The Extremal Quotient

Abstract

The extremal quotient is defined as the ratio of the largest to the absolute value of the smallest observation. Its analytical properties for symmetrical, continuous and unlimited distributions are obtained from a study of the auto-quotient defined as the ratio of two non-negative variates with identical distributions. The relation of the two statistics is established by proving that, for sufficiently large samples from an initial distribution with median zero, the largest (or smallest) value may be assumed to be positive (or negative) and that the extremes are independent. It follows that the distribution and the probability of the extremal quotient possess certain symmetries, and that its median is unity. As many moments exist for the extremal quotient as moments and reciprocal moments exist simultaneously for the initial variate. The logarithm of the extremal quotient is symmetrically distributed. These properties hold for all continuous symmetrical unlimited variates which possess a monotonically increasing probability function. For the exponential type, the asymptotic distribution of the extremal quotient can only be expressed by an integral. In this case, no moments exist. For the Cauchy type, the asymptotic distribution is very simple, and the logarithm of the extremal quotient has the same distribution as the midrange for initial distributions of the exponential type. It is not necessary to consider asymmetrical distributions since, in this case, for sufficiently large samples, one of the extremes will outweigh the other, unless the distribution is nearly symmetrical or has rapidly varying tails.