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The equally distributed welfare equivalent income associated with an Atkinson Index with an inequality aversion parameter of 1.0 is simply the geometric mean of incomes. For values other than one, the equivalent value is an Lp norm divided by the number of elements, with p equal to one minus the inequality aversion parameter.
The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values. If all the weights are equal, the weighted geometric mean simplifies to the ordinary unweighted geometric mean. [1]
The arithmetic mean, or less precisely the average, of a list of n numbers x 1, x 2, . . . , x n is the sum of the numbers divided by n: + + +. The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:
Then the maximum spacing estimator of θ 0 is defined as a value that maximizes the logarithm of the geometric mean of sample spacings: ^ = (), = + + = + = + (). By the inequality of arithmetic and geometric means , function S n ( θ ) is bounded from above by −ln( n +1), and thus the maximum has to exist at least in the supremum sense.
This estimate is sometimes referred to as the "geometric CV" (GCV), [19] [20] due to its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean. The parameters μ and σ can be obtained, if the arithmetic mean and the arithmetic variance are known:
A power mean serves a non-linear moving average which is shifted towards small signal values for small p and emphasizes big signal values for big p. Given an efficient implementation of a moving arithmetic mean called smooth one can implement a moving power mean according to the following Haskell code.
The output image G(x,y) of a geometric mean is given by (,) = [, (,)] Where S(x,y) is the original image, and the filter mask is m by n pixels. Each pixel of the output image at point (x,y) is given by the product of the pixels within the geometric mean mask raised to the power of 1/mn.
In mathematics, the arithmetic–geometric mean (AGM or agM [1]) of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential , trigonometric functions , and other special functions , as well as some ...