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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 $$(f_{0},\tilde f_{1})$$.
##### MSC:
 62L10 Sequential statistical analysis 62L15 Optimal stopping in statistics
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##### References:
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