Summary: A. Wald’s [Sequential analysis. NY: Wiley and Sons (1947; Zbl 0029.15805)] sequential probability ratio test (SPRT) and group sequential probability ratio test (GSPRT) remain relevant in addressing a wide range of practical problems. The area of clinical trials owes a great debt to the theory and methodologies of SPRT and GSPRT. In recent years, there has been a surge of practical applications of these methodologies in many areas including low frequency sonar detection, tracking of signals, early detection of abrupt changes in signals, computer simulations, agricultural sciences, pest management, educational testing, economics, and finance.
But, obviously, there are circumstances where sampling one observation at a time may not be practical. In contexts of monitoring “inventory” or “queues”, observations may appear sequentially but in groups where the group sizes may be ideally treated as random variables themselves. For example, one may sequentially record the number of stopped cars \((M_{i})\) and the number of cars \(\sum_{j=1}^{M_j}X_{ij}\) without “working brake lights” when a traffic signal changes from green to red, \(i=1,2,\dots \). This can be easily accomplished since every “working brake light” must glow bright red when a driver applies brakes. In this example, one notes that (i) it may be reasonable to model \(M_{i}\)’s as random, but (ii) it would appear impossible to record data sequentially on the status of brake lights (that is, \(X_{ij}=0\) or \(1\)) individually for car # 1, and then for car # 2, and so on when a group of cars rush to stop at an intersection!
In order to address these kinds of situations, we start with an analog of the concept of a best ”fixed-sample-size” test based on data \(\{M_{i},X_{i_1},\dots ,X_{i_{M_{i}}}\), \(i=1,\dots ,k\}\). Then, a random sequential probability ratio test (RSPRT) is developed for deciding between a simple null hypothesis and a simple alternative hypothesis with preassigned Type I and II errors \(\alpha ,\beta \). The RSPRT and the best “fixed-sample-size” test with \(k\equiv k_{\min}\) associated with the same errors \(\alpha ,\beta \) are compared. Illustrations of RSPRT include a one-parameter exponential family followed by substantive computer simulations that lead to a broad set of guidelines. Two real applications and data analysis are highlighted.