Risk aggregation with empirical margins: Latin hypercubes, empirical copulas, and convergence of sum distributions

TL;DR

This paper investigates the convergence properties of multivariate distributions constructed with empirical margins and copulas, focusing on risk aggregation, and establishes uniform consistency and convergence rates for the sum distribution.

Contribution

It provides the first strong uniform consistency results and convergence rate criteria for the sum distribution in empirical copula models, extending beyond classic CLTs.

Findings

Strong uniform consistency of the estimated sum distribution function.

A sufficient criterion for the convergence rate of $O(n^{-1/2})$ in probability.

Applicability to copulas with bounded and certain unbounded densities.

Abstract

This paper studies convergence properties of multivariate distributions constructed by endowing empirical margins with a copula. This setting includes Latin Hypercube Sampling with dependence, also known as the Iman--Conover method. The primary question addressed here is the convergence of the component sum, which is relevant to risk aggregation in insurance and finance. This paper shows that a CLT for the aggregated risk distribution is not available, so that the underlying mathematical problem goes beyond classic functional CLTs for empirical copulas. This issue is relevant to Monte-Carlo based risk aggregation in all multivariate models generated by plugging empirical margins into a copula. Instead of a functional CLT, this paper establishes strong uniform consistency of the estimated sum distribution function and provides a sufficient criterion for the convergence rate $O(n^{-1/2})$ in probability. These convergence results hold for all copulas with bounded densities. Examples with unbounded densities include bivariate Clayton and Gauss copulas. The convergence results are not specific to the component sum and hold also for any other componentwise non-decreasing aggregation function. On the other hand, convergence of estimates for the joint distribution is much easier to prove, including CLTs. Beyond Iman--Conover estimates, the results of this paper apply to multivariate distributions obtained by plugging empirical margins into an exact copula or by plugging exact margins into an empirical copula.