The distribution of Brownian motion on linear stopping boundaries. (English) Zbl 0884.62093
Sequential Anal. 16, No. 4, 345-352 (1997); addendum ibid. 17, No. 1, 123-124 (1998).
Summary: We consider Brownian motion with drift and stopping boundaries: linear upper and lower boundaries, and possibly a vertical boundary at a truncation point, all under conditions assuring a finite stopping time. T. W. Anderson [Ann. Math. Stat. 31, 165-197 (1960; Zbl 0089.35501)] derived formulas for the distributions of the stopped process along these boundaries and for the associated expected stopping times. We present simpler formulas, and briefer derivations.
MSC:
62M02 | Markov processes: hypothesis testing |
62L10 | Sequential statistical analysis |
62E15 | Exact distribution theory in statistics |
60J65 | Brownian motion |
60G40 | Stopping times; optimal stopping problems; gambling theory |
Keywords:
sequential tests; Brownian bridge; likelihood ratio identity; inverse Gaussian distributionCitations:
Zbl 0089.35501References:
[1] | DOI: 10.1214/aoms/1177705996 · Zbl 0089.35501 · doi:10.1214/aoms/1177705996 |
[2] | Aritage P., Sequential Medical Trials (1975) |
[3] | Chhikara R.S., The Inverse Gaussian Distribution: Theory, Methodology and Applications (1989) · Zbl 0701.62009 |
[4] | Freedman D., Brownian Motion and Diffusion (1971) · Zbl 0501.60070 |
[5] | Liu A., Uniform minimum variance unbiased estimation of the drift of Brownian motion with linear boundaries (1989) |
[6] | Siegmund D., Sequential Analysis Tests and Confidence Intervals (1985) · Zbl 0573.62071 |
[7] | Whitehead J., The Design and Analysis of Sequential Clinical Trials (1992) · Zbl 0747.62109 |
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