Shuffle of min’s random variable approximations of bivariate copulas’ realization

ABSTRACT

The comonotonicity and countermonotonicity provide intuitive upper and lower dependence relationship between random variables. This paper constructs the shuffle of min’s random variable approximations for a given Uniform [0, 1] random vector. We find the two optimal orders under which the shuffle of min’s random variable approximations obtained are shown to be extensions of comonotonicity and countermonotonicity. We also provide the rate of convergence of these random vectors approximations and apply them to compute value-at-risk.

KEYWORDS:

• Copula
• Narrow bounds of copula
• Shuffle of Min approximation.

Mathematics Subject Classification:

• 65C10, 62P20, 41A99

Acknowledgments

We thank reviewers for their valuable comments.

Appendix A

We first prove the following lemma before the proof of Theorem 3.1.

Lemma 4.1.

If , then we have (A1) If , then we have (A2)

Proof.

Based on the fact that , it is easy to obtain Equations (A1(A1) ) and (A2(A2) ) from Equation (3.5(3.5) ).

Proof of Theorem 3.1.

We only give the proof of the case . It is similar to prove the case . It is equal to show the inverse negative proposition. That is, we will prove that if there exists h, k such that , , ∀i ≠ h, j ≠ k and , i, j ∈ {0, 1, …, m − 1}, denoting , then is not the maximum Spearman’s rho. In fact, we will show .

Combining Equation (3.7(3.7) ) with Lemma 4.1, we have Then (A3) Suppose . By comparing with , there are only σ⁽¹⁾_h(k) ≠ σ_h⁽¹⁾*(k) and σ⁽¹⁾_h(l) ≠ σ_h⁽¹⁾*(l). Thus, Equation (A3(A3) ) becomes Based on the end points in Equation (3.4(3.4) ), it is easy to compute that where D₁ = ∑^{l − 1}_{q = k + 1}P(A_{h, q}) + 2∑^{k − 1}_{q = 0}P(A_{h, q}). Thus . The conclusion comes.

Additional information

Funding

Zheng’s research was supported by the Young Program of National Natural Science Foundation of China [grant number 11201012]. Yang’s research was partly supported by the Key Program of National Natural Science Foundation of China [grant number 11131002 ] and the National Natural Science Foundation of China [grant number 11271033]. Huang was partly supported by the NSF [grant number DMS-1208952].