In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean [1] 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.
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 .
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. [2] A similar result is available for Bajraktarević means, which are generalizations of quasi-arithmetic means. [3]
Scale-invariance: The quasi-arithmetic mean is invariant with respect to offsets and scaling of : .
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).
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.
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