  In this paper we propose a new aggregation method for constructing composite
indicators that is based on a penalization of the power means. The idea
underlying this approach consists in multiplying the power mean by a factor
that takes into account for the horizontal heterogeneity among indicators with
the aim of penalizing the units with larger heterogeneity. In order to measure
this heterogeneity, we scale the vector of normalized indicators by their power
means, we compute the variance of the scaled normalized indicators transformed
by means of the appropriate Box-Cox function, and we measure the heterogeneity
as the counter image of this variance through the Box-Cox function. The
resulting penalization factor can be interpreted as the relative error, or the
loss of information, that we obtain substituting the vector of the normalized
indicators with their power mean. This penalization approach has the advantage
to be fully data-driven and to be coherent with the same principle underlying
the power mean approach, that is the minimum loss of information principle as
well as to allow for a more refined rankings. The penalized power mean of order
one coincides with the Mazziotta Pareto Index.
