Mathematical Sciences: Application of Interval Analysis in Statistical Computing

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9500831 Kennedy Abstract The IEEE proposed standard for binary floating-point computer arithmetic has been adopted by most computer manufacturers. Included among the features of this floating-point number system is programmer selectable rounding mode. Two rounding modes, round to plus or minus infinity, give hardware support for implementations of interval analysis computations. Substantive areas outside statistics are employing interval analysis and interval computations to good advantage in research. Research workers in statistical computing have been slow to recognize or evaluate the potential utility of interval analysis. The overall objective of this research is to identify areas in statistical computing where interval analysis can be used to good advantage, and to develop needed methods and algorithms in these areas. Specifically, a library of routines for self- validating approximation of cdf and inverse cdf for most of the usual univariate continuous probability distributions, and selected multivariate distributions will be developed. Accuracy essentially to machine precision over very broad areas of the variable/parameter space will be the goal. Self-validating computational methods for certain kinds of global optimization problems will also be developed. Scientific computations are usually done using a digital computer. Computers are not, however, capable of expressing exactly most numbers that constitute intermediate and final results for a computing problem. A part of the inexactness inherent in computer arithmetic is due to the need to round to a specific number of numeric digits after each arithmetic operation. Therefore answers produced by a computer for difficult computing tasks are often only rough approximations to the unknown true answers which would be obtained if exact computations could be performed. The objective of this research project is to develop alternative computational methods that yield more accurate answers in difficult statistical computing ap plications. The price of increased accuracy is that substantially more computer resources are consumed to produce answers. However, today there are enough computer resources available to support this very computationally intensive approach. The main motivation for the research is that it is best to spend extra time and effort to insure good answers, instead of settling for poor answers that are obtained quickly.