Saddlepoint Approximations for Various Statistics of Dependent, Non- Gaussian Random Variables: Applications to the Maximum Variate and the Range Variate

摘要

The determination of the statistics of a set of random variables (RVs) is frequently achieved by assuming the RVs to be joint Gaussian or to be statistically independent of each other. Although this assumption greatly simplifies the analysis, it can lead to very misleading probability measures, especially on the tails of the distributions, where the exact details of the particular RVs can be important This report presents a new method for deriving saddlepoint approximations (SPAS) for a number of very useful statistics of a general set of M dependent RVs, including the joint moments, the joint cumulative distribution function (CDF), the joint exceedance distribution function (EDF), and the joint probability density function (PDF). In particular, application to the maximum RV of a set of M dependent non-Gaussian RVs will be thoroughly investigated for two different examples. Also, the statistics of the range variate (maximum - minimum) are determined for the same two examples. The calculation of these M-dimensional statistics sometimes requires that multiple M-dimensional SPs be determined in order to evaluate an EDF or PDF at a single point in probability space. Also, the storage requirements and/or execution times can rapidly grow unmanageable as the number of dimensions, M, increases. Finally, care must be taken to ensure that the SP is located in the correct region of M-dimensional space; this allowed region of analyticity varies with the particular statistic under investigation.