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 . 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 of the real line to the real numbers, and is both continuous and injective, the f-mean of numbers is defined as , which can also be written

We require f to be injective in order for the inverse function to exist. Since is defined over an interval, lies within the domain of .

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 nor smaller than the smallest number in .

Examples

If = ℝ, the real line, and , (or indeed any linear function , not equal to 0) then the f-mean corresponds to the arithmetic mean.
If = ℝ⁺, the positive real numbers and , 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 = ℝ⁺ and , then the f-mean corresponds to the harmonic mean.
If = ℝ⁺ and , then the f-mean corresponds to the power mean with exponent .
If = ℝ and , 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), . The 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 for any single function :

Symmetry: The value of is unchanged if its arguments are permuted.

Idempotency: for all x, .

Monotonicity: is monotonic in each of its arguments (since is monotonic).

Continuity: is continuous in each of its arguments (since 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 of two variables: .

Mediality: For any quasi-arithmetic mean of two variables:.

Balancing: For any quasi-arithmetic mean 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, which are generalizations of quasi-arithmetic means.

Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and scaling of : .

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 to be an analytic function then the answer is positive.

Homogeneity

Means are usually homogeneous, but for most functions , 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 .

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

Generalizations

Consider a Legendre-type strictly convex function . Then the gradient map is globally invertible and the weighted multivariate quasi-arithmetic mean is defined by , where 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 a symmetric positive-definite matrix. The pair of matrix quasi-arithmetic means yields the matrix harmonic mean: