utmeans

Utmeans, short for utility-transformed means, are a class of generalized means defined by applying a monotone transformation U to each data value, computing the arithmetic mean of the transformed values, and then applying the inverse transformation to return to the original scale. Formally, for a data set x1, x2, ..., xn in the domain of U, the utmean is M_U(x) = U^{-1}((1/n) ∑_{i=1}^n U(x_i)). If U is the identity function, M_U is the ordinary arithmetic mean.

The utmean framework unifies several known means through the choice of U. If U(x) = x, M_U is the arithmetic mean. If U(x) = log x (with x > 0), M_U becomes the geometric mean: exp((1/n) ∑ log x_i). If U(x) = x^p (p ≠ 0), M_U yields the power (or Hölder) means: ( (1/n) ∑ x_i^p )^{1/p}. With U(x) = 1/x (for positive data), M_U yields the harmonic mean. The method requires that U be strictly increasing and continuous, with a well-defined inverse on the transformed range, and data must lie in the domain of U.

Properties of utmeans include monotonicity in each argument and symmetry with respect to data order. They provide a flexible way to tailor aggregation to context by selecting U to emphasize or downweight certain values, making them useful in decision theory, economics, and robust data fusion. Computation is straightforward but depends on the numerical stability of U and its inverse.

Example: for x = (1, 4, 9) and U = log (positive values), M_U = exp((0 + 1.386 + 2.197)/3) ≈ 3.30.