ON COMPARISON OF STOPPING-TIMES IN SEQUENTIAL-PROCEDURES FOR EXPONENTIAL-FAMILIES OF STOCHASTIC-PROCESSES

For curved (k + 1, k)-exponential families of stochastic processes a n atural and often studied sequential procedure is to stop observation w hen a linear combination of the coordinates of the canonical process c rosses a prescribed level. For such procedures the model is, approxima tely or exactly, a non-curved exponential family. Subfamilies of these stopping rules defined by having the same Fisher (expected) informati on are considered. Within a subfamily the Bartlett correction for a po int hypothesis is also constant. Methods for comparing the durations o f the sampling periods for the stopping rules in such a subfamily are discussed. It turns out that some stopping times tend to be smaller th an others. For exponential families of diffusions and of counting proc esses the probability that one such stopping time is smaller than anot her can be given explicity. More generally, an Edgeworth expansion of this probability is given.