Kolmogorov's Generalized Mean

By Kardi Teknomo, PhD.

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Phillips (2000) suggests the following quasi-arithmetic generalized mean which has been available through the work Andrey Kolmogorov in 1930. The mean is quite general that it can cover arithmetic mean, quadratic mean, harmonic mean and others. Suppose we have a continuous monotonic function from positive real numbers to positive real number and the inverse function of exist, then we can make infinite number of means using

Extending the generalized mean to input numbers is straightforward.

I hope you remember the definition of inverse function that if then . Simple inverse function can be obtained by exchange and in the expression. Then the following are two properties of inverse function:

Example:

, then , then we have which is an arithmetic mean

Example:

, then , then we have which can be simplified into Geometric mean by

Example:

, then , then we get which is a quadratic mean

Example:

, then , then we obtain which is a harmonic mean

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See also: Minkowski mean, Lehmer mean, arithmetic mean, harmonic mean, geometric mean, quadratic mean
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Teknomo, Kardi. Mean and Average. http:\\people.revoledu.com\kardi\ tutorial\BasicMath\Average\