Numerical methods for self-validating computations of probabilities and percentiles in selected distributions using interval analysis

Abstract

<p>Most scientific computations are carried out on computers which employ fixed-precision floating-point number systems. Therefore, the accuracy of the values produced by an scalar algorithm without given associated error estimators still pose a problem in today's software. Self-validating numerical methods which not only produce an answer but also produce a guaranteed error bound would be of interest, especially for the following situations: (1) an essentially true answer is required for an accuracy comparison study among several competing algorithms or an accuracy study of a newly developed algorithm, and (2) the computed result has to satisfy given accuracy requirements because it is to be used in subsequent computations;In this study, we use four different numerical tools--interval arithmetic, automatic differentiation, continued fraction, and Taylor series expansion--to develop self-validating numerical integration methods. Then we apply these methods to the computations of probabilities and percentiles in selected distributions;Our software was developed in IBM compatible personal computers equipped with INTEL 80287 NPX. This software includes a support library and several algorithms which compute the probabilities and percentiles of selected distributions. The support library includes basic rounded interval arithmetic operations and some utility routines such as interval complete gamma function and interval tan[superscript]-1 function. These software are available upon request from the authors.</p>