability for the bessel process

AU - Betensky, Rebecca

PY - 1998/1/1

Y1 - 1998/1/1

N2 - 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.

AB - 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.

KW - First passage time

KW - O’Brien–Fleming test

KW - Sequential test

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UR - http://www.scopus.com/inward/citedby.url?scp=85037922038&partnerID=8YFLogxK

U2 - 10.1239/aap/1035228130

DO - 10.1239/aap/1035228130

M3 - Article

VL - 30

SP - 807

EP - 830

JO - Advances in Applied Probability

JF - Advances in Applied Probability

SN - 0001-8678

IS - 3

ER -