Gaussian Double Mean Process

By Kardi Teknomo, PhD.

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We can view double mean as a general process to find means of two input numbers using as a recurrence formula of two means.

Philips (2000) proposed Gaussian double mean process is a difference equation that involves two means and with two inputs and , which relates to each other using the following formula

Where

Compare this difference equation formula with Archimedean Double mean process.

The Gaussian double mean has a very nice property that the sequence and will converge to a common limit and it converges very fast with quadratic speed.

Example: (Arithmetic Geometric mean)

We use arithmetic and geometric mean, thus the name of this mean is AGM = Arithmetic Geometric mean

and

Philips (2000) shows that the sequence and will converge to a common limit where

For instance:

,

You may notice that AGM is symmetric mean.

Example: ( Harmonic-geometric mean )

We use harmonic mean and geometric mean

and

Compare this with the Archimedean harmonic-geometric mean, the (Gaussian) Harmonic-geometric mean is symmetric.

For instance:

,

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See also: Archimedean Double mean process
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Preferable reference for this tutorial is

Teknomo, Kardi. Mean and Average. http:\\people.revoledu.com\kardi\ tutorial\BasicMath\Average\