On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables
On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables
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DOI：
10.1007/s00362-012-0429-2
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发表时间：
2013-05-01
影响因子：
1.3
通讯作者：
Rubio, Francisco J.
中科院分区：
文献类型：
--
作者：
Diaz-Frances, Eloisa;Rubio, Francisco J.
两个独立的正态随机变量X和Y的比值分布是重尾分布，没有矩。其密度的形状可以是单峰、双峰、对称、不对称，甚至类似于接近其模态的正态分布。据我们所知，科学文献中只有通过模拟和经验结果才提出了Z = X/Y分布的合理正态近似的条件。在以beta = E(X) /E(Y)为中心的区间I中，对于X和Y是独立的，具有正均值，并且它们的变异系数满足某些条件的情况，给出了Z分布的正态近似存在的证明。此外，还提出了一种评估特定比率X/Y分布与所提出的正态近似的接近程度的图形信息方法，即通过接收者工作特征（ROC）曲线。
The distribution of the ratio of two independent normal random variables X and Y is heavy tailed and has no moments. The shape of its density can be unimodal, bimodal, symmetric, asymmetric, and/or even similar to a normal distribution close to its mode. To our knowledge, conditions for a reasonable normal approximation to the distribution of Z = X/Y have been presented in scientific literature only through simulations and empirical results. A proof of the existence of a proposed normal approximation to the distribution of Z, in an interval I centered at beta = E(X) /E(Y), is given here for the case where both X and Y are independent, have positive means, and their coefficients of variation fulfill some conditions. In addition, a graphical informative way of assessing the closeness of the distribution of a particular ratio X/Y to the proposed normal approximation is suggested by means of a receiver operating characteristic (ROC) curve.