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Characterization of optimality in classes of “truncatable” stopping rules. (English) Zbl 1311.62124
Summary: Let \(X_1\), \(X_2,\dots\), \(X_n,\dots\) be a discrete-time stochastic process. The following optimal stopping problem is considered. We observe \(X_1\), \(X_2,\dots\) on the one-by-one basis getting successively the data \(x_1\), \(x_2,\dots\). At each stage \(n\), \(n=1,2,\dots\), after the data \(x_1,\dots,x_n\) have been observed, we may stop, and if we stop, we gain \(g_n(x_1,\dots,x_n)\). In this article, we characterize the structure of all optimal (randomized) stopping times \(\tau\) that maximize the average gain value \(G(\tau )=E g_\tau (X_1,\dots , X_\tau )\) in some natural classes of stopping times \(\tau\) we call truncatable: \(\tau \) is called truncatable if \(G(\tau \wedge N)\to G(\tau )\) as \(N\to \infty\). It is shown that under some additional conditions on the structure of \(g_n\) (suitable for statistical applications) every finite (with probability 1) stopping time is truncatable.
MSC:
62L10 Sequential statistical analysis
62L15 Optimal stopping in statistics
62C10 Bayesian problems; characterization of Bayes procedures
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