ABSTRACT
Sequential tests for composite hypotheses in general form a topic of fairly complex nature. In order to review some of these advanced topics, one may look at the books of Wald (1947), Ghosh (1970), Govindarajulu (1981) and Siegmund (1985), among others. Our objective here is very modest. We simply want to expose the readers to sequential tests of composite hypotheses, only in a few special but interesting cases with the help of examples. We deliberately do not discuss any optimality issues. We include sequential tests for (i) σ2 in a N(µ, σ2) distribution, (ii) µ in a N(µ, σ2) distribution when σ2 is unknown, (iii) the correlation coefficient ρ in a bivariate normal distribution with all parameters unknown and (iv) the shape parameter p in a gamma distribution when the scale parameter µ is unknown. In the literature, sequential analogs of the t-test, χ2-text and F -test are
available. Ideas of sequential t (and t2) tests were developed by Rushton (1950), Arnold (1951), Hoel (1954), Ghosh (1960a,b) and Sacks (1965). The sequential χ2, t and F tests can be viewed as special cases from a larger class of tests known as the invariant SPRTs. Hall et al. (1965) and Wijsman (1971) gave excellent accounts of this area. Details can be obtained from Ghosh (1970), Govindarajulu (1981) and Siegmund (1985), among others. We merely focus on building some appreciation of the breadth of useful problems which can be resolved with sequential sampling.
