Boundary crossing probabilities for the cumulative sample mean. (English) Zbl 1411.60048
Summary: We develop a new measure of reliability for the mean behavior of a process by calculating the probability the cumulative sample mean will stay within a given distance from the true mean over a period of time. This probability is derived using boundary-crossing properties of Brownian bridges. We derive finite sample results for independent and identically distributed normal data, limiting results for data meeting a functional central limit theorem, and draw parallels to standard normal confidence intervals. We deliver numerical results for i.i.d., dependent, and queueing processes.
MSC:
| 60F17 | Functional limit theorems; invariance principles |
| 60K10 | Applications of renewal theory (reliability, demand theory, etc.) |
| 62E20 | Asymptotic distribution theory in statistics |
| 62G15 | Nonparametric tolerance and confidence regions |
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