Abstract
Calculation of exceedance probabilities or the inverse problem of finding the level corresponding to a given exceedance probability occurs in many practical applications. For instance, it is often of interest in offshore engineering to evaluate the wind, wave, current, and sea ice properties with annual exceedance probabilities of, e.g., 10−1, 10−2, and 10−3, or so-called 10-year, 100-year, and 1000-year values. A methodology is provided in this article to calculate a tight upper bound of the exceedance probability, given any probability distribution from a wide range of commonly used distributions. The approach is based on a generalization of the Chebyshev inequality for the class of distributions with a logarithmically concave cumulative distribution function, and has the potential to relieve the often-debated exercise of determining an appropriate probability distribution function based on limited data, particularly in terms of tail behavior. Two numerical examples are provided for illustration.
Highlights
The exceedance probability is the probability of an uncertain parameter exceeding a certain threshold
The results provided by the log-concave probability density function (PDF) class essentially bound the results of the tested probability distributions
In the distributionally robust methodology, stationarity in time means that the probability distribution must remain in the specified distribution class as time goes by, and the provided probability bound is the upper bound of exceedance probability given any precise probability distribution from the specified class
Summary
The exceedance probability is the probability of an uncertain parameter exceeding a certain threshold. The common approach for calculating the exceedance or non-exceedance probabilities is based on a description of uncertainty by a probability density function. Methods such as the method of moments or maximum likelihood are often used to estimate the parameters of an assumed or justified probability model based on a set of data points. The problem becomes a matter of calculating the upper bound of exceedance probability, given any probability distribution belonging to this set In this paper, this is called a “distributionally robust” approach, inspired by the terminology in the optimization community for problems involving uncertainty descriptions of a similar spirit—i.e., distributionally robust optimization. In civil and mechanical engineering domains, this perspective is often known and categorized under the theory of “imprecise probabilities”; see, e.g., [2] and [3]
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