In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean[1] is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function
It is also called Kolmogorov mean after Soviet mathematician Andrey Kolmogorov.
If f is a function which maps an interval
of the real line to the real numbers, and is both continuous and injective, the f-mean of
, which can also be written We require f to be injective in order for the inverse function
Since f is injective and continuous, it follows that f is a strictly monotonic function, and therefore that the f-mean is neither larger than the largest number of the tuple
nor smaller than the smallest number in
is unchanged if its arguments are permuted.
Replacement: Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained.
it holds: Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks:
Self-distributivity: For any quasi-arithmetic mean
Mediality: For any quasi-arithmetic mean
Balancing: For any quasi-arithmetic mean
Central limit theorem : Under regularity conditions, for a sufficiently large sample,
[2] A similar result is available for Bajraktarević means and deviation means, which are generalizations of quasi-arithmetic means.
[3][4] Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and scaling of
There are several different sets of properties that characterize the quasi-arithmetic mean (i.e., each function that satisfies these properties is an f-mean for some function f).
Means are usually homogeneous, but for most functions
Indeed, the only homogeneous quasi-arithmetic means are the power means (including the geometric mean); see Hardy–Littlewood–Pólya, page 68.
The homogeneity property can be achieved by normalizing the input values by some (homogeneous) mean
However this modification may violate monotonicity and the partitioning property of the mean.
Consider a Legendre-type strictly convex function
is globally invertible and the weighted multivariate quasi-arithmetic mean[9] is defined by
is a normalized weight vector (
by default for a balanced average).
From the convex duality, we get a dual quasi-arithmetic mean
a symmetric positive-definite matrix.
The pair of matrix quasi-arithmetic means yields the matrix harmonic mean:
[10] MR4355191 - Characterization of quasi-arithmetic means without regularity condition Burai, P.; Kiss, G.; Szokol, P. Acta Math.
[11] MR4574540 - A dichotomy result for strictly increasing bisymmetric maps Burai, Pál; Kiss, Gergely; Szokol, Patricia J.