Applications of interval analysis to selected topics in statistical computing.

摘要

In interval analysis, an interval is treated not only as a set of numbers, but as a number in and of itself. The development of interval analysis is closely connected to the development of electronic digital computers. Conventional electronic computation is typically performed using a fixed-precision, floating-point processor. This approach is a finite approximation to calculations with real numbers of infinite precision. The finite approximation leads to errors of various types. While the fundamental operations of addition, subtraction, multiplication and division are typically accurate to one-half unit-last-place in floating-point computations, the effect of cumulative error in repeated calculations is usually unknown and too-frequently ignored. Using interval analysis, an interval is constructed which (after each computation) is guaranteed to contain the true value. By seeking ways to keep the interval narrow, it is possible to obtain results which are of guaranteed accuracy.; This dissertation uses interval analysis in topics of statistical computing. Two major topics are addressed: bounding computational errors and global optimization.; For bounding computational errors, series are used which yield a bound on the truncation error which results from a finite series approximation to an infinite series. By evaluating the series with intervals to bound rounding errors and by using the bound on the truncation error, an interval is obtained which is guaranteed to contain the true value. For some series, interval numerical quadrature rules are also employed. These ideas are applied to the computation of tail probabilities and critical points of several statistical distributions such as Bivariate Chi-Square and Bivariate F distributions.; As regards to global optimization, the EM algorithm is one tool frequently used for optimization in statistics. The EM algorithm is fairly flexible and is able to handle missing data. However, as with most optimization algorithms, there is no guarantee of finding a global optimum. Interval analysis can be used to compute an enclosure of the range of a function over a specified domain. By enclosing the range of the gradient of the loglikelihood, those parts of the parameter space where the gradient is nonzero can be eliminated as not containing stationary points. An algorithm proceeds by repeatedly bisecting an initial region into smaller regions which are evaluated for the possibility of the gradient being nonzero. Upon termination, all stationary points of the loglikelihood are contained in the remaining regions.