Measuring the influence of the kth largest variable on functions over the unit hypercube

Authors:
Jean-Luc Marichal;Pierre Mathonet
Affiliations:
Mathematics Research Unit, FSTC, University of Luxembourg, Luxembourg, Grand Duchy of Luxembourg;Mathematics Research Unit, FSTC, University of Luxembourg, Luxembourg, Grand Duchy of Luxembourg
Venue:
MDAI'10 Proceedings of the 7th international conference on Modeling decisions for artificial intelligence
Year:
2010

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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 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.