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On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables
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Díaz-Francés, Eloísa and Rubio, Francisco J. (2013) On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables. Statistical Papers, Volume 54 (Number 2). pp. 309-323. doi:10.1007/s00362-012-0429-2 ISSN 0932-5026.
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Official URL: http://dx.doi.org/10.1007/s00362-012-0429-2
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.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||
Journal or Publication Title: | Statistical Papers | ||||
Publisher: | Springer | ||||
ISSN: | 0932-5026 | ||||
Official Date: | May 2013 | ||||
Dates: |
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Volume: | Volume 54 | ||||
Number: | Number 2 | ||||
Page Range: | pp. 309-323 | ||||
DOI: | 10.1007/s00362-012-0429-2 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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