Distribution-free risk analysis

Gray, Ander and Ferson, Scott and Kreinovich, Vladik and Patelli, Edoardo (2022) Distribution-free risk analysis. International Journal of Approximate Reasoning, 146. pp. 133-156. ISSN 0888-613X (https://doi.org/10.1016/j.ijar.2022.04.001)

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Abstract

Elementary formulas for propagating information about means and variances through mathematical expressions have long been used by analysts. Yet the precise implications of such information are rarely articulated. This paper explores distribution-free techniques for risk analysis that do not require simulation, sampling or approximation of any kind. We describe best-possible bounds on risks that can be inferred given only information about the range, mean and variance of a random variable. These bounds generalise the classical Chebyshev inequality in an obvious way. We also collect in convenient tables several formulas for propagating range and moment information through calculations involving 7 binary convolutions (addition, subtraction, multiplication, division, powers, minimum, and maximum) and 9 unary transformations (scalar multiplication, scalar translation, exponentiation, natural and common logarithms, reciprocal, square, square root and absolute value) commonly encountered in risk expressions. These formulas are rigorous rather than approximate, and in most cases are either exact or mathematically best-possible. The formulas can be used effectively even when only interval estimates of the moments are available. Although most discussions of moment propagation assume stochastic independence among variables, this paper shows the assumption to be unnecessary and generalises formulas for the case when no assumptions are made about dependence, and when correlations are partially known. Along with partial means and variances, we show how interval covariances may be propagated and tracked through expressions.

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