Heavy-Tailed Density Estimation

A novel statistical method is proposed and investigated for estimating a heavy tailed density under mild smoothness assumptions. Statistical analyses of heavy-tailed distributions are susceptible to the problem of sparse information in the tail of the distribution getting washed away by unrelated features of a hefty bulk. The proposed Bayesian method avoids this problem by incorporating smoothness and tail regularization through a carefully specified semiparametric prior distribution, and is able to consistently estimate both the density function and its tail index at near minimax optimal rates of contraction. A joint, likelihood driven estimation of the bulk and the tail is shown to help improve uncertainty assessment in estimating the tail index parameter and offer more accurate and reliable estimates of the high tail quantiles compared to thresholding methods. Supplementary materials for this article are available online.

本文提出并研究了一种全新的统计方法，用于在温和光滑性假设下估计重尾密度（heavy-tailed density）。重尾分布的统计分析极易面临如下问题：分布尾部的稀疏信息会被分布主体中的非相关特征所淹没。所提出的贝叶斯方法通过精心设定的半参数先验分布（semiparametric prior distribution）引入光滑性与尾部正则化，从而规避了上述问题，能够以近似极小极大最优收缩速率一致估计密度函数及其尾部指数。相较于阈值法，对分布主体与尾部进行似然驱动的联合估计，有助于提升尾部指数参数估计中的不确定性评估效果，并能给出更为精准可靠的高尾部分位数估计结果。本文的补充材料可在线获取。