Sklar’s theorem, copula products, and ordering results in factor models

Abstract

We consider a completely specified factor model for a risk vector X = (X 1, . . ., Xd ), where the joint distributions of the components of X with a risk factor Z and the conditional distributions of X given Z are specified. We extend the notion of *-product of d-copulas as introduced for d = 2 and continuous factor distribution in Darsow et al. [6] and Durante et al. [8] to the multivariate and discontinuous case. We give a Sklar-type representation theorem for factor models showing that these *-products determine the copula of a completely specified factor model. We investigate in detail approximation, transformation, and ordering properties of *-products and, based on them, derive general orthant ordering results for completely specified factor models in dependence on their specifications. The paper generalizes previously known ordering results for the worst case partially specified risk factor models to some general classes of positive or negative dependent risk factor models. In particular, it develops some tools to derive sharp worst case dependence bounds in subclasses of completely specified factor models.

Estimate a nonparametric copula density function based on probit and wavelet transforms

“…A copula is a function that illustrate modeling the dependency of random variables. Sklar's created and initially utilized the copula [16]. This function has several advantages for modeling dependencies in multivariate data.…”

تقدير كثافة الرابطة باستعمال التحويل المويجي

“…In this form, the coefficients 𝛼 𝑗 0 𝑘 and 𝛽 𝑗𝑘 (1) , 𝛽 𝑗𝑘 (2) and 𝛽 𝑗𝑘 (3) with 𝑗 ≥ 𝑗 0 are unique for each choice of 𝑗 0 ∈ 𝑁. For all 𝑗 ∈ 𝑁 and = (𝑘 1 , 𝑘 2 ) ∈ 𝑧 2 , the functions 𝜑 𝑗0𝑘 and 𝜓 𝑗𝑘…”

“…Copula functions are very useful tools for analyzing dependencies between random variables. For the strong boundary effects, the copula density estimator was used according to Sklar's theorem 1 . Copulas are quickly becoming popular as multivariate data modeling tools.…”

Improved Robust Price Bounds for Multi-Asset Derivatives under Market-Implied Dependence Information

“…(b) A similar result as the equalitity in (4.9) holds true for lower bounds. However, an analogon of (4.10) for lower bounds cannot be obtained for general dimension since upper product ordering results are different from ordering results for lower products which correspond to lower bounds in partially specified factor models, see [8].…”