Saddlepoint Methods in Statistics

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DMS 9625396 Butler This research is concerned with the application of saddlepoint methods in four areas: (a) First they are used to construct intractable likelihoods in dynamic stochastic systems models so as to allow for statistical inference. This leads to the computation of performance characteristics for these systems with applications in production management and systems reliability. (b) Secondly, saddlepoint methods are used to approximate special functions arising in Statistics and applied Mathematics including Bessel and hypergeometric functions of scalar and matrix argument. The matrix argument functions cannot always be accurately approximated and, when they can, the computational times needed are often prohibitive. By contrast , the saddlepoint approximations are highly accurate and can always be computed in a couple seconds. (c) Saddlepoint approximations are developed for the multivariate cumulative distribution functions arising in sampling theory. (d) Saddlepoint theory related to the generalized inverse Gaussian distributions is also investigated. Concepts and models for stochastic systems and networks are now found in all areas of science. Examples include various models for computer system networks, ecosystems, production management methods in manufacturing, and reliability testing. The first part of this project develops methods that will allow for statistical inference with such models so as to assess various performance characteristics related to such systems. These evaluations in turn allow for the description and prediction of outcomes for the system as well as for system construction and design. Ultimately these model-based descriptions and predictions pertain to the real phenomena that underly the models. The second major part of this proposal concerns the approximation of important special functions that arise in the use of statistical methods and applied mathematics. Special functions were created by mathematicians because they repeatedly arise in the solution of a wide range of important scientific problems. The special functions addressed here are called Bessel and hypergeometric functions and are perhaps the most important, widely encompassing and general of the special functions in Mathematics. These functions historically have played a central role in the physical and mathematical sciences and are extremely difficult to compute. The proposal suggests highly accurate approximations to these functions that can be computed in a fraction of the time that is currently needed for their approximation.