Buying with exact confidence. (English) Zbl 0753.62051
The following stopping time problem arises e.g. in software testing: let \((X_ k)\) be a sequence of independent times with known common (continuous) distribution \(F\). The \(X_ k\) represent the times at which events occur and a finite number of such times is observed. Let \(K(t)\) be the number of events observed before \(t\), \(K(t)=\) number of indices \(\{j\mid X_ j\leq t\}\). Then consider an increasing sequence \((b_ k)\) and stop at \(\tau=b_ J\) where \(J=\) smallest \(j\) with \(K(b_ j)<j\). Then for \(m\geq 0\), \(\alpha\in(0,1)\), \((b_ j)\) can be chosen such that for all \(n>m\), \(P(n-K(\tau)>m\mid n)=\alpha\).
In other words, if the random variable \(N\) represents the number of events with distribution supported on \((m+1,\infty)\), then \(P(N- K(\tau)>m)=\alpha\). The properties of this stopping time are studied and compared to earlier results. Finally, numerical examples are given.
In other words, if the random variable \(N\) represents the number of events with distribution supported on \((m+1,\infty)\), then \(P(N- K(\tau)>m)=\alpha\). The properties of this stopping time are studied and compared to earlier results. Finally, numerical examples are given.
Reviewer: M.Kohlmann (St.Augustin)
MSC:
62L15 | Optimal stopping in statistics |
60G40 | Stopping times; optimal stopping problems; gambling theory |
62P30 | Applications of statistics in engineering and industry; control charts |
62G30 | Order statistics; empirical distribution functions |
62N99 | Survival analysis and censored data |