Kybernetika

We study the stability of the classical optimal sequential probability ratio test based on independent identically distributed observations X₁, X₂, ... when testing two simple hypotheses about their common density f : f=f₀ versus f=f₁. As a functional to be minimized, it is used a weighted sum of the average (under f=f₀) 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 X₁, X₂, ... with the density f₀. For t_* being the corresponding optimal stopping time we consider a situation when this rule is applied for testing between f₀ and an alternative ^~f₁, where ^~f₁ is some approximation to f₁. An inequality is obtained which gives an upper bound for the expected cost excess, when t_* is used instead of the rule ^~t_* optimal for the pair (f₀,^~f₁). The inequality found also estimates the difference between the minimal expected costs for optimal tests corresponding to the pairs (f₀, f₁) and (f₀,^~f₁).