Development of a kernel density estimation with hybrid estimated bounded data

Uncertainty quantification, which identifies a probabilistic distribution for uncertain data, is important for yielding accurate and reliable results in reliability analysis and reliability-based design optimization. Sufficient data are needed for accurate uncertainty quantification, but data is very limited in engineering fields. For statistical modeling using insufficient data, kernel density estimation (KDE) with estimated bounded data (KDE-ebd) has been recently developed for more accurate and conservative estimation than the original KDE by combining given data and bounded data within estimated intervals of random variables from the given data. However, the estimated density function using KDE-ebd is modeled beyond the domain of random variables due to conservative estimation of the density function with long and thick tails. To overcome this problem, this paper proposes kernel density estimation with hybrid estimated bounded data (KDE-Hebd), which does not violate the domain of the random variables, and uses point or interval estimation of the bounds for generating the bounded data. KDE-ebd often yields too wide bounds for very insufficient data or large variations because it uses only the estimated intervals of random variables. The proposed KDE with hybrid estimated bounded data alternatively selects a point estimator or interval estimator according to whether the estimated intervals violate the domain of the random variables. The performance of the proposed method was evaluated by comparing the estimation accuracy from statistical simulation tests for mathematically derived sample data and real experimental data using KDE, KDE-ebd and KDE-Hebd. As a result, it was demonstrated that KDE-Hebd was more accurate than KDE-ebd without the violation of the domain of random variables, especially for a large coefficient of variation.