# Bivariate Kumaraswamy Models via Modified FGM Copulas: Properties and Applications

## Abstract

**:**

## 1. Introduction

- Symmetry: $C(u,\text{}v)=C(v,\text{}u)$, $\forall (u,\text{}v)\in {[0,\text{}1]}^{2}$, and have the lower and upper tail dependence coefficients equal to zero.
- It is positive quadrant dependent (PQD) for $\theta \in (0,\text{}1]$ and negative quadrant dependent (NQD) for $\theta \in [-1,\text{}0).$

## 2. Modified Bivariate FGM Copula

- $\mathsf{\Phi}\left(0\right)=\mathsf{\Psi}\left(0\right)=\mathsf{\Phi}\left(1\right)=\mathsf{\Psi}\left(1\right)=0$. This is known as a boundary condition.
- $\mathrm{min}\{\alpha \beta ,\text{}\xi \eta \}\ge 1,$ where $\alpha =inf\{\frac{\partial \tilde{\mathsf{\Phi}\left(u\right)}}{\partial u}:u\in {A}_{1}\}<0,$where $\frac{\partial \tilde{\mathsf{\Phi}\left(u\right)}}{\partial u}=\mathsf{\Phi}\left(u\right)+u\frac{\partial \mathsf{\Phi}\left(u\right)}{\partial u},\text{}$$\beta =sup\{\frac{\partial \tilde{\mathsf{\Phi}\left(u\right)}}{\partial u}:u\in {A}_{1}\}>0.$Again, $\xi =inf\{\frac{\partial \tilde{\mathsf{\Psi}\left(v\right)}}{\partial v}:v\in {A}_{2}\}<0,\text{}$ and$\eta =sup\{\frac{\partial \tilde{\mathsf{\Psi}\left(v\right)}}{\partial v}:v\in {A}_{2}\}>0,\text{}$ where$${A}_{1}=\{u\in (0,\text{}1):\frac{\partial \tilde{\mathsf{\Phi}\left(u\right)}}{\partial u}exists\},$$$${A}_{2}=\{v\in (0,\text{}1):\frac{\partial \tilde{\mathsf{\Psi}\left(v\right)}}{\partial v}exists\}.$$

**Theorem**

**1.**

**Proof.**

- The associated bivariate copula density from Equation (2) will be$$c(u,\text{}v)=\frac{\partial C(u,\text{}v)}{\partial u\partial v}=1+\theta \mathsf{\Phi}\left(u\right)\mathsf{\Psi}\left(v\right)\left\{\left[1+u\frac{\partial \mathsf{\Phi}\left(u\right)}{\partial u}\right]\left[1+v\frac{\partial \mathsf{\Phi}\left(v\right)}{\partial v}\right]\right\}.$$
- The conditional copula density of U given $V=v,\text{}$ from Equation (3), will be$$c\left(u\right|v)=\frac{\partial C(u,\text{}v)}{\partial v}=u\left\{1+\theta \mathsf{\Phi}\left(u\right)\mathsf{\Psi}\left(v\right)\left(1+v\right)\right\}.$$

- Kendall’s $\tau $: This measures the amount of concordance present in a bivariate distribution. Suppose that $(X,\text{}Y)$ and $(\tilde{X},\text{}\tilde{Y})$ are two independent pairs of random variables from a joint distribution function. We say that these pairs are concordant if “large values of one tend to be associated” with “large values of the other”, and “small values of one” tend to be associated with “small values of the other”. The pairs are called discordant if large goes with small or vice versa. Algebraically we have concordant pairs if $(X-\tilde{X})(Y-\tilde{Y})>0$ and discordant pairs if we reverse the inequality. Let X and Y be continuous random variables with copula $C.$ Then, Kendall’s $\tau $ is given by$${\tau}_{s}(X,\text{}Y)=4{\phantom{\rule{0.277778em}{0ex}}\int \int \phantom{\rule{0.277778em}{0ex}}}_{{[0,1]}^{2}}C(u,\text{}v)dC(u,\text{}v)-1.$$
- Spearman’s $\rho $: For two random variables, X and Y are equal to the linear correlation coefficient between ${F}_{1}\left(X\right)$ and ${F}_{2}\left(Y\right),\text{}$ where ${F}_{1}$ and ${F}_{2}$ are the marginal distributions of X and Y, respectively. Then, Spearman’s ${\rho}_{s}$ is given by$${\rho}_{s}(X,\text{}Y)=\rho \left(U={F}_{1}\left(X\right),\text{}V={F}_{2}\left(Y\right)\right)=12{\phantom{\rule{0.277778em}{0ex}}\int \int \phantom{\rule{0.277778em}{0ex}}}_{{[0,1]}^{2}}uvdC(u,\text{}v)-3,\text{}$$Alternatively, ${\rho}_{s}(X,\text{}Y)$ can be written as ${\rho}_{s}=12{\int}_{0}^{1}{\int}_{0}^{1}\left[C(u,\text{}v)-uv\right]dudv$. Also, as mentioned earlier, one can equivalently show that ${\rho}_{s}(U,\text{}V)=\rho \left({F}_{1}\left(X\right),\text{}{F}_{2}\left(V\right)\right).$ For details on such copula based measures of dependence, see Nelsen (2006).

**Proposition**

**1.**

- ${\rho}_{s}(X,\text{}Y)=\theta A(u,\text{}v),\text{}$ where $A(u,\text{}v)=12\left[{\int}_{0}^{1}u\mathsf{\Phi}\left(u\right)du\right]\left[{\int}_{0}^{1}v\mathsf{\Psi}\left(v\right)dv\right],$
- $$\begin{array}{ccc}\hfill {\tau}_{s}(X,\text{}Y)& =& \frac{2}{3}{\rho}_{\theta}+{\int}_{0}^{1}{v}^{2}\frac{{\mathsf{\Psi}}^{\prime}\left(v\right)}{\mathsf{\Psi}\left(v\right)}dv\left\{2+{\int}_{0}^{1}{u}^{2}\frac{{\mathsf{\Phi}}^{\prime}\left(u\right)}{\mathsf{\Phi}\left(u\right)}du\right\}\hfill \\ & & +{\theta}^{2}\left\{\left({\int}_{0}^{1}u{\mathsf{\Phi}}^{\prime}\left(u\right)\mathsf{\Phi}\left(u\right)du\right)\left({\int}_{0}^{1}v{\mathsf{\Psi}}^{\prime}\left(v\right)\mathsf{\Psi}\left(v\right)dv\right)\right\},\hfill \end{array}$$

**Proof.**

## 3. Bivariate KW-FGM Type Models

**Bivariate KW-FGM (Type I) Model:**

- $\mathsf{\Phi}\left(u\right)=u{(1-{u}^{{a}_{1}})}^{{b}_{1}},\text{}$ for $({a}_{1},\text{}{b}_{1})0,$
- $\mathsf{\Psi}\left(v\right)=v{(1-{v}^{{a}_{2}})}^{{b}_{2}},\text{}$ for $({a}_{2},\text{}{b}_{2})0.$

**Bivariate KW-FGM (Type II) Model:**

- $\mathsf{\Phi}\left(u\right)={u}^{{\delta}_{1}}{(1-u)}^{1-{\delta}_{1}},\text{}$ for ${\delta}_{1}>0,$
- $\mathsf{\Psi}\left(v\right)={v}^{{\delta}_{2}}{(1-v)}^{1-{\delta}_{2}},$ for ${\delta}_{2}>0.$

**Bivariate KW-FGM (Type III) Model:**

- $\mathsf{\Phi}\left(u\right)=u\left[log\left(1+(1-u)\right)\right],\text{}$
- $\mathsf{\Psi}\left(v\right)=v\left[log\left(1+(1-v)\right)\right].$

**Bivariate KW-FGM (Type-IV) Copula:**

## 4. Some Properties of the Bivariate KW-FGM Type Copulas

- For the BK-FGM (Type I) bivariate copula
- Closed form expression for Kendall’s $\tau $ is not available.
- Spearman’s correlation coefficient will be$${\rho}_{\theta}=\theta {\left({a}_{1}{a}_{2}\right)}^{-1},$$

- For the BK-FGM (Type II) bivariate copula
- Kendall’s $\tau $ will be$$\begin{array}{ccc}\hfill {\tau}_{s}(X,\text{}Y)& =& B({\delta}_{1}+2,\text{}2-{\delta}_{1})B({\delta}_{1}+3,\text{}2-{\delta}_{1})+({\delta}_{1}-1)[B({\delta}_{1}+2,\text{}2-{\delta}_{1})-B({\delta}_{1}+1,\text{}1-{\delta}_{1})]\hfill \\ & & +{\delta}_{1}B({\delta}_{1}+2,\text{}1-{\delta}_{1})\left({\delta}_{1}-\frac{1}{2}\right)-\frac{B({\delta}_{1}+1,\text{}1-{\delta}_{1})}{2},\text{}\hfill \end{array}$$
- Corresponding Spearman’s correlation coefficient will be$${\rho}_{s}(X,\text{}Y)=\theta {\left(B({\delta}_{1}+2,\text{}2-{\delta}_{1})\right)}^{2},\text{}$$

- For the BK-FGM (Type III) copula, no closed form expressions for Kendall’s $\tau $ and Spearman’s $\rho $ are available. They need to be evaluated numerically.
- For the BK-FGM (Type III) copula
- Kendall’s $\tau $ will be$${\tau}_{s}(X,\text{}Y)=4\left(1-\frac{\mathsf{\Gamma}(1+1/a)\mathsf{\Gamma}(1+b)}{\mathsf{\Gamma}(1+1/a+b)}-{\left(1-\frac{\mathsf{\Gamma}(1+1/a)\mathsf{\Gamma}(1+b)}{\mathsf{\Gamma}(1+1/a+b)}\right)}^{2}\right)-1$$
- Spearman’s correlation coefficient will be$${\rho}_{s}(X,\text{}Y)=12\left(1-\frac{\mathsf{\Gamma}(1+1/a)\mathsf{\Gamma}(1+b)}{\mathsf{\Gamma}(1+1/a+b)}-{\left(1-\frac{\mathsf{\Gamma}(1+1/a)\mathsf{\Gamma}(1+b)}{\mathsf{\Gamma}(1+1/a+b)}\right)}^{2}\right)-3.$$

#### Dependence Properties

**Tail Dependence Property:**

- In our case (for the bivariate KW-FGM (type I) copula model),$$\begin{array}{ccc}\hfill {\lambda}_{L}& =& \underset{u\downarrow 0}{lim}\frac{C(u,\text{}u)}{u}\hfill \\ & =& \underset{u\downarrow 0}{lim}{u}^{2}\left(1+\theta \left({u}^{2}{(1-{u}^{{a}_{1}})}^{{b}_{1}}{(1-{u}^{{a}_{2}})}^{{b}_{2}}\right)\right)\hfill \\ & =& 0.\hfill \end{array}$$$$\begin{array}{ccc}\hfill \tilde{C}(u,\text{}u)& =& 1-2u+C(u,\text{}u)\hfill \\ & =& 1-2u+{u}^{2}\left(1+\theta \left({u}^{2}{\left[1-\left({(1-u)}^{{a}_{1}}\right)\right]}^{{b}_{1}}{\left[1-\left({(1-u)}^{{a}_{2}}\right)\right]}^{{b}_{2}}\right)\right).\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\lambda}_{U}& =& \underset{u\uparrow 1}{lim}\frac{1-2u+C(u,\text{}u)}{1-u}\hfill \\ & =& \underset{u\uparrow 1}{lim}\frac{2(1-u)}{1-u}-\underset{u\to 1}{lim}\frac{1-C(u,\text{}u)}{1-u}\hfill \\ & =& 2-\underset{u\uparrow 1}{lim}\frac{1-{u}^{2}\left(1+\theta \left({u}^{2}{\left[1-\left({(1-u)}^{{a}_{1}}\right)\right]}^{{b}_{1}}{\left[1-\left({(1-u)}^{{a}_{2}}\right)\right]}^{{b}_{2}}\right)\right)}{1-u}\hfill \\ & =& 0.\hfill \end{array}$$
- For the bivariate KW-FGM (type II) copula model,$$\begin{array}{ccc}\hfill {\lambda}_{L}& =& \underset{u\downarrow 0}{lim}\frac{C(u,\text{}u)}{u}\hfill \\ & =& \underset{u\downarrow 0}{lim}{u}^{2}\left(1+\theta \left({u}^{{\delta}_{1}+{\delta}_{2}}{\left(1-u\right)}^{2-({\delta}_{1}+{\delta}_{2})}\right)\right)\hfill \\ & =& 0,\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\lambda}_{U}& =& \underset{u\uparrow 1}{lim}\frac{1-2u+C(u,\text{}u)}{1-u}\hfill \\ & =& \underset{u\uparrow 1}{lim}\frac{2(1-u)}{1-u}-\underset{u\to 1}{lim}\frac{1-C(u,\text{}u)}{1-u}\hfill \\ & =& 2-\underset{u\uparrow 1}{lim}\frac{1-{u}^{2}\left(1+\theta \left({u}^{{\delta}_{1}+{\delta}_{2}}{\left(1-u\right)}^{2-({\delta}_{1}+{\delta}_{2})}\right)\right)}{1-u}\hfill \\ & =& 0,\hfill \end{array}$$

**Positive Quadrant Dependent (PQD) and Left-Tail Decreasing (LTD) Property:**

- X and Y are PQD if and only if either ∀$u\in (0,\text{}1)$ and ∀$u\in (0,\text{}1)$, $\mathsf{\Phi}\left(u\right)\left[\mathsf{\Psi}\left(v\right)\right]\ge 0$ or $\mathsf{\Phi}\left(u\right)\left[\mathsf{\Psi}\left(v\right)\right]\le 0,\text{}$
- X and Y are LTD if and only if $\frac{\mathsf{\Phi}\left(u\right)}{u}$ and $\frac{\mathsf{\Psi}\left(v\right)}{v}$ is monotone. Next, consider the following:
**Proposition****2.**The BK-FGM (Type I, Type II and Type III) copulas are PQD**Proof.**For the modified BK-FGM (Type I) copula, we have $\mathsf{\Phi}\left(u\right)={u}^{{a}_{1}}{(1-{u}^{{a}_{1}})}^{{b}_{1}}$ and $\mathsf{\Psi}\left(v\right)={v}^{{a}_{2}}{(1-{v}^{{a}_{2}})}^{{b}_{2}}$. Note that, for any real $({a}_{1},\text{}{a}_{2},\text{}{b}_{1},\text{}{b}_{2})0,$ $\mathsf{\Phi}\left(u\right)\ge 0,$ for all $u\in (0,\text{}1)$ as well as $\mathsf{\Psi}\left(v\right)\ge 0,\text{}$ for all $v\in (0,\text{}1).$ Hence, $(X,\text{}Y)$ are PQD. ☐Similarly, one can easily check the PQD property for the other two copula models.**Proposition****3.**The BK-FGM (Type I and Type III) copula exhibits LTD properties, while, for the BK-FGM (Type II), it is indeterministic.**Proof.**For the modified BK-FGM (Type I) copula, consider the ratio $\frac{\mathsf{\Phi}\left(u\right)}{u}={u}^{{a}_{1}}{(1-{u}^{{a}_{1}})}^{{b}_{1}}$. It is monotonically decreasing provided, ${a}_{1}>1$ and for any ${b}_{1}>0,\text{}$ and it is also true for any $u\in (0,\text{}1).$ Similar results hold for the other ratio $\frac{\mathsf{\Psi}\left(v\right)}{v},\text{}$ for any $v\in (0,\text{}1).$ Hence, it is LTD for only ${a}_{1}>1$ and for any ${b}_{1}>0,\text{}$ but not for any other possible choices of the constants ${a}_{1}$ and ${b}_{1}$. ☐Again, for the modified BK-FGM (Type III)copula, the ratio $\frac{\mathsf{\Phi}\left(u\right)}{u}=log\left(1+(1-u)\right).$ It is monotonically decreasing for any $u\in (0,\text{}1).$ Similar results will hold for the other ratio $\frac{\mathsf{\Psi}\left(v\right)}{v},$ for any $v\in (0,\text{}1).$ Hence, it is LTD.However, for the modified BK-FGM (Type II) copula, these ratios are not uniformly increasing and/or decreasing. This is why it is indeterministic in this sense.

## 5. Simulation from a Bivariate Copula

## 6. Applications

#### 6.1. Application in Risk Management

- Simulate U, V and W independently from standard uniform distribution,
- If $U\le {\lambda}_{s}$, for the given bivariate KW-FGM (Type II) copula (say, ${C}_{{\rho}_{s},\text{}1}$), take (${X,\text{}Y)}^{T}={\left({F}_{1}^{-1}(V),\text{}{F}_{2}^{-1}({C}_{{\rho}_{s},\text{}1,\text{}U}^{-1}(W))\right)}^{T}.$
- If $U>{\lambda}_{s}$, for the given bivariate KW-FGM (Type II) copula (say, ${C}_{{\rho}_{s},\text{}2}$, ), take ${(X,\text{}Y)}^{T}={\left({F}_{1}^{-1}\left(V\right),\text{}{F}_{2}^{-1}({C}_{{\rho}_{s},\text{}2,\text{}U}^{-1}(W))\right)}^{T}.$

#### 6.2. An Application to Insurance Data

- ${X}_{1}$: an indemnity payment,
- ${X}_{1}$: an allocated loss adjustment expense (comprising lawyers’ fees and claim investigation process).

## 7. Conclusions

- Extension to the multivariate case and study several associated properties. It is noteworthy to mention that, albeit complex nature of these type of models (involving several parameters), we expect that multivariate KW distribution construction via such type of copula models will be much more interesting and computationally will be more easy to handle.
- For modeling large losses, asymmetric copulas are more useful as compared to symmetric copulas. Thus, we will consider a family of asymmetric copulas as introduced in Nelsen (2006), Chapter 4, which has the following form:$$C(u,\text{}v)=uv+\theta a\left(u\right)b\left(v\right),\text{}\phantom{\rule{1.em}{0ex}}\theta \in [-1,\text{}1].$$
- Since a convex combination of any two (or more) valid copulas is also a copula, we would be interested in studying the role of such a mixture of copula in developing bivariate, and sub- sequently multivariate, Kumaraswamy type distributions. For example, one may start with the following:$${C}^{mixture}(u,\text{}v)={\theta}_{1}{C}^{symmetric}(u,\text{}v)+(1-{\theta}_{1}){C}^{asymmetric}(u,\text{}v)$$
- A natural multivariate extension of the above asymmetric copula would be$$C({u}_{1},\text{}{u}_{2},\text{}\cdots ,\text{}{u}_{p})=\prod _{i=1}^{p}{u}_{1}+\theta \prod _{i=1}^{p}{a}_{i}\left({u}_{i}\right),$$

## Acknowledgments

## Conflicts of Interest

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Statistics | Nasdaq | S&P 500 |
---|---|---|

Mean | 0.00038 | 0.00030 |

Mean (annualized) | 10.141% | 7.857% |

Standard Deviation | 0.01694 | 0.01076 |

Minimum | −0.10168 | −0.07113 |

Median | 0.00122 | 0.00028 |

Maximum | 0.13255 | 0.05574 |

Excess of Kurtosis | 4.91481 | 3.78088 |

Asymmetry | 0.01490 | −0.10267 |

**Table 2.**Proportion of observations (number of observations in brackets), for t = 751 to 2971, where the portfolio loss exceeded the estimated Value at Risk for $\alpha =0.05$.

Copula | GARCH-N | GARCH-t | GARCH-E |
---|---|---|---|

Nelsen–Ten | 0.0675 (167) | 0.0698 (122) | 0.0322 (63) |

Gumbel–Hougaard | 0.0666 (128) | 0.0207 (46) | 0.0312 (69) |

Bivariate Gaussian | 0.0693 (117) | 0.0359 (92) | 0.0281 (78) |

Bivariate KW-FGM (Type I) copula | 0.0828 (82) | 0.0244 (42) | 0.0206 (46) |

Bivariate KW-FGM (Type II) copula | 0.2141 (77) | 0.0286 (48) | 0.0153 (52) |

Bivariate KW-FGM (Type III) copula | 0.0287 (37) | 0.0126 (30) | 0.0103 (28) |

Bivariate KW-FGM (Type IV) copula | 0.1354 (54) | 0.0329 (47) | 0.0189 (39) |

Bivariate Copula | $\mathit{\theta}$ | ${\mathit{S}}_{\mathit{n}}$ | ${\mathit{T}}_{\mathit{n}}$ | ${\mathit{S}}_{0\mathit{n}}$ | p-Value (in %) | Critical Value (${\mathit{c}}_{2\mathit{n}}$) |
---|---|---|---|---|---|---|

Bivariate KW-FGM (Type I) copula | 0.623 | 3.0755 | 2.643 | 1.036 | 45.3 | 0.422 |

Bivariate KW-FGM (Type II) copula | 1.233 | 2.189 | 3.547 | 0.427 | 0.18 | 0.163 |

Bivariate KW-FGM (Type III) copula | 1.026 | 0.147 | 0.564 | 0.117 | 78.3 | 0.795 |

Bivariate KW-FGM (Type IV) copula | 0.342 | 0.422 | 0.642 | 0.137 | 88.2 | 0.831 |

FGM copula | 0.589 | 1.567 | 2.034 | 0.493 | 37.2 | 0.327 |

Nelsen-Ten | 1.253 | 2.384 | 4.031 | 0.622 | 43.4 | 0.285 |

Gumbel–Hougaard | 0.783 | 1.657 | 2.842 | 0.842 | 18.9 | 0.638 |

Bivariate Gaussian | 0.732 | 1.268 | 2.416 | 0.715 | 44.8 | 0.483 |

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**MDPI and ACS Style**

Ghosh, I.
Bivariate Kumaraswamy Models via Modified FGM Copulas: Properties and Applications. *J. Risk Financial Manag.* **2017**, *10*, 19.
https://doi.org/10.3390/jrfm10040019

**AMA Style**

Ghosh I.
Bivariate Kumaraswamy Models via Modified FGM Copulas: Properties and Applications. *Journal of Risk and Financial Management*. 2017; 10(4):19.
https://doi.org/10.3390/jrfm10040019

**Chicago/Turabian Style**

Ghosh, Indranil.
2017. "Bivariate Kumaraswamy Models via Modified FGM Copulas: Properties and Applications" *Journal of Risk and Financial Management* 10, no. 4: 19.
https://doi.org/10.3390/jrfm10040019