Measuring the influence of the k-th largest variable on functions over the unit hypercube

Abstract

By considering a least squares approximation of a given square integrable function f:[0,1]^n --> R by a shifted L-statistic function (a shifted linear combination of order statistics), we define an index which measures the global influence of the k-th largest variable on f. We show that this influence index has appealing properties and we interpret it as an average value of the difference quotient of f in the direction of the k-th largest variable or, under certain natural conditions on f, as an average value of the derivative of f in the direction of the k-th largest variable. We also discuss a few applications of this index in statistics and aggregation theory.