Open Access

ARTICLE

An Approximate Method for Expansion of the Cumulative Distribution Function in the Asymptotic Tails

Lyuben D. Ivanov¹, Justin Y-T. Wu²

1 Retired, formerly with American Bureau of Shipping, Houston – USA
2 Comprel, Inc. Raleigh, NC 27614

Structural Longevity 2012, 8(1), 53-70. https://doi.org/10.3970/sl.2012.008.053

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.

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Keywords

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