On the property of reduction of variability for mean functions

Mean functions, formally introduced by Cauchy in 1821, are common tools both in the estimation of location measures of unknown distributions and in the aggregation of given data with the purpose of reducing their variability before application of prediction algorithms. However, apart for few specific cases, for this family of statistics there is a lack of existence of formal proofs confirming the reduction in variability with respect to original data. In this paper we provide several results dealing with this property, providing conditions such that the components of a sample and the corresponding mean function are comparable in the convex stochastic order. These results refer to different families of mean functions, such as Ordered Weighted Averagings (OWA), weighted quasi-arithmetic means and the nullnorms class of operators, which are aggregation functions of interest in the field of fuzzy neural networks and in machine learning algorithms. Additional results dealing with the closure under distortion of the convex stochastic order are provided as well.