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

Definition

If ''f'' is a function which maps an interval $I$ of the real line to the real numbers, and is both continuous and injective, the ''f''-mean of $n$ numbers
$x_1, \dots, x_n \in I$
is defined as $M_f(x_1, \dots, x_n) = f^\left( \fracn \right)$, which can also be written
:$M_f(\vec x)= f^\left(\frac \sum_^f(x_k) \right)$

We require ''f'' to be injective in order for the inverse function $f^$ to exist. Since $f$ is defined over an interval, $\fracn$ lies within the domain of $f^$.

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 $x$ nor smaller than the smallest number in $x$.

Examples

* If $I = \mathbb$, the real line, and $f(x) = x$, (or indeed any linear function $x\mapsto a\cdot x + b$, $a$ not equal to 0) then the ''f''-mean corresponds to the arithmetic mean.
* If $I = \mathbb^+$, the positive real numbers and $f(x) = \log(x)$, then the ''f''-mean corresponds to the geometric mean. According to the ''f''-mean properties, the result does not depend on the base of the logarithm as long as it is positive and not 1.
* If $I = \mathbb^+$ and $f(x) = \frac$, then the ''f''-mean corresponds to the harmonic mean.
* If $I = \mathbb^+$ and $f(x) = x^p$, then the ''f''-mean corresponds to the power mean with exponent $p$.
* If $I = \mathbb$ and $f(x) = \exp(x)$, then the ''f''-mean is the mean in the log semiring, which is a constant shifted version of the LogSumExp (LSE) function (which is the logarithmic sum), $M_f(x_1, \dots, x_n) = \mathrm(x_1, \dots, x_n)-\log(n)$. The $-\log(n)$ corresponds to dividing by , since logarithmic division is linear subtraction. The LogSumExp function is a smooth maximum: a smooth approximation to the maximum function.

Properties

The following properties hold for $M_f$ for any single function $f$:

Symmetry: The value of $M_f$ is unchanged if its arguments are permuted.

Idempotency: for all ''x'', $M_f(x,\dots,x) = x$.

Monotonicity: $M_f$ is monotonic in each of its arguments (since $f$ is monotonic).

Continuity: $M_f$ is continuous in each of its arguments (since $f$ is continuous).

Replacement: Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. With $m=M_f(x_1,\dots,x_k)$ it holds:

:$M_f(x_1,\dots,x_k,x_,\dots,x_n) = M_f(\underbrace_,x_,\dots,x_n)$

Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks:$M_f(x_1,\dots,x_) = M_f(M_f(x_1,\dots,x_), M_f(x_,\dots,x_), \dots, M_f(x_,\dots,x_))$

Self-distributivity: For any quasi-arithmetic mean $M$ of two variables: $M(x,M(y,z))=M(M(x,y),M(x,z))$.

Mediality: For any quasi-arithmetic mean $M$ of two variables:$M(M(x,y),M(z,w))=M(M(x,z),M(y,w))$.

Balancing: For any quasi-arithmetic mean $M$ of two variables:$M\big(M(x, M(x, y)), M(y, M(x, y))\big)=M(x, y)$.

Central limit theorem : Under regularity conditions, for a sufficiently large sample, $\sqrt\$ is approximately normal.
A similar result is available for Bajraktarević means and deviation means, which are generalizations of quasi-arithmetic means.

Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and scaling of $f$: $\forall a\ \forall b\ne0 ((\forall t\ g(t)=a+b\cdot f(t)) \Rightarrow \forall x\ M_f (x) = M_g (x)$.

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 0for 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, but that if one additionally assumes $M$ to be an analytic function then the answer is positive.

Homogeneity

Means are usually homogeneous, but for most functions $f$, the ''f''-mean is not.
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 $C$.
:$M_ x = C x \cdot f^\left( \frac \right)$
However this modification may violate monotonicity and the partitioning property of the mean.

Generalizations

Consider a Legendre-type strictly convex function $F$. Then the gradient map $\nabla F$ is globally invertible and the weighted multivariate quasi-arithmetic mean is defined by
$M_(\theta_1,\ldots,\theta_n;w) = ^\left(\sum_^n w_i \nabla F(\theta_i)\right)$, where $w$ is a normalized weight vector ($w_i=\frac$ by default for a balanced average). From the convex duality, we get a dual quasi-arithmetic mean $M_$ associated to the quasi-arithmetic mean $M_$.
For example, take $F(X)=-\log\det(X)$ for $X$ a symmetric positive-definite matrix.
The pair of matrix quasi-arithmetic means yields the matrix harmonic mean:
$M_(\theta_1,\theta_2)=2(\theta_1^+\theta_2^)^.$

See also

* Generalized mean
* Jensen's inequality

References

* Andrey Kolmogorov (1930) "On the Notion of Mean", in "Mathematics and Mechanics" (Kluwer 1991) — pp. 144–146.
* Andrey Kolmogorov (1930) Sur la notion de la moyenne. Atti Accad. Naz. Lincei 12, pp. 388–391.
* 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.

{{DEFAULTSORT:Quasi-Arithmetic Mean
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MR4355191 - Characterization of quasi-arithmetic means without regularity conditionBurai, P.; Kiss, G.; Szokol, P.Acta Math. Hungar. 165 (2021), no. 2, 474–485.

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MR4574540 - A dichotomy result for strictly increasing bisymmetric maps

Burai, Pál; Kiss, Gergely; Szokol, Patricia

J. Math. Anal. Appl. 526 (2023), no. 2, Paper No. 127269, 9 pp.