The Warranty Problem: Its Statistical and Game Theoretic Aspects

消费产品的保修通常是一维的(通过使用时间或使用量来索引)或二维的(由这两个变量来索引)。这个项目的重点是二维保证，这实际上是一个没有解决的话题。保修方案提出了至少两类或研究问题：最优价格/保修组合的具体说明，以及为满足未来索赔而预测的储备基金。前者是多学科的，涉及经济学、博弈论、营销学、运筹学和概率建模等方面。后者涉及两个维度的时间序列和点过程的分析，对于这一点，使用贝叶斯方法是合适的。该项目解决了这些问题的以下方面。1.将第一个问题正规化，以促进各单件的数学发展。2.开发按时间和使用情况索引的事件(故障)发生的概率模型。3.由所建立的模型生成的平面上点的更新理论。4.基于描述制造商和消费者之间以及制造商之间的竞争行为的主观概率考虑，开发有意义的效用函数。5.开发一种进行二维时间序列分析的技术。6.平面上点过程的统计推断。虽然项目的所有方面都受到保修问题的推动，但第2、3、5和6项都有其自身的科学价值，并将为决策科学提供新的工具。

Consumer products often carry warranties which are either one- dimensional (indexed by time in service or amount of usage) or two- dimensional (indexed by both of these variables). The focus of this project is on two-dimensional warranties, virtually an unaddressed topic. The warranty scenario raises at least two classes or research issues: the specification of an optimum price/warranty combination, and the forecast of a reserve fund to meet future claims. The former is multidisciplinary, involving aspects of economics, game theory, marketing, operations research, and probabilistic modeling. The latter involves the analysis of time series and point processes in two dimensions, for which the use of Bayesian methods is appropriate. This project addresses the following aspects of these issues. 1. Formalization of the first issue to facilitate mathematical development of the individual pieces. 2. Development of probabilistic models for the occurrence of events (failures) indexed by time and usage. 3. Renewal theory for points in a plane generated by the models developed. 4. Development of meaningful utility functions based on subjective probability considerations which describe the competitive behavior between manufacturers and consumers, and between manufacturers. 5. Development of a technology for undertaking times series analysis in two dimensions. 6. Statistical inference for point processes in a plane. Although all aspects of the project are motivated by the warranty problem, items 2, 3, 5, and 6 have scientific merit of their own and will provide new tools for the decision sciences.