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 f.
It is also called Kolmogorov mean after Russian 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 = 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 = 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 = ℝ+ and f(x) = \frac{1}{x}, then the f-mean corresponds to the harmonic mean.
If I = ℝ+ and f(x) = x^p, then the f-mean corresponds to the power mean with exponent p.
If I = 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 , 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_fis 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, 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.
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 = C x \cdot f^{-1}\left( \frac{f\left(\frac{x_1}{C x}\right) + \cdots + f\left(\frac{x_n}{C x}\right)}{n} \right)
However this modification may violate monotonicity and the partitioning property of the mean.
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.
See also
Generalized mean
Jensen's inequality
