The inverse transformation of the explicit fourth-moment standardization for structural reliability

In practical engineering, the probability distributions of some random variables are often unknown, and the only available information about these may be their statistical moments. To conduct structural reliability assessment without the exclusion of random variables with unknown probability distributions, an explicit fourth-moment standardization function has been proposed, and a single expression of its inverse transformation, that is, normal transformation, with limitations of inputting sets of the third and fourth moments (skewness and kurtosis) of random variables was derived. However, the clear definition of the complete expressions of the inverse transformation of fourth-moment standardization function under different combinations of skewness and kurtosis of random variables has not been provided yet. It is in this regard that four criteria are proposed to derive the complete inverse transformation of fourth-moment standardization function, and then the complete expressions of the inverse transformation are formulated. Through the numerical examples presented, the proposed complete expressions are found to be quite efficient for normal transformations and to be sufficiently accurate to include random variables with unknown probability distributions in structural reliability assessment.