Quasi-arithmetic mean

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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 . It is also called Kolmogorov mean after Russian mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.

DefinitionEdit

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  .

ExamplesEdit

  • 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.

PropertiesEdit

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.[1] A similar result is available for Bajraktarević means, which are generalizations of quasi-arithmetic means.[2]

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

CharacterizationEdit

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.[3]:chapter 17
  • Self-distributivity is essentially sufficient to characterize quasi-arithmetic means.[3]:chapter 17
  • Replacement: Kolmogorov proved that the five properties of symmetry, fixed-point, monotonicity, continuity, and replacement fully characterize the quasi-arithmetic means.[4]
  • 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,[5] but that if one additionally assumes   to be an analytic function then the answer is positive.[6]

HomogeneityEdit

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.

ReferencesEdit

  1. ^ de Carvalho, Miguel (2016). "Mean, what do you Mean?". The American Statistician. 70 (3): 764‒776. doi:10.1080/00031305.2016.1148632.
  2. ^ Barczy, M. and Burai, P. (2019). "Limit theorems for Bajraktarević and Cauchy quotient means of independent identically distributed random variables". arXiv:1909.02968 [math.PR].CS1 maint: multiple names: authors list (link)
  3. ^ a b Aczél, J.; Dhombres, J. G. (1989). Functional equations in several variables. With applications to mathematics, information theory and to the natural and social sciences. Encyclopedia of Mathematics and its Applications, 31. Cambridge: Cambridge Univ. Press.CS1 maint: multiple names: authors list (link)
  4. ^ Grudkin, Anton (2019). "Characterization of the quasi-arithmetic mean". Math stackexchange.
  5. ^ Aumann, Georg (1937). "Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften". Journal für die reine und angewandte Mathematik. 1937 (176): 49–55. doi:10.1515/crll.1937.176.49. S2CID 115392661.
  6. ^ Aumann, Georg (1934). "Grundlegung der Theorie der analytischen Analytische Mittelwerte". Sitzungsberichte der Bayerischen Akademie der Wissenschaften: 45–81.
  • 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 alsoEdit