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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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  • Eloísa Díaz-Francés
  • Francisco Rubio

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. Copyright Springer-Verlag 2013

Suggested Citation

  • Eloísa Díaz-Francés & Francisco Rubio, 2013. "On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables," Statistical Papers, Springer, vol. 54(2), pages 309-323, May.
    • Handle: RePEc:spr:stpapr:v:54:y:2013:i:2:p:309-323
    DOI: 10.1007/s00362-012-0429-2
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    References listed on IDEAS

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    1. Jack Hayya & Donald Armstrong & Nicolas Gressis, 1975. "A Note on the Ratio of Two Normally Distributed Variables," Management Science, INFORMS, vol. 21(11), pages 1338-1341, July.
    2. Marsaglia, George, 2006. "Ratios of Normal Variables," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 16(i04).
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    Cited by:

    1. Caginalp, Carey & Caginalp, Gunduz, 2019. "Price equations with symmetric supply/demand; implications for fat tails," Economics Letters, Elsevier, vol. 176(C), pages 79-82.
    2. Caginalp, Carey & Caginalp, Gunduz, 2020. "Derivation of non-classical stochastic price dynamics equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 560(C).
    3. Carey Caginalp & Gunduz Caginalp, 2019. "Derivation of non-classical stochastic price dynamics equations," Papers 1908.01103, arXiv.org, revised Aug 2020.
    4. Caginalp, Carey & Caginalp, Gunduz, 2019. "Stochastic asset price dynamics and volatility using a symmetric supply and demand price equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 807-824.
    5. Gunduz Caginalp, 2020. "Fat tails arise endogenously in asset prices from supply/demand, with or without jump processes," Papers 2011.08275, arXiv.org, revised Mar 2021.
    6. Carey Caginalp & Gunduz Caginalp, 2018. "Asset Price Volatility and Price Extrema," Papers 1802.04774, arXiv.org, revised Jul 2018.
    7. Thomas Vasileiou & Leopold Summerer, 2020. "A biomimetic approach to shielding from ionizing radiation: The case of melanized fungi," PLOS ONE, Public Library of Science, vol. 15(4), pages 1-16, April.
    8. Caginalp, Carey & Caginalp, Gunduz, 2018. "The quotient of normal random variables and application to asset price fat tails," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 499(C), pages 457-471.

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