Quasi-arithmetic mean

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, the f-mean of ${\displaystyle n}$ numbers ${\displaystyle x_{1},\dots ,x_{n}\in I}$ is defined as ${\displaystyle M_{f}(x_{1},\dots ,x_{n})=f^{-1}\left({\frac {f(x_{1})+\cdots +f(x_{n})}{n}}\right)}$, which can also be written

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

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

• 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 the mean in the log semiring, which is a constant shifted version of the LogSumExp (LSE) function (which is the logarithmic sum), ${\displaystyle M_{f}(x_{1},\dots ,x_{n})=\mathrm {LSE} (x_{1},\dots ,x_{n})-\log(n)}$. The ${\displaystyle -\log(n)}$ 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.

The following properties hold for ${\displaystyle M_{f}}$ for any single function ${\displaystyle f}$:

Symmetry: The value of ${\displaystyle M_{f}}$is unchanged if its arguments are permuted.

Idempotency: for all x, ${\displaystyle M_{f}(x,\dots ,x)=x}$.

Monotonicity: ${\displaystyle M_{f}}$ is monotonic in each of its arguments (since ${\displaystyle f}$ is monotonic).

Continuity: ${\displaystyle M_{f}}$ is continuous in each of its arguments (since ${\displaystyle f}$ is continuous).

Replacement: Subsets of elements can be averaged a priori, without altering the mean, given that the multiplicity of elements is maintained. With ${\displaystyle m=M_{f}(x_{1},\dots ,x_{k})}$ it holds:

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

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

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

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

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

Central limit theorem : Under regularity conditions, for a sufficiently large sample, ${\displaystyle {\sqrt {n}}\{M_{f}(X_{1},\dots ,X_{n})-f^{-1}(E_{f}(X_{1},\dots ,X_{n}))\}}$ is approximately normal.[1]

Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and scaling of ${\displaystyle f}$: ${\displaystyle \forall a\ \forall b\neq 0((\forall t\ g(t)=a+b\cdot f(t))\Rightarrow \forall x\ M_{f}(x)=M_{g}(x)}$.

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.[2]^{:chapter 17}
• Self-distributivity is essentially sufficient to characterize quasi-arithmetic means.[2]^{:chapter 17}
• Replacement: Kolmogorov proved that the five properties of symmetry, fixed-point, monotonicity, continuity, and replacement fully characterize the quasi-arithmetic means.[3]
• 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,[4] but that if one additionally assumes ${\displaystyle M}$ to be an analytic function then the answer is positive.[5]

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 (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 ${\displaystyle C}$.

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

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

• 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.

• Generalized mean
• Jensen's inequality