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. A new statistical method, Independent Approximates (IAs), is defined and proven to enable closed-form estimation of the parameters of heavy-tailed distributions. The pdf of the IAs is proven to be the normalized nth-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 is conducted for the Student's t distribution using between 1000 to 100,000 samples. Closed-form estimates of the location and scale are determined from the mean of the IA pairs and the variance 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. The theoretical precision is finite for a limited range of the shape but can be extended by using higher-order groupings for a given moment.
翻译:独立、 相同分布的单维分布样本中, 独立、 独立、 相同分布的样本, 独立、 独立、 独立、 独立、 独立、 同一分布的样本, 通过将样本分割成双对、 三进制或 n- 级组合, 并保留这些组合的中位值。 新的统计方法, 独立、 近似( IAs) 定义并被证明能够对重尾分布参数进行封闭式估计。 独立、 独立、 同一分布的pdf 被证明是原始密度的正常 nth的力量。 从此属性中, 重、 重零售分布的分布被证明具有定义明确的工具, 其 IA 有限、 三进制组合的限第二时间点, 其直径比值的缩略度( n-1), 其直径比值的缩略度的缩略度和缩略度的缩略度是I 的缩略图, 其缩略图的缩略图的缩略图和缩略图的缩略图由I 的缩略图的I 的缩略图和缩略图的缩略图的缩略图确定。