Measuring the distance between single random inputs and OWA operators

Considered a random sample of fixed cardinality extracted from a population with unknown distribution, this paper deals with the ability of an Ordered Weighted Averaging (OWA) operator to approximate the distance between the values it assumes and the single observation of the sample. To this purpose, a measure of distance between random quantities recently introduced in the literature is considered, which takes into account both their mutual dependence and the shape of their distributions. Conditions are identified on the weights of the OWA operator that minimize this distance, or for which the different distances are ordered as the weights of two different operators vary. The paper also considers the case of operators defined as mixture of order statistics and, subsequently, the case of input values from populations with different distributions, showing conditions on these distributions that highlight the importance of the inputs in the value assumed by this distance.