重尾数据高斯化的Lambert方法及Tukey h的逆作为特例
The Lambert Way to Gaussianize heavy tailed data with the inverse of Tukey's h as a special case
摘要：
我提出了一个参数化的双射变换来生成任意RV X ~ F的重尾版本Y。所谓的“重尾兰伯特W x F”RV Y的尾部行为取决于尾部参数delta >= 0：对于delta = 0，Y = X，对于delta > 0，Y的尾部比X重。当X是高斯分布时，这个重尾分布的元族简化为Tukey的h分布。Lambert的W函数提供了一个显式的逆变换，可以通过最大似然估计。该逆可以从数据中去除重尾，并且还提供累积分布（cdf）和密度函数（pdf）的解析表达式。作为一种特殊情况下，这些产量明确的公式Tukey的H PDF和CDF -作者的知识首次在文献中。模拟和应用S&P 500的日志回报和太阳耀斑数据证明了所介绍的方法的实用性。实现所提出方法的R包“LambertW”（cran.r-project.org/web/packages/LambertW）已在CRAN上公开提供。
英文摘要：
I present a parametric, bijective transformation to generate heavy tail versions Y of arbitrary RVs X ~ F. The tail behavior of the so-called 'heavy tail Lambert W x F' RV Y depends on a tail parameter delta >= 0: for delta = 0, Y = X, for delta > 0 Y has heavier tails than X. For X being Gaussian, this meta-family of heavy-tailed distributions reduces to Tukey's h distribution. Lambert's W function provides an explicit inverse transformation, which can be estimated by maximum likelihood. This inverse can remove heavy tails from data, and also provide analytical expressions for the cumulative distribution (cdf) and probability density function (pdf). As a special case, these yield explicit formulas for Tukey's h pdf and cdf - to the author's knowledge for the first time in the literature. Simulations and applications to S&P 500 log-returns and solar flares data demonstrate the usefulness of the introduced methodology. The R package "LambertW" (cran.r-project.org/web/packages/LambertW) implementing the presented methodology is publicly available at CRAN.
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