Local asymptotic efficiency of a sequential probability ratio test for d-guarantee discrimination of composite hypotheses

Abstract

A sequential Wald test for discrimination of two simple hypotheses θ = θ1 and θ = θ2 with boundaries A and B is applied to distinguish composite hypotheses θ < θ0 and θ > θ0, the parameters θ1, θ2, A, and B being chosen in such a way that d-posteriori probabilities of errors do not exceed the given restrictions β0 and β1. An asymptotic behavior of boundaries A, B and the average observation time are studied when β= max{β0, β1} → 0. An asymptotic (β → 0) comparison is made of Eθv with the least given number of observations necessary for discrimination of composite hypotheses with the same restrictions β0, β1 on d-posteriori probabilities of errors. It is shown that the minimum (in a neighborhood of the point θ = θ0) gain of the average observation time makes up 25%. Therefore, there are sequential tests within the bounds of a d-posteriori approach that give a gain in the size of observations for every value of a parameter tested.