Contributions to engineering statistics: X -testing for *reliability and optimal partitioning strategies.

Abstract

The research presented in this thesis was largely motivated by two important problems in manufacturing: reliability demonstration and selective assembly. In addition to being important in the assurance of product quality and dependability, the exploration of these problems has uncovered a number of interesting theoretical results concerning hypotheses testing for binary data and optimal partitioning. These results have implications beyond the two original motivating problems. Selective assembly is a technique in manufacturing that reduces overall variation and thus improves the quality of an assembled product. In this process, one or both of the individual components of a mating pair are separated into several classes, and the final product is assembled by selecting pairs of components from appropriate classes. The problem of optimal partitioning of the component distribution to minimize the expected value of a given loss function is considered. In many cases this problem is equivalent to a more general optimal partitioning problem that arises in various statistical applications as well as in scalar quantization. Solutions to this general problem are derived which have a simple characterization provided the density is strongly unimodal. In proving this, the concepts of stochastic dominance to the left and uniform stochastic dominance of a random variable are introduced. Reliability demonstration plans are frequently used to formally verify that the reliability of a product exceeds a certain specified value with a certain degree of confidence. When the value of the reliability to be demonstrated is close to one, traditional reliability demonstration plans are problematic since they require extremely large sample sizes and have low power. One solution to this problem is to induce failure in the testing process by testing products under conditions in which they are more likely to fail. A general framework to describe this type of extreme testing, or X-testing is provided. Specifically, the effect of X-testing on sample size and power of reliability demonstration plans based on binary data is considered. The properties of X-tests are studied with respect to zero failure, fixed sample size, and fixed power plans, and conditions are derived under which X-tests are inadmissible or universally efficient.