A Combinatorial Approach for Analyzing Start-Up Demonstration Tests

There exist various approaches to evaluate the statistical properties of a set of start-up demonstration tests. Among them is the function generating method and the Markov chain imbedding approach. A different approach stems from using basic combinatorics. It has been presented in various articles and it is intended here to give a short summary. Various criteria are suggested for determining whether to accept or reject the tested equipment. These include the older and simpler ones like CSTF (continuous successes total failures) according to which acceptance is achieved if a certain length of run of successes comes before a specified total number of failures, and vice versa for rejection. A more advanced test criterion is for instance TSCSTFCF (total successes continuous successes total failures continuous failures) where the equipment is accepted if either there appears a certain length of run of successes or a specified total number of successes is encountered before a pre-specified length of a run of failures and a certain total number of failures. It is rejected if the opposite case is observed. Using a combinatorial approach, features like the expected number of tests and the probability of accepting the tested unit are evaluated. Further on, an optimization procedure for determining the various parameters that are involved in the process will be presented. The reasonable objective of minimizing the number of required tests is considered. This is carried out subject to confidence level constraints. The theory has been expanded to include multi-valued tests and to possibly assuming some dependence between the tests. A recent generalization to the two-dimensional case will also be included.