ABSTRACT
In this chapter we use the notation of Subsection 2.7.1, assuming that θn = θ , where θ is an unknown parameter or more generally a situation (not necessarily random) taking on the two values 0 or 1, so that Θn = Θ = {0,1} and pθ=i(Xn1) = pi(Xn1), i = 0,1 are two distinct densities. The goal is to test two hypotheses H0 : θ = 0 and H1 : θ = 1, i.e., to identify which one of the two densities p0 or p1 is the true density. In applications, often {pθ (Xn1)} is a parametric family, and the hypotheses are of the form H0 : θ = θ0 and H1 : θ = θ1. We would like to test the hypotheses sequentially, taking advantage of the fact that as an observation arrives, our knowledge of the true state of the process becomes more refined, so that we may decide whether or not more data are needed to make a final decision. Accordingly if we decide to stop at the stage n the terminal decision takes on two values dn = 0 or 1, where dn = i means that the hypothesis Hi is accepted and therefore the alternative hypothesis is rejected.
