# Statistical Power Analysis in Reliability Demonstration Testing: The Probability of Test Success
## 1 Introduction
### Assessment of Recent Work
The resulting posterior density can be used to carry out the reliability demonstration. For planningpurposes, however, assuming a successful test (no failures occur), the posterior density can be used to calculate the required sample size for the test. These procedures differ partly in the determination of the prior distribution and in the proportion of prior knowledge used. [PERSON] and [PERSON] [12] do not use a distribution of reliability in their method, instead using a nominal reliability value with a fixed confidence level of 63.2% of the predecessor product. [PERSON] et al. [13] remove this restriction, permitting variable confidence levels as well as accelerated tests. When employing the [PERSON] and [PERSON] methods [15; 16], failure distributions can be used as prior knowledge. [PERSON] et al. [15] use a beta and a uniform distribution, which are weighted according to the similarity of the products. The method introduced by [PERSON] [17; 18; 19] allows prior knowledge from different populations and sources, such as test results and field data, to be used. Additionally, [PERSON]'s approach [13] permits the use of prior knowledge from fatigue and lifetime calculations. The main challenge of these methods is the primarily subjective assessment of the similarity of two products, which makes the reliability demonstration vulnerable in the event of doubt [22]. Although [PERSON] [23] describes approaches for determining similarity, uncertainty remains in the application of these methods, because even very small changes in the production process can lead to profound changes in failure behavior. However, it is difficult to objectively attest a certain similarity of the products in this case. None of the methods for reducing the required sample size of SR tests takes the type II error into account for test planning; in fact, only the type I error is considered by the confidence level. Consequently, no statement on the result of the test is possible, which means that no estimate can be made about either the success of the test nor about a successful reliability demonstration. However, some approaches do exist which address the issue of the two types of risks during zero-failure testing. [PERSON] et al. [24] use the Bayesian approach presented in [25] in order to assign a success probability to the failure-free reliability demonstration test. However, they make use of an indifference region, which uses an interval of reliability values instead of a single reliability target. This results in a decoupling of the planning and assessment aspects of the test from the reliability target. The reliability target should be very much integrated as an integral part of reliability demonstration test planning. The very similar approach in [26] uses the assurance of [25] in order to calculate a probability for the outcome of the SR test. The main drawback, other than the use of two reliability values instead of one reliability target, is the sole focus on the SR test. No other test types can be analysed using these approaches and no comments about lifetime ratios or acceleration factors are made. A transfer to the context of failure-based tests is not trivial, as these approaches rely on the binary classification on which the SR test is based. [PERSON] and [PERSON] [27] propose a Monte Carlo simulation (MCS) for assurance calculation of failure-based tests. However, the computational burden is very high, and the indifference region is used. The problem with these Bayesian approaches of [24; 25; 26; 27] is that they result in a credibility interval instead of a confidence interval, which is required for a reliability demonstration with a required confidence level. In addition, the indifference region does not comply with the demonstrated reliability target. The literature contains several approaches that address the optimization of end-of-life testing (EoL test, called failure-based tests) strategies in a frequentist manner. However, the type II error is neglected within most approaches.