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On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables

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 ZX/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 βE(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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Correspondence to Eloísa Díaz-Francés.

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Díaz-Francés, E., Rubio, F.J. On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables. Stat Papers 54, 309–323 (2013). https://doi.org/10.1007/s00362-012-0429-2

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Keywords

  • Coefficient of variation
  • Ratio of normal means
  • ROC curve

Mathematics Subject Classification (2000)

  • 62E17