Extended Order-p Means and Modes of Continuous Densities

Abstract

The order-p mean of a finite multiset of real numbers is the minimiser of the sum of p-th powers of the distances to all numbers in the multiset. From Gauß' and Legendre's work on least squares estimation and the arithmetic mean (p=2), this concept has developed in progressing generality. First, it was established for positive integer p, with p=1 yielding the median as proven by Glaisher and Fechner. The further generalisation to arbitrary positive real p was considered by Jackson, Barral Souto and
Fr´echet who also established that the limits p→0 and p→∞ yield the mode and mid-range value of the data, respectively. Replacing sums by integrals, order-p means can be defined for continuous densities of real numbers; Fr´echet contributed substantially to the study of this concept. Whereas the inclusion of the arithmetic mean and median in the scale of order-p means as well as the mid-range limit transfer straightforward to the continuous case, this is not the case for the mode limit, as was pointed out by Fr´echet. Inspired by applications in signal and image processing, we present in this work an extension of order-p means of continuous densities to negative exponents between -1 and 0. We show that for densities with a unique global maximum (mode) the order-p means approach the mode for p→-1. We demonstrate the relation between the extended order-p means and modes by numerical examples.