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.
