On Start-Up Demonstration Tests Under Exchangeability

Consecutive successes total failures (CSTF) is a well known start-up demonstration test procedure in which a unit under test is accepted when successive start-up attempts produce a specified number of consecutive successes before a specified number of failures; otherwise the unit is rejected, and in both cases testing (the experiment) is terminated. The CSTF procedure is studied here assuming that the probability of a successful start-up is a random variable. Under this assumption, the outcomes of the attempted start-ups are dependent random variables following an exchangeable model. The unconditional, and the conditional probability distributions of the waiting time (length of the test) are derived and studied in this situation. The first two moments of unconditional waiting time distribution are calculated. Numerical and graphical illustrations are provided, and comparisons are made with the corresponding results for the i.i.d. model.