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 {\displaystyle 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 {\displaystyle \ f\ } is a function that maps some continuous interval I {\displaystyle \ I\ } of the real line to some other continuous subset J ≡ f ( I ) {\displaystyle \ J\equiv f(I)\ } of the real numbers, and f {\displaystyle \ f\ } is both continuous, and injective (one-to-one).
(We require f {\displaystyle \ f\ } to be injective on I {\displaystyle \ I\ } in order for an inverse function f − 1 {\displaystyle \ f^{-1}\ } to exist. We require I {\displaystyle \ I\ } and J {\displaystyle \ J\ } to both be continuous intervals in order to ensure that an average of any finite (or infinite) subset of values within J {\displaystyle \ J\ } will always correspond to a value in I {\displaystyle \ I\ } .) Subject to those requirements, the f mean of n {\displaystyle \ n\ } numbers x 1 , … , x n ∈ I {\displaystyle \ x_{1},\ldots ,x_{n}\in I\ } is defined to be
M f ( x 1 , … , x n ) ≡ f − 1 ( 1 n ( f ( x 1 ) + ⋯ + f ( x n ) ) ) , {\displaystyle \ M_{f}(x_{1},\dots ,x_{n})\;\equiv \;f^{-1}\!\left(\ {\frac {1}{n}}{\Bigl (}\ f(x_{1})+\cdots +f(x_{n})\ {\Bigr )}\ \right)\ ,}
or equivalently
M f ( x → ) = f − 1 ( 1 n ∑ k = 1 n f ( x k ) ) . {\displaystyle \ M_{f}({\vec {x}})\;=\;f^{-1}\!\!\left(\ {\frac {1}{n}}\sum _{k=1}^{n}f(x_{k})\ \right)~.}
A consequence of f {\displaystyle \ f\ } being defined over some selected interval, I , {\displaystyle \ I\ ,} mapping to yet another interval, J , {\displaystyle \ J\ ,} is that 1 n ( f ( x 1 ) + ⋯ + f ( x n ) ) {\displaystyle \ {\frac {1}{n}}\left(\ f(x_{1})+\cdots +f(x_{n})\ \right)\ } must also lie within J . {\displaystyle \ J\ ~.} And because J {\displaystyle \ J\ } is the domain of f − 1 , {\displaystyle \ f^{-1}\ ,} so in turn f − 1 {\displaystyle \ f^{-1}\ } must produce a value inside the same domain the values originally came from, I . {\displaystyle \ I~.}
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