基于矩的概率分布差异度量

DIVERGENCE MEASURE BETWEEN THE PROBABILITY DISTRIBUTIONS BASED ON MOMENTS

在模型不确定环境下,决策者不完全信任参考分布,经常用参考分布的某邻域来刻画真实分布以进行稳健决策、鉴于未知的真实分布的具体形式难以刻画,并且真实分布与参考分布的各阶矩的偏差一般不会太大,旨在建立一个不依赖分布的具体形式,仅仅依赖分布的各阶矩的非参数指标来度量参考分布与真实分布的偏离程度.首先,构造了一个概率分布差异矩度量指标,并做了无量纲改进,分析了该度量指标的不变性和收敛性;其次,从实际需要出发,提出了四阶概率分布差异矩度量,与相对熵作了比较分析,并利用Bootstrap方法分别给出正态分布总体和帕累托分布总体中该度量在不同置信度下的临界值.

Under model uncertainty, decision-makers do not trust the reference distribution completely, and characterize the true distribution with a set of distributions deviating from the reference distribution. Since it is difficult to characterize the specific form of the unknown true distribution, and its deviation of the moments from the reference distribution are generally not too large, the paper aims to establish a distribution divergence measure depending only on the moments and not on the specific forms of distributions to measure the deviation of the reference distribution from the true distribution. First, based on the moment generating function, we construct a probability distributions divergence measure based only on moments, then improve it to a imensionless measure, and analyze its invariance and convergence. Then, to meet the practical need we put forward a probability distributions divergence measure based only on the first four moments, and make a comparative analysis to the relative entropy, and give the critical values under different confidence levels for the normal distributions and Pareto distributions by Bootstrap method. The probability distributions divergence measure based only on moments can offer a powerful tool for robust control and decision-making.