Approximations to optimal stopping rules for exponential random variables
Let \(X_ 1,X_ 2,..\). be i.i.d. random variables with finite mean and \(Y_ n=\max(X_ 1,...,X_ n)-cn\) where c is a positive constant. The problem to maximize the reward \(E(Y_{\tau})\) over all stopping rules \(\tau\) with \(E(Y^-_{\tau})<\infty\) has the following solution: the optimal \(\tau^*_ c\) is in\(f\{\) \(n\geq 1: X_ n\geq \gamma \}\) where \(\gamma =const\) such that \(E(X_ 1-\gamma)^+=c\), see the book by \textit{Y. S. Chow, H. Robbins} and \textit{D. Siegmund}, Great expectations: The theory of optimal stopping, pp. 56-58 (1971; Zbl 0233.60044). Evidently, for this answer we must know completely the distribution of \(X_ 1.\) However, it can happen that only partial information about this distribution is available. This is just the case treated in this paper. The author supposes that \(X_ 1\) has an exponential distribution with unknown mean \(\mu\). If the exact solution of the above optimal stopping problem does not exist, then it is natural to look for finding ''good'' approximate solutions. Following this line the author introduces the time \({\hat \tau}_ c=\inf \{n\geq n_ c:\quad X_ n\geq -\bar X_ n\log(c/\bar X_ n)\}\) where \(n_ c\) is a positive integer and \(\bar X_ n=(X_ 1+...+X_ n)/n.\) After establishing some useful properties of \({\hat \tau}{}_ c\) the author proves the asymptotic optimality of the rule \({\hat \tau}{}_ c\) in the following sense: if \(c\to 0\) and \(\delta c^{-\alpha}\leq n_ c=o(c^{-1})\) for some \(\delta>0\) and \(0<\alpha<1\), then \(E(Y_{{\hat \tau}_ c})-E(Y_{\tau^*_ c})\to 0.\)
scientific article; zbMATH DE number 3866423