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 $M_{f}(x_{1},\dots ,x_{n})=f^{-1}\left({\frac {f(x_{1})+\cdots +f(x_{n})}{n}}\right)$, which can also be written

$M_{f}({\vec {x}})=f^{-1}\left({\frac {1}{n}}\sum _{k=1}^{n}f(x_{k})\right)$

We require f to be injective in order for the inverse function $f^{-1}$ to exist. Since $f$ is defined over an interval, ${\frac {f(x_{1})+\cdots +f(x_{n})}{n}}$ lies within the domain of $f^{-1}$.

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 $I=\mathbb {R}$, 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 $I=\mathbb {R} ^{+}$, 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 $I=\mathbb {R} ^{+}$ and $f(x)={\frac {1}{x}}$, then the f-mean corresponds to the harmonic mean.
• If $I=\mathbb {R} ^{+}$ and $f(x)=x^{p}$, then the f-mean corresponds to the power mean with exponent $p$.
• If $I=\mathbb {R}$ 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), $M_{f}(x_{1},\dots ,x_{n})=\mathrm {LSE} (x_{1},\dots ,x_{n})-\log(n)$. The $-\log(n)$ corresponds to dividing by n, 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:

$M_{f}(x_{1},\dots ,x_{k},x_{k+1},\dots ,x_{n})=M_{f}(\underbrace {m,\dots ,m} _{k{\text{ times}}},x_{k+1},\dots ,x_{n})$

Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks:$M_{f}(x_{1},\dots ,x_{n\cdot k})=M_{f}(M_{f}(x_{1},\dots ,x_{k}),M_{f}(x_{k+1},\dots ,x_{2\cdot k}),\dots ,M_{f}(x_{(n-1)\cdot k+1},\dots ,x_{n\cdot k}))$

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, ${\sqrt {n}}\{M_{f}(X_{1},\dots ,X_{n})-f^{-1}(E_{f}(X_{1},\dots ,X_{n}))\}$ 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 $f$: $\forall a\ \forall b\neq 0((\forall t\ g(t)=a+b\cdot f(t))\Rightarrow \forall x\ M_{f}(x)=M_{g}(x)$.

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 [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, 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 $C$.

$M_{f,C}x=Cx\cdot f^{-1}\left({\frac {f\left({\frac {x_{1}}{Cx}}\right)+\cdots +f\left({\frac {x_{n}}{Cx}}\right)}{n}}\right)$

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 $M_{\nabla F}(\theta _{1},\ldots ,\theta _{n};w)={\nabla F}^{-1}\left(\sum _{i=1}^{n}w_{i}\nabla F(\theta _{i})\right)$, where $w$ is a normalized weight vector ($w_{i}={\frac {1}{n}}$ by default for a balanced average). From the convex duality, we get a dual quasi-arithmetic mean $M_{\nabla F^{*}}$ associated to the quasi-arithmetic mean $M_{\nabla F}$. 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: $M_{\nabla F}(\theta _{1},\theta _{2})=2(\theta _{1}^{-1}+\theta _{2}^{-1})^{-1}.$

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.

[10] MR4355191 - Characterization of quasi-arithmetic means without regularity condition

Burai, P.; Kiss, G.; Szokol, P. Acta Math. Hungar. 165 (2021), no. 2, 474–485.

[11]

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.