Quasi-arithmetic mean


In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean 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. It is a broader generalization than the regular generalized mean.

Definition

If is a function that maps some continuous interval of the real line to some other continuous subset of the real numbers, and is both continuous, and injective.
Subject to those requirements, the of numbers is defined to be
or equivalently
A consequence of being defined over some selected interval, mapping to yet another interval, is that must also lie within And because is the domain of so in turn must produce a value inside the same domain the values originally came from,
Because is injective and continuous, it necessarily follows that is a strictly monotonic function, and therefore that the is neither larger than the largest number of the tuple nor smaller than the smallest number contained in hence contained somewhere among the values of the original sample.

Examples

Properties

The following properties hold for for any single function :
Symmetry: The value of is unchanged if its arguments are permuted.
Idempotency: for all the repeated average
Monotonicity: is monotonic in each of its arguments.
Continuity: is continuous in each of its arguments.
Replacement: Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. With 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 of two variables:
Mediality: For any quasi-arithmetic mean of two variables:
Balancing: For any quasi-arithmetic mean of two variables:
Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and non-trivial scaling of quasi-arithmetic For any with and constants, and a quasi-aritmetic function, and are always the same. In mathematical notation:
Central limit theorem : Under certain regularity conditions, and for a sufficiently large sample,
is approximately normally distributed. A similar result is available for Bajraktarević means and deviation means, which are generalizations of quasi-arithmetic means.

Characterization

There are several different sets of properties that characterize the quasi-arithmetic mean.Mediality is essentially sufficient to characterize quasi-arithmetic means.Self-distributivity is essentially sufficient to characterize quasi-arithmetic means.Replacement: Kolmogorov proved that the five properties of symmetry, fixed-point, monotonicity, continuity, and replacement fully characterize the quasi-arithmetic means.
  • Continuity is superfluous in the characterization of two variables quasi-arithmetic means. See for the details.Balancing: An interesting problem is whether this condition implies that the mean is quasi-arithmetic. Georg Aumann showed in the 1930s that the answer is no in general, but that if one additionally assumes to be an analytic function then the answer is positive.

Homogeneity

Means are usually homogeneous, but for most functions, the f-mean is not.
Indeed, the only homogeneous quasi-arithmetic means are the power means ; see Hardy-Littlewood-Pólya, page 68.
The homogeneity property can be achieved by normalizing the input values by some mean.
However this modification may violate monotonicity and the partitioning property of the mean.

Generalizations

Consider a Legendre-type strictly convex function. Then the gradient map is globally invertible and the weighted multivariate quasi-arithmetic mean is defined by
, where is a normalized weight vector. From the convex duality, we get a dual quasi-arithmetic mean associated to the quasi-arithmetic mean.
For example, take for a symmetric positive-definite matrix.
The pair of matrix quasi-arithmetic means yields the matrix harmonic mean: