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Stability estimating in optimal sequential hypotheses testing. (English) Zbl 1165.62052
Summary: We study the stability of the classical optimal sequential probability ratio test based on independent, identically distributed observations \(X_{1}, X_{2}, \dots\) when testing two simple hypotheses about their common density \(f: f=f_{0}\) versus \(f=f_{1}\). As a functional to be minimized, a weighted sum of the average (under \(f=f_{0}\)) sample number and the two types of error probabilities is used. We prove that the problem is reduced to stopping time optimization for a ratio process generated by \(X_{1}, X_{2}, \dots\) with the density \(f_{0}\). For \(\tau _{*}\) being the corresponding optimal stopping time we consider a situation when this rule is applied for testing between \(f_{0}\) and an alternative \(\tilde f_{1}\), where \(\tilde f_{1}\) is some approximation to \(f_{1}\). An inequality is obtained which gives an upper bound for the expected cost excess, when \(\tau _{*}\) is used instead of the rule \(\tilde \tau _{*}\) optimal for the pair \((f_{0},\tilde f_{1})\). The inequality found also estimates the difference between the minimal expected costs for optimal tests corresponding to the pairs \((f_{0}, f_{1})\) and \((f_{0},\tilde f_{1})\).
62L10 Sequential statistical analysis
62L15 Optimal stopping in statistics
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