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. It is also called Kolmogorov mean after Soviet mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.

Definition

of the real numbers is both continuous, and injective (one-to-one).

1. in order for an inverse function.)

Subject to those requirements, the numbers is defined to be

or equivalently

is a strictly monotonic function, and therefore that the is neither larger than the largest number of the tuple hence contained somewhere among the values of the original sample.

Examples

• the real line otherwise any and any) then the corresponds to the arithmetic mean.

  the strictly positive real numbers then the corresponds to the geometric mean. (The result is the same for any logarithm; it does not depend on the base of the logarithm, as long as that base is strictly positive but not .)

  then the corresponds to the harmonic mean.

  then the corresponds to the power mean one gets the root mean square

  then the is the mean in the log semiring, which is a constant-shifted version of the LogSumExp in the expression corresponds to dividing by , since logarithmic division is linear subtraction.) The LogSumExp function is a smooth maximum: It is a smooth approximation to the maximum function.

Properties

:

Symmetry: is unchanged if its arguments are permuted.

Idempotency:

Monotonicity is monotonic).

Continuity is continuous).

Replacement it holds:

Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks:

Self-distributivity: For any quasi-arithmetic of two variables:

Mediality: For any quasi-arithmetic of two variables:

Balancing: For any quasi-arithmetic of two variables:

Scale-invariance 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 (i.e., each function that satisfies these properties is an f-mean for some function f).

• 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 [10] for the details.

• Balancing: An interesting problem is whether this condition (together with symmetry, fixed-point, monotonicity and continuity properties) implies that the mean is quasi-arithmetic. Georg Aumann showed in the 1930s that the answer is no in general, to be an analytic function then the answer is positive.

Homogeneity

Means are usually homogeneous, the f-mean is not. Indeed, the only homogeneous quasi-arithmetic means are the power means (including the geometric mean); see HardyLittlewoodPólya, page 68.

.

However this modification may violate monotonicity and the partitioning property of the mean.

Generalizations

Consider a Legendre-type strictly convex function. Then the gradient is globally invertible and the weighted multivariate quasi-arithmetic mean a symmetric positive-definite matrix

See also

• Generalized mean

• Jensen's inequality

References

• Andrey Kolmogorov (1930) "On the Notion of Mean", in "Mathematics and Mechanics" (Kluwer 1991) — pp.144146.

• Andrey Kolmogorov (1930) Sur la notion de la moyenne. Atti Accad. Naz. Lincei 12, pp.388391.

• John Bibby (1974) "Axiomatisations of the average and a further generalisation of monotonic sequences," Glasgow Mathematical Journal, vol. 15, pp.63–65.

• Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952) Inequalities. 2nd ed. Cambridge Univ. Press, Cambridge, 1952.

• B. De Finetti, "Sul concetto di media", vol. 3, p. 36996, 1931, istituto italiano degli attuari.