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On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables. (English) Zbl 1364.62039
Summary: 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:
zoverw
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