On approximation of optimal stopping of Bayesian sequential test for a normal mean (Q1891678)

We present a simple and direct approach in which supermartingales are used to approximate the optimal stopping sets associated with the Bayesian sequential test for normal population means. Serveral conclusions are given. The sequential tests for normal population means often arise from or are created through approximations or transformations in practice. A standard case is as follows. Let \(X_ 1, X_ 2, X_ 3,\dots\) be a sequence of mutually independent observations from a normal population \(X\) with distribution \(N(\theta, \sigma^ 2)\). We want to test \(H_ 0: \theta\leq 0\) versus \(H_ 1: \theta>0\). Suppose that the cost of a wrong decision is \(r(\theta)= k| \theta|\), and the cost of sampling by time \(n\) is \(c_ 0 n\). When does the parameter \(\theta\) have the prior distribution \(N(\theta_ 0, \sigma^ 2_ 0)\), and what is the optimal Bayes sequential strategy (provided \(\sigma^ 2\), \(\sigma^ 2_ 0\), \(\theta_ 0\), \(k\) and \(c_ 0\) are known)? The numerical optimal solution of the problem can directly be obtained via backward induction, but analytic approximation is still necessary. A classical analytic approximation is related to the solution of a heat equation with free boundary. In Section 2, we establish a proposition based on which one can select certain supermartingales to encompass the approximation of the optimal stopping sets directly. This proposition is applied to the Bayesian sequential test for a normal population mean. Certain approximate optimal stopping sets are given in Section 3. Finally we compare our approximations with previous ones.