Buying with exact confidence
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.
scientific article; zbMATH DE number 97558