This project has two major goals: (a) to find explanations for the remarkable accuracy of saddlepoint approximations, and (b) to continue development of a framework for implementing nonparametric statistical inference in stochastic systems. Consideration of objective (a) uses two mathematical tools: the Ikehara-Weiner theorem, more commonly used in analytic number theory to prove the prime number theorem, and complex integration methods applied to inversion formulas of moment generating functions (MGFs). Both methods focus on the analytic continuation of MGFs and their properties outside of the convergence strip. These two tools will be used to streamline and extend what is known concerning the uniformly relative accuracy of saddlepoint approximations. Objective (b) continues previous work on the development of a framework for implementing bootstrap inference in stochastic models that are finite-state semi-Markov processes. These models include most of the commonly used stochastic models in reliability, multi-state survival analysis, epidemic modeling, and communication and manufacturing systems. Three tools are required to complete the framework: cofactor rules specifying the Laplace transforms for performance characteristics, saddlepoint approximations to invert these transforms, and the bootstrap to provide statistical inference in conjunction with the two previous tools.
Modern statistical methods use models that involve complicated distributions from which the computation of probabilities can be a formidable task. This task is often simplified by using saddlepoint approximations. Such approximations generally provide probabilities with very little effort and most often achieve 2-3 significant digit accuracy. Explanations for this remarkable accuracy have continued to elude researchers. Part (a) of this proposal outlines two new approaches the investigator will consider to explain this accuracy. Among the modern methods that require probability computations from complicated distributions are the procedures the investigator considers in part (b) of the proposal. This work concerns the development of a framework for implementing nonparametric statistical inference in complex stochastic systems some of which began with the complex systems formulated in engineering during the cybernetics movement. These stochastic systems include most of the standard stochastic models used in reliability, multi-state survival analysis, epidemic modeling, and communication and manufacturing systems. No such general methodology currently exists for implementing statistical inference in the context of general stochastic systems models so the framework proposed by the investigator would provide tools that are currently unavailable. The proposal also addresses significant questions in other disciplines where answers are lacking due to certain computational difficulties. In ocean and electrical engineering accurate approximations are given for distributions of extreme hull stress during heavy seas and distributions for extreme responses in signal processing; in quantum physics, approximations are proposed for "gauge" functions, the computation of which are fundamental in quantum theory, and whose computation is difficult even in the simplest cases.
