Search boundaries of the truncated discrete sequential test

Abstract

The theme of this paper is improved planning of binomial sequential probability ratio tests in the context of comparison of two objects as to their time between failures or to failure, assumed to be exponentially distributed. The authors’ earlier works established that the probabilities of I- and II- type errors (α and β) are discrete in character and do not lend themselves to analytical expression. Accordingly, the choice of the optimal parameters for the decision boundaries necessitates a search for extrema in discrete sets. The present work outlines a procedure that involves application of the continued-fractions theory, and permits finding the set of boundary positions in which the test characteristics undergo changes. It was established, that in the domains described in the earlier papers, the relationships of α and β versus these positions are close to planar and – within narrow limits – stepwise. The step sizes are highly variable, so that the standard minimum search procedures are either cumbersome or actually useless. On the basis of these relationships~– and others – a search algorithm is proposed for the optimal test boundaries. An example is presented – planning and implementation of this test in the integrated-circuit industry.