# A Note on Upper Tail Behavior of Liouville Copulas

## Abstract

**:**

## 1. Introduction

## 2. Notation

## 3. Tail Order of a Scale Mixture Model

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

## 4. Tail Order Function of Liouville Copulas

**Proposition**

**2**(Upper tail order of Liouville copulas).

**Proof.**

**Proposition**

**3**(Tail order functions of Liouville copulas).

- If $0<\alpha <\xi $, then$$\begin{array}{c}\hfill C(u{w}_{1},\cdots ,u{w}_{d})\sim u\times \frac{B(\xi ,d\xi -\xi )}{B(\xi -\alpha ,d\xi -\xi )}{\int}_{\mathit{s}\ge \mathbf{0},\left|\right|\mathit{s}\left|\right|=1}\underset{i}{min}\{{s}_{i}^{-\alpha}{w}_{i}\}{F}_{\mathit{S}}(d\mathit{s}),\phantom{\rule{1.em}{0ex}}u\to {0}^{+}.\end{array}$$
- If $0<\xi <\alpha $, then$$\begin{array}{c}\hfill C(u{w}_{1},\cdots ,u{w}_{d})\sim {u}^{\alpha /\xi}\ell (u)\times {\left[\frac{B(\xi ,d\xi -\xi )}{\xi \mathbb{E}[{T}^{\xi}]}\right]}^{\alpha /\xi}{\int}_{\mathit{s}\ge \mathbf{0},\left|\right|\mathit{s}\left|\right|=1}\underset{i}{min}\{{s}_{i}^{-\alpha}{w}_{i}\}{F}_{\mathit{S}}(d\mathit{s}),\phantom{\rule{1.em}{0ex}}u\to {0}^{+},\end{array}$$
- If $0<\xi =\alpha $, and moreover, $\mathbb{E}[{T}^{\xi}]<\infty $, then expression (6) holds.

**Proof.**

**Example**

**1**(Gamma-Liouville copula: tail dependence).

**Example**

**2**(Gamma-Liouville copula: intermediate and negative tail dependence).

## 5. Tail Order Density of Liouville Copulas

**Proposition**

**4.**

- 1.
- If ${C}_{1}(x,v):={D}_{x}C(x,v)$ is ultimately monotone as $x\to {0}^{+}$ for any $0<v\le 1$, and $g({w}_{1},{w}_{2}):={lim}_{u\to {0}^{+}}\frac{{C}_{1}(u{w}_{1},u{w}_{2})}{{u}^{\kappa -1}\ell (u)}$ exists and continuous in ${w}_{1}$, then ${D}_{{w}_{1}}b({w}_{1},{w}_{2})$ exists and $g({w}_{1},{w}_{2})={D}_{{w}_{1}}b({w}_{1},{w}_{2})$.
- 2.
- Further, if $c(u,x):={D}_{x}{C}_{1}(u,x)$ is ultimately monotone as $x\to {0}^{+}$ for any $0<u\le 1$, and $h({w}_{1},{w}_{2}):={lim}_{u\to {0}^{+}}\frac{c(u{w}_{1},u{w}_{2})}{{u}^{\kappa -2}\ell (u)}$ exists and continuous in ${w}_{2}$, then ${D}_{{w}_{2}}{D}_{{w}_{1}}b({w}_{1},{w}_{2})$ exists and $h({w}_{1},{w}_{2})={D}_{{w}_{2}}{D}_{{w}_{1}}b({w}_{1},{w}_{2})$.

**Proof.**

**Example**

**3**(Lower tail of Gumbel copula).

**Proposition**

**5.**

**Proof.**

**Remark**

**3.**

**Example**

**4.**

## 6. Concluding Remark

## Acknowledgments

## Conflicts of Interest

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Hua, L. A Note on Upper Tail Behavior of Liouville Copulas. *Risks* **2016**, *4*, 40.
https://doi.org/10.3390/risks4040040

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Hua L. A Note on Upper Tail Behavior of Liouville Copulas. *Risks*. 2016; 4(4):40.
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Hua, Lei. 2016. "A Note on Upper Tail Behavior of Liouville Copulas" *Risks* 4, no. 4: 40.
https://doi.org/10.3390/risks4040040