Measuring the Influence of the kth Largest Variable on Functions over the Unit Hypercube

By considering a least squares approximation of a given square integrable function f : [0, 1]n → IR by a shifted L-statistic function (a shifted linear combination of order statistics), we define an index which measures the global influence of the kth 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 kth largest variable or, under certain natural conditions on f , as an average value of the derivative of f in the direction of the kth largest variable. We also discuss a few applications of this index in statistics and aggregation theory.