Computing the real solutions of Fleishman's equations for simulating non-normal data.
Computing the real solutions of Fleishman's equations for simulating non-normal data.
复制标题
计算弗莱什曼方程的实数解以模拟非正态数据。
DOI：
10.1111/bmsp.12259
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Helwig,NathanielE
中科院分区：
文献类型：
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作者：
Helwig,NathanielE
Fleishman幂方法经常用于模拟具有期望偏度和峰度的非正态数据。Fleishman的方法需要求解一个非线性方程组来找到三阶多项式权重，将标准正态变量转换为具有期望矩的非正态变量。大多数幂方法的使用者似乎没有意识到，对于偏度和峰度的典型组合，Fleishman方程有多个解。此外，研究人员缺乏一种简单的方法来探索Fleishman方程的多解，因此大多数应用程序只考虑一个单一的解决方案。在本文中，我们提出了新的方法来寻找所有的真实的-值解的Fleishman方程。此外，我们的解决方案的特点在高阶矩的差异。我们的幂方法的理论分析表明，通常存在两个解决方案的Fleishman的方程有显着的差异，在高阶矩。使用模拟的例子，我们证明了这些差异可以对非正态分布的形状以及从数据计算的统计量的抽样分布产生显着影响。讨论了选择解决方案的一些考虑因素，并提出了改进报告标准的一些建议。
Fleishman's power method is frequently used to simulate non‐normal data with a desired skewness and kurtosis. Fleishman's method requires solving a system of nonlinear equations to find the third‐order polynomial weights that transform a standard normal variable into a non‐normal variable with desired moments. Most users of the power method seem unaware that Fleishman's equations have multiple solutions for typical combinations of skewness and kurtosis. Furthermore, researchers lack a simple method for exploring the multiple solutions of Fleishman's equations, so most applications only consider a single solution. In this paper, we propose novel methods for finding all real‐valued solutions of Fleishman's equations. Additionally, we characterize the solutions in terms of differences in higher order moments. Our theoretical analysis of the power method reveals that there typically exists two solutions of Fleishman's equations that have noteworthy differences in higher order moments. Using simulated examples, we demonstrate that these differences can have remarkable effects on the shape of the non‐normal distribution, as well as the sampling distributions of statistics calculated from the data. Some considerations for choosing a solution are discussed, and some recommendations for improved reporting standards are provided.