Non-exchangeability of copulas arising from shock models

TL;DR

This paper investigates the non-exchangeability of copulas derived from shock models, providing explicit bounds on their asymmetry measures and interpreting the shocks at maximal asymmetry points, aiding better model selection.

Contribution

It introduces the maximal asymmetry function for shock-based copulas, computes sharp bounds for key families, and offers practical interpretations for model selection.

Findings

Computed maximal asymmetry functions for major shock copula families.

Established sharp bounds for asymmetry measure $_$ for Marshall, maxmin, and RMM copulas.

Provided practical examples illustrating the interpretation of shocks at maximal asymmetry.

Abstract

When choosing the right copula for our data a key point is to distinguish the family that describes it at the best. In this respect, a better choice of the copulas could be obtained through the information about the (non)symmetry of the data. Exchangeability as a probability concept (first next to independence) has been studied since 1930's, copulas have been studied since 1950's, and even the most important class of copulas from the point of view of applications, i.e. the ones arising from shock models s.a. Marshall's copulas, have been studied since 1960's. However, the point of non-exchangeability of copulas was brought up only in 2006 and has been intensively studied ever since. One of the main contributions of this paper is the maximal asymmetry function for a family of copulas. We compute this function for the major families of shock-based copulas, i.e. Marshall, maxmin and reflected maxmin (RMM for short) copulas and also for some other important families. We compute the sharp bound of asymmetry measure $\mu_\infty$, the most important of the asymmetry measures, for the family of Marshall copulas and the family of maxmin copulas, which both equal to $\frac{4}{27}\ (\approx 0.148)$. One should compare this bound to the one for the class of PQD copulas to which they belong, which is $3-2\sqrt{2}\ \approx 0.172)$, and to the general bound for all copulas that is $\frac13$. Furthermore, we give the sharp bound of the same asymmetry measure for RMM copulas which is $3-2\sqrt{2}$, compared to the same bound for NQD copulas, where they belong, which is $\sqrt{5}-2\ (\approx 0.236)$. One of our main results is also the statistical interpretation of shocks in a given model at which the maximal asymmetry measure bound is attained. These interpretations for the three families studied are illustrated by examples that should be helpful to practitioners when choosing the model for their data.