Kernel density maximum entropy method with generalized moments for evaluating probability distributions, including tails, from a small sample of data

Abstract

AbstractIn this paper, a novel method to determine the distribution of a random variable from a sample of data is presented. The approach is called generalized kernel density maximum entropy method, because it adopts a kernel density representation of the target distribution, while its free parameters are determined through the principle of maximum entropy (ME). Here, the ME solution is determined by assuming that the available information is represented from generalized moments, which include as their subsets the power and the fractional ones. The proposed method has several important features: (1) applicable to distributions with any kind of support, (2) computational efficiency because the ME solution is simply obtained as a set of systems of linear equations, (3) good trade‐off between bias and variance, and (4) good estimates of the tails of the distribution, in the presence of samples of small size. Moreover, the joint application of generalized kernel density maximum entropy with a bootstrap resampling allows to define credible bounds of the target distribution. The method is first benchmarked through an example of stochastic dynamic analysis. Subsequently, it is used to evaluate the seismic fragility functions of a reinforced concrete frame, from the knowledge of a small set of available ground motions.