@Article{sl.2012.008.053,
AUTHOR = {Lyuben D. Ivanov, Justin Y-T. Wu},
TITLE = {An Approximate Method for Expansion of the Cumulative Distribution Function in the Asymptotic Tails},
JOURNAL = {Structural Longevity},
VOLUME = {8},
YEAR = {2012},
NUMBER = {1},
PAGES = {53--70},
URL = {http://www.techscience.com/sl/v8n1/42889},
ISSN = {1944-6128},
ABSTRACT = {An efficient approximate method is developed for calculation of very
small values of a cumulative distribution function (CDF) or probability of exceedance (POE) located in the asymptotic tails of a continuous distribution. Only
three properly selected CDF points are needed to model each of the two tails, while
a total of nine points are sufficient to interpolate and extrapolate to cover the entire
range of the distribution. The approximated CDF is fast to calculate and is exact at
the fitting points while providing smooth transitions from point to point as well as
from the end points to the extreme tails. The method is most suitable when suffi-
cient data is unavailable or difficult to compute. In general, the method has three
areas of potential applications. The first is the calculation of the very small POE to
characterize variables such as long-term extreme loads for risk analysis and design.
The second area is calculation of the very small CDF value to characterize variables
such as material strength and flaw size. The third case is calculating all ordinates
of the entire probabilistic distribution, including CDF and PDF functions. Detailed
formulae are provided and the performance demonstrated using commonly used
distributions such as Weibull, Gaussian, Lognormal, etc.},
DOI = {10.3970/sl.2012.008.053}
}