# Quasi-arithmetic mean

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In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function ${\displaystyle f}$. It is also called Kolmogorov mean after Russian mathematician Andrey Kolmogorov.

## Definition

If f is a function which maps an interval ${\displaystyle I}$ of the real line to the real numbers, and is both continuous and injective then we can define the f-mean of two numbers

${\displaystyle x_{1},x_{2}\in I}$

as

${\displaystyle M_{f}(x_{1},x_{2})=f^{-1}\left({\frac {f(x_{1})+f(x_{2})}{2}}\right).}$

For ${\displaystyle n}$ numbers

${\displaystyle x_{1},\dots ,x_{n}\in I}$,

the f-mean is

${\displaystyle M_{f}(x_{1},\dots ,x_{n})=f^{-1}\left({\frac {f(x_{1})+\cdots +f(x_{n})}{n}}\right).}$

We require f to be injective in order for the inverse function ${\displaystyle f^{-1}}$ to exist. Since ${\displaystyle f}$ is defined over an interval, ${\displaystyle {\frac {f\left(x_{1}\right)+f\left(x_{2}\right)}{2}}}$ lies within the domain of ${\displaystyle f^{-1}}$.

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 ${\displaystyle x}$ nor smaller than the smallest number in ${\displaystyle x}$.

## Examples

• If ${\displaystyle I}$ = ℝ, the real line, and ${\displaystyle f(x)=x}$, (or indeed any linear function ${\displaystyle x\mapsto a\cdot x+b}$, ${\displaystyle a}$ not equal to 0) then the f-mean corresponds to the arithmetic mean.
• If ${\displaystyle I}$ = ℝ+, the positive real numbers and ${\displaystyle f(x)=\log(x)}$, 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 ${\displaystyle I}$ = ℝ+ and ${\displaystyle f(x)={\frac {1}{x}}}$, then the f-mean corresponds to the harmonic mean.
• If ${\displaystyle I}$ = ℝ+ and ${\displaystyle f(x)=x^{p}}$, then the f-mean corresponds to the power mean with exponent ${\displaystyle p}$.
• If ${\displaystyle I}$ = ℝ and ${\displaystyle f(x)=\exp(x)}$, then the f-mean is a constant shifted version of the LogSumExp (LSE) function, ${\displaystyle M_{f}(x_{1},\dots ,x_{n})=LSE(x_{1},\dots ,x_{n})-\log(n)}$. The LogSumExp function is used as a smooth approximation to the maximum function.

## Properties

• Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks.
${\displaystyle M_{f}(x_{1},\dots ,x_{n\cdot k})=M_{f}(M_{f}(x_{1},\dots ,x_{k}),M_{f}(x_{k+1},\dots ,x_{2\cdot k}),\dots ,M_{f}(x_{(n-1)\cdot k+1},\dots ,x_{n\cdot k}))}$
• Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained.
With ${\displaystyle m=M_{f}(x_{1},\dots ,x_{k})}$ it holds
${\displaystyle M_{f}(x_{1},\dots ,x_{k},x_{k+1},\dots ,x_{n})=M_{f}(\underbrace {m,\dots ,m} _{k{\text{ times}}},x_{k+1},\dots ,x_{n})}$
• The quasi-arithmetic mean is invariant with respect to offsets and scaling of ${\displaystyle f}$:
${\displaystyle \forall a\ \forall b\neq 0((\forall t\ g(t)=a+b\cdot f(t))\Rightarrow \forall x\ M_{f}(x)=M_{g}(x)}$.
• If ${\displaystyle f}$ is monotonic, then ${\displaystyle M_{f}}$ is monotonic.
• Any quasi-arithmetic mean ${\displaystyle M}$ of two variables has the mediality property ${\displaystyle M(M(x,y),M(z,w))=M(M(x,z),M(y,w))}$ and the self-distributivity property ${\displaystyle M(x,M(y,z))=M(M(x,y),M(x,z))}$. Moreover, any of those properties is essentially sufficient to characterize quasi-arithmetic means; see Aczél–Dhombres, Chapter 17.
• Any quasi-arithmetic mean ${\displaystyle M}$ of two variables has the balancing property ${\displaystyle M{\big (}M(x,M(x,y)),M(y,M(x,y)){\big )}=M(x,y)}$. An interesting problem is whether this condition (together with fixed-point, symmetry, monotonicity and continuity properties) implies that the mean is quasi-arithmetic. Georg Aumann showed in the 1930s that the answer is no in general,[1] but that if one additionally assumes ${\displaystyle M}$ to be an analytic function then the answer is positive.[2]
• Under regularity conditions, a central limit theorem can be derived for the generalised f-mean, thus implying that for a large sample ${\displaystyle {\sqrt {n}}\{M_{f}(X_{1},\dots ,X_{n})-f^{-1}(E_{f}(X_{1},\dots ,X_{n}))\}}$ is approximately normal.[3]

## Homogeneity

Means are usually homogeneous, but for most functions ${\displaystyle f}$, the f-mean is not. Indeed, the only homogeneous quasi-arithmetic means are the power means and the geometric mean; see Hardy–Littlewood–Pólya, page 68.

The homogeneity property can be achieved by normalizing the input values by some (homogeneous) mean ${\displaystyle C}$.

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

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

## References

1. ^ Aumann, Georg (1937). "Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften". Journal für die reine und angewandte Mathematik. 176: 49–55. doi:10.1515/crll.1937.176.49.
2. ^ Aumann, Georg (1934). "Grundlegung der Theorie der analytischen Analytische Mittelwerte". Sitzungsberichte der Bayerischen Akademie der Wissenschaften: 45–81.
3. ^ de Carvalho, Miguel (2016). "Mean, what do you Mean?". The American Statistician. 70: 764‒776. doi:10.1080/00031305.2016.1148632.
• Aczél, J.; Dhombres, J. G. (1989) Functional equations in several variables. With applications to mathematics, information theory and to the natural and social sciences. Encyclopedia of Mathematics and its Applications, 31. Cambridge Univ. Press, Cambridge, 1989.
• 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.