Probability Approximations by Log Likelihood Maximization

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Abstract
The computation of multivariate integrals is an important mathematical problem in reliability theory. Various approximation methods have been developed for this task. In the so‐called FORM and SORM methods, it is assumed that all variables have been transformed into independent standard normal ones. Only then can the methods be applied. Here, it is shown that such transformations are not necessary. It is sufficient to maximize the log likelihood function of the probability distribution in the original space, then to approximate the function and limit‐state function near the maximum points by second‐order Taylor expansions to obtain asymptotic approximations. In the same way, asymptotic sensitivity factors for the parameter dependence of the failure probability can be found. The advantages of this method are that the often‐complicated numerical transformation into the normal space is avoided and that the results have a natural interpretation in terms of the parameters of the original random variables and limit‐state function.

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