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Computing the real solutions of Fleishman's equations for simulating non-normal data
British Journal of Mathematical and Statistical Psychology ( IF 2.6 ) Pub Date : 2021-11-15 , DOI: 10.1111/bmsp.12259
Nathaniel E Helwig 1, 2
Affiliation  

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.

中文翻译:

计算用于模拟非正态数据的 Fleishman 方程的实解

Fleishman 幂法经常用于模拟具有所需偏度和峰度的非正态数据。Fleishman 的方法需要求解非线性方程组,以找到将标准正态变量转换为具有所需矩的非正态变量的三阶多项式权重。大多数幂法用户似乎不知道 Fleishman 方程对于偏度和峰度的典型组合有多个解。此外,研究人员缺乏一种简单的方法来探索 Fleishman 方程的多个解,因此大多数应用程序只考虑单个解。在本文中,我们提出了寻找 Fleishman 方程的所有实值解的新方法。此外,我们根据高阶矩的差异来描述解决方案。我们对幂方法的理论分析表明,通常存在两个 Fleishman 方程组的解,它们在高阶矩上有显着差异。通过模拟示例,我们证明了这些差异会对非正态分布的形状以及从数据计算的统计数据的抽样分布产生显着影响。讨论了选择解决方案的一些注意事项,并提供了一些改进报告标准的建议。以及从数据计算的统计数据的抽样分布。讨论了选择解决方案的一些注意事项,并提供了一些改进报告标准的建议。以及从数据计算的统计数据的抽样分布。讨论了选择解决方案的一些注意事项,并提供了一些改进报告标准的建议。
更新日期:2021-11-15
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