736 IEEE TRANSACTIONS ON RELIABILITY, VOL. 62, NO. 3, SEPTEMBER 2013 A Start-Up Demonstration Test Based on Exchangeable Binary Trials Athanasios C. Rakitzis and Demetrios L. Antzoulakos Abstract—We investigate several aspects of the recently intro- duced consecutive successes distance failures start-up demonstra- tion test under an exchangeable model. By assuming that the prob- ability of a successful start-up attempt is a random variable, in- stead of a fixed value, the outcomes of the successive start-ups be- come -dependent exchangeable binary random variables. From a practical point of view, this is a more realistic model than the or- dinary model of -independent start-ups. Critical quantities of the test such as the expected length of the test, the probability of ac- ceptance of the equipment under test, as well as the distribution of the length of the test, are derived. Illustrative numerical examples based on a Beta-mixing distribution for , and comparisons with the corresponding i.i.d. model, are presented. Furthermore, a com- parison study is performed between the consecutive successes dis- tance failures and other competitive start-up demonstration tests. Finally, inferential procedures for the estimation of the unknown parameter(s) of the Beta-mixing distribution are also discussed. Index Terms—Exchangeable random variables, maximum likeli- hood estimation, run statistics, scan statistics, start-up demonstra- tion tests, statistical inference, waiting time distribution. ACRONYM cdf cumulative distribution function CS consecutive successes CSDF consecutive successes distance failures CSTF consecutive successes total failures CSCF consecutive successes consecutive failures TSTF total successes total failures i.i.d. -independent and identically distributed MLE maximum likelihood estimation (estimator) r.v. random variable(s) pgf probability generating function pmf probability mass function SDT start-up demonstration test Manuscript received June 29, 2012; revised December 25, 2012; accepted April 04, 2013. Date of publication July 02, 2013; date of current version August 28, 2013. Associate Editor: R. H. Yeh. A. C. Rakitzis is with the Department of Mathematics and Statistics, Univer- sity of Cyprus, 1678 Nicosia, Cyprus (e-mail: arakitz@ucy.ac.cy). D. L. Antzoulakos is with the Department of Statistics and Insurance Science, University of Piraeus, Piraeus 18534, Greece (e-mail: dantz@unipi.gr). Digital Object Identifier 10.1109/TR.2013.2270428 NOTATION outcome of the th start-up number of consecutive successful start-ups required for the acceptance of the equipment maximum distance between two unsuccessful start-ups required for the rejection of the equipment probability of a successful start-up parameters of the Beta distribution number of start-up attempts until termination of the test number of start-up attempts until termination of the test with acceptance of the equipment number of start-up attempts until termination of the test with rejection of the equipment probability of accepting the equipment number of start-up attempts until termination of the test given that the equipment is accepted number of start-up attempts until termination of the test given that the equipment is rejected probability mass function of probability generating function of expected number of start-up attempts until the termination of the test standard deviation of the number of start-ups until the termination of the test I. INTRODUCTION A start-up demonstration test (SDT) is a procedure by which a vendor demonstrates to a customer the reliability of equipment with regard to its starting. Successive start-up at- tempts are conducted, and the decision whether to accept or re- ject the equipment is taken according to specific stopping rules. In the early research on start-up demonstration testing, as dis- cussed by Hahn and Gage [11], the equipment is accepted as soon as a pre-specified number of consecutive successful start-ups occur in a series of attempted start-ups of the equip- ment (consecutive successes (CS) scheme). They considered the case where the individual start-ups are i.i.d. ( -independent and identically distributed) with successful start-up probability . Viveros and Balakrishnan [19] derived further results for the 0018-9529 © 2013 IEEE