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 is defined as, which can also be written We require f to be injective in order for the inverse function f^{-1} to exist. Since f is defined over an interval, 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

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,. 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 it holds: Partitioning: The computation of the mean can be split into computations of equal sized sub-blocks: Self-distributivity: For any quasi-arithmetic mean M of two variables:. Mediality: For any quasi-arithmetic mean M of two variables:. Balancing: For any quasi-arithmetic mean M of two variables:. Central limit theorem : Under regularity conditions, for a sufficiently large sample, 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:.

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).

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. 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, where w is a normalized weight vector ( by default for a balanced average). From the convex duality, we get a dual quasi-arithmetic mean associated to the quasi-arithmetic mean. For example, take for X a symmetric positive-definite matrix. The pair of matrix quasi-arithmetic means yields the matrix harmonic mean: