Extreme Tail Ratios and Overrepresentation among Subpopulations with Normal Distributions

Theodore P. Hill () and Ronald F. Fox
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Theodore P. Hill: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
Ronald F. Fox: School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA

Stats, 2022, vol. 5, issue 4, 1-8

Abstract: Given several different populations, the relative proportions of each in the high (or low) end of the distribution of a given characteristic are often more important than the overall average values or standard deviations. In the case of two different normally-distributed random variables, as is shown here, one of the (right) tail ratios will not only eventually be greater than 1 from some point on, but will even become infinitely large. More generally, in every finite mixture of different normal distributions, there will always be exactly one of those distributions that is not only overrepresented in the right tail of the mixture but even completely overwhelms all other subpopulations in the rightmost tails. This property (and the analogous result for the left tails), although not unique to normal distributions, is not shared by other common continuous centrally symmetric unimodal distributions, such as Laplace, nor even by other bell-shaped distributions, such as Cauchy (Lorentz) distributions.

Keywords: tail ratio; normal distribution; overrepresentation; finite mixture of distributions (search for similar items in EconPapers)
JEL-codes: C1 C10 C11 C14 C15 C16 (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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