Quasi-arithmetic mean

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.

Definition

If f is a function which maps an interval of the real line to the real numbers, and is both continuous and injective then we can define the f-mean of two numbers

as

For numbers

,

the f-mean is

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

Properties

With it holds
.

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 and the geometric mean; see HardyLittlewoodPólya, 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.

References

  1. Aumann, Georg (1937). "Vollkommene Funktionalmittel und gewisse Kegelschnitteigenschaften". Journal für die reine und angewandte Mathematik. 176: 49–55. doi:10.1515/crll.1937.176.49.
  2. Aumann, Georg (1934). "Grundlegung der Theorie der analytischen Analytische Mittelwerte". Sitzungsberichte der Bayerischen Akademie der Wissenschaften: 45–81.
  3. de Carvalho, Miguel (2016). "Mean, what do you Mean?". The American Statistician. 70: 764‒776. doi:10.1080/00031305.2016.1148632.

See also

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