Abstract
By considering a least squares approximation of a given square integrable function \(f : [0,1]^n \rightarrow {I \kern -3pt \mathcal R}\) 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.
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Marichal, JL., Mathonet, P. (2010). Measuring the Influence of the kth Largest Variable on Functions over the Unit Hypercube. In: Torra, V., Narukawa, Y., Daumas, M. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2010. Lecture Notes in Computer Science(), vol 6408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16292-3_4
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DOI: https://doi.org/10.1007/978-3-642-16292-3_4
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