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Keywords:
sequential hypotheses test; simple hypothesis; optimal stopping; sequential probability ratio test; likelihood ratio statistic; stability inequality
sequential hypotheses test; simple hypothesis; optimal stopping; sequential probability ratio test; likelihood ratio statistic; stability inequality
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, it is used a weighted sum of the average (under $f_0$) sample number and the two types error probabilities. 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 $(f_0,\tilde f_1)$.
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