Fractional statistical moments are utilized for various tasks of uncertainty quantification, including the estimation of probability distributions. However, an estimation of fractional statistical moments of costly mathematical models by statistical sampling is challenging since it is typically not possible to create a large experimental design due to limitations in computing capacity. This paper presents a novel approach for the analytical estimation of fractional moments, directly from polynomial chaos expansions. Specifically, the first four statistical moments obtained from the deterministic PCE coefficients are used for an estimation of arbitrary fractional moments via H\"{o}lder's inequality. The proposed approach is utilized for an estimation of statistical moments and probability distributions in three numerical examples of increasing complexity. Obtained results show that the proposed approach achieves a superior performance in estimating the distribution of the response, in comparison to a standard Latin hypercube sampling in the presented examples.

翻译：分数统计矩被用于不确定性量化的各种任务，包括概率分布的估计。然而，对于计算成本高昂的数学模型，通过统计抽样估计分数统计矩具有挑战性，因为计算能力的限制通常无法生成大规模实验设计。本文提出了一种新方法，可直接从多项式混沌展开中解析估计分数矩。具体而言，利用确定性PCE系数获得的前四个统计矩，通过Hölder不等式估计任意分数矩。所提方法被用于三个复杂度递增的数值示例中的统计矩和概率分布估计。结果表明，与标准拉丁超立方抽样相比，所提方法在估计响应分布方面表现出优越性能。