Statistical inference for start-up demonstration tests with rejection of units upon observing d failures

Abstract

In this paper, we consider the statistical inference for the success probability in the case of start-up demonstration tests in which rejection of units is possible when a pre-fixed number of failures is observed before the required number of consecutive successes are achieved for acceptance of the unit. Since the expected value of the stopping time is not a monotone function of the unknown parameter, the method of moments is not useful in this situation. Therefore, we discuss two estimation methods for the success probability: (1) the maximum likelihood estimation (MLE) via the expectation-maximization (EM) algorithm and (2) Bayesian estimation with a beta prior. We examine the small-sample properties of the MLE and Bayesian estimator. Finally, we present an example to illustrate the method of inference discussed here.

Keywords:

• start-up demonstration test
• maximum likelihood estimator
• EM-algorithm
• runs
• Bayesian estimation
• probability generating function

Acknowledgements

We express our sincere thanks to the editor and the anonymous reviewer for their constructive comments. The first and second authors were supported by The Chinese University of Hong Kong Faculty of Science Direct Grant No. 2060333.