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

TL;DR

This paper introduces an influence index for the k-th largest variable in functions over the unit hypercube, based on least squares approximation and order statistics, with applications in statistics and aggregation theory.

Contribution

It defines a new influence measure for order statistics using least squares approximation, providing interpretability and applications in various fields.

Findings

The influence index can be interpreted as an average difference quotient or derivative.

The index has desirable mathematical properties and interpretability.

Applications demonstrate its usefulness in statistics and aggregation theory.

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.