A new statistical estimation method, Independent Approximates (IAs), is defined and proven to enable closed-form estimation of the parameters of heavy-tailed distributions. Given independent, identically distributed samples from a one-dimensional distribution, IAs are formed by partitioning samples into pairs, triplets, or nth-order groupings and retaining the median of those groupings that are approximately equal. The pdf of the IAs is proven to be the normalized n^th power of the original density. From this property, heavy-tailed distributions are proven to have well-defined means for their IA pairs, finite second moments for their IA triplets, and a finite, well-defined (n-1)^th moment for the nth grouping. Estimation of the location, scale, and shape (inverse of degree of freedom) of the generalized Pareto and Student's t distributions are possible via a system of three equations. Performance analysis of the IA estimation methodology for the Student's t distribution demonstrates that the method converges to the maximum likelihood estimate. Closed-form estimates of the location and scale are determined from the mean of the IA pairs and the second moment of the IA triplets, respectively. For the Student's t distribution, the geometric mean of the original samples provides a third equation to determine the shape, though its nonlinear solution requires an iterative solver. With 10,000 samples the relative bias of the parameter estimates is less than 0.01 and the relative precision is less than +/- 0.1. Statistical physics applications are carried out for both a small sample (331) astrophysics dataset and a large sample (2 x 10^8) standard map simulation.
Statistical inference for heavy tailed series with extremal independence
本文考虑平稳时间序列$\{X_j, j \in Z\} whose finite dimensional distributions are regularly varying with extremal independence. We assume that for each $ h \geq 1 $, conditionally on $ X 0 $ to exceed a threshold tending to infinity, the conditional distribution of $ X h$适当归一化弱收敛到一个非退化分布.在本文中，我们考虑估计的正常化和极限分布。
Modelling outliers and structural breaks in dynamic linear models with a novel use of a heavy tailed prior for the variances: An alternative to the Inverted Gamma
The Spurious Factor Dilemma: Robust Inference in Heavy-Tailed Elliptical Factor Models
KLASMOE and School of Mathematics and Statistics, Northeast Normal University（东北师范大学数学与统计学院）；Department of Statistics and Data Science, National University of Singapore（新加坡国立大学统计与数据科学系）；Department of Mathematics, Shanghai University（上海大学数学系）
Heavy tailed priors: an alternative to non-informative priors in the estimation of proportions on small areas
我们探索柯西和一个新的重尾（Fuquene，Perez和Pericchi（2011））先验估计小区域的比例。使用分层模型和指数族形式的二项式似然。我们相信，在调查抽样设置重尾先验可能比选择无信息先验更有效地消除对涉及主观因素或假设的方法的反感。为了说明稳健的贝叶斯方法，我们将此方法应用于一个流行的例子：“贝叶斯问题”。最后，我们建议在小区域内不存在或存在离群值时使用Cauchy先验，当离群值是特定的小区域时使用Fuquene et al.（2011）先验。
