A Sequential Framework for Composite Hypothesis Testing

This paper addresses the problem of classification via composite sequential hypothesis testing. We focus on two possible schemes for the hypotheses: non-nested and nested. For the first case, we present the generalized sequential probability ratio test (GSPRT) and provide an analysis of its asymptotic optimality. Yet, for the nested case, this algorithm is shown to be inconsistent. Consequently, an alternative solution is derived based on Bayesian considerations, similar to the ones used for the Bayesian information criterion and asymptotic maximum a posteriori probability criterion for model order selection. The proposed test, named penalized GSPRT (PGSPRT), is based on restraining the exponential growth of the GSPRT with respect to the sequential probability ratio test. Furthermore, the commonly used performance measure for sequential tests, known as the average sample number, is evaluated for the PGSPRT under each of the hypotheses. Simulations are carried out to compare the performance measures of the proposed algorithms for two nested model order selection problems.