On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables |
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Authors: | Eloísa Díaz-Francés Francisco J. Rubio |
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Affiliation: | 1. Centro de Investigación en Matemáticas (CIMAT), A.P. 402, 36000, Guanajuato, GTO, Mexico 2. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK
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Abstract: | 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 β = 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. |
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