On generalized multi-state start-up demonstration tests

In view of the fact that start-up results of products are usually multi-state, based on the traditional two-state start-up demonstration tests, such as consecutive successes total failures (CSTF), total successes consecutive failures (TSCF), consecutive successes consecutive failures (CSCF) and total successes total failures (TSTF), four multi-state start-up demonstration tests are proposed in this paper. By using finite Markov chain imbedding approach, the acceptance and rejection probabilities, the probability mass function, the distribution function and the expectation of the number of the start-up tests to be terminated are given. We also provide a procedure to select the optimal parameter values. Besides, the estimations of possibly unknown probabilities are given by using maximum likelihood estimation. Finally, a numerical example that contains two tables is given to illustrate the advantages of multi-state start-up demonstration tests. The first table is presented to illustrate that the multi-state start-up demonstration tests are superior to two-state start-up demonstration tests. The second one is to illustrate that the four-state models (proposed in this paper) are superior to CSTF and CS ⁽¹⁾ CS ^(1,2) TF (proposed by Smith and Griffith) with the same values of α and β set by Smith and Griffith.