# Invariance Explains Multiplicative and Exponential Skedactic Functions

## Publication Date

8-2015

Technical Report: UTEP-CS-15-63

Published in: Van Nam Huynh, Vladik Kreinovich, and Songsak Sriboonchitta (eds.), Causal Inference in Econometrics, Springer Verlag, Cham, Switzerland, 2016, pp. 119-131.

## Abstract

In many situation, we have an (approximately) linear dependence between several quantities y = f(x1, ..., xn). The variance v of the corresponding approximation error often depends on the values of the quantities x1, ..., xn: v = v(x1, ..., xn); the function describing this dependence is known as the skedactic function. Empirically, two classes of skedactic functions are most successful: multiplicative functions v = c * |x1|γ1 * ... * |xn|γn and exponential functions v = exp(α + γ1 * x1 + ... + γn * xn). In this paper, we use natural invariance ideas to provide a possible theoretical explanation for this empirical success; we explain why in some situations multiplicative skedactic functions work better and in some exponential ones. We also come up with a general class of invariant skedactic function that includes both multiplicative and exponential functions as particular cases.

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