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

*(English)*Zbl 1364.62039Summary: 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.

##### MSC:

62E17 | Approximations to statistical distributions (nonasymptotic) |

62P10 | Applications of statistics to biology and medical sciences; meta analysis |

##### Software:

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\textit{E. Díaz-Francés} and \textit{F. J. Rubio}, Stat. Pap. 54, No. 2, 309--323 (2013; Zbl 1364.62039)

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##### References:

[1] | Brody, JP; Williams, BA; Wold, BJ; Quake, SR, Significance and statistical errors in the analysis of DNA microarray data, Proc Natl Acad Sci USA, 99, 12975-12978, (2002) |

[2] | Chamberlin, SR; Sprott, DA, Some logical aspects of the linear functional relationship, Stat Pap, 28, 291-299, (1987) |

[3] | Díaz-Francés, E; Sprott, DA, Statistical analysis of nuclear genome size of plants with flow cytometer data, Cytometry, 45, 244-249, (2001) |

[4] | Geary, RC, The frequency distribution of the quotient of two normal variates, J R Stat Soc B Methodol, 93, 442-446, (1930) · JFM 56.1094.01 |

[5] | Hayya, J; Armstrong, D; Gressis, N, A note on the ratio of two normally distributed variables, Manag Sci, 21, 1338-1341, (1975) · Zbl 0309.62011 |

[6] | Hinkley, DV, On the ratio of two correlated normal variables, Biometrika, 56, 635-639, (1969) · Zbl 0183.48101 |

[7] | Kotz S, Lumelskii Y, Pensky M (2003) The stress-strength model and its generalizations: theory and applications. World Scientific Publishing, River Edge · Zbl 1017.62100 |

[8] | Kuethe, DO; Caprihan, A; Gach, HM; Lowe, IJ; Fukushima, E, Imaging obstructed ventilation with NMR using inert fluorinated gases, J Appl Physiol, 88, 2279-2286, (2000) |

[9] | Lisák, MA; Doležel, J, Estimation of nuclear DNA content in sesleria (poaceae), Caryologia, 52, 123-132, (1998) |

[10] | Marsaglia, G, Ratios of normal variables and ratios of sums of uniform variables, J Am Stat Assoc, 60, 193-204, (1965) · Zbl 0126.35302 |

[11] | Marsaglia, G, Ratios of normal variables, J Stat Softw, 16, 1-10, (2006) |

[12] | Merrill, AS, Frequency distribution of an index when both the components follow the normal law, Biometrika, 20A, 53-63, (1928) |

[13] | Palomino, G; Doležel, J; Cid, R; Brunner, I; Méndez, I; Rubluo, A, Nuclear genome stability of mammillaria san-angelensis (cactaceae) regenerants, Plant Sci, 141, 191-200, (1999) |

[14] | Schneeweiss, H; Sprott, DA; Viveros, R, An approximate conditional analysis of the linear functional relationship, Stat Pap, 28, 183-202, (1987) · Zbl 0631.62081 |

[15] | Sklar LA (2005) Flow cytometry for biotechnology. Oxford University Press, New York |

[16] | Watson JV (1992) Flow cytometry data analysis: basic concepts and statistics. Cambridge University Press, New York |

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